#  Optimal fire allocation in a combat model of mixed NCW type

My A. Vu† , Nam H. Nguyen‡, Hanh Le T. Nguyen§, Anh N. Ta♡, Mong H. Nguyen♠
Corresponding author.
Received: / Revised: / Accepted:
DOI:

© 2021 University of Guilan http://jmm.guilan.ac.ir

Abstract. In this work, we introduce a nonlinear Lanchester model of NCW-type
and study a problem of finding the optimal fire allocation for this model. A
Blue party $B$ will fight against a Red party consisting of $A$ and $R$, where
$A$ is an independent force and $R$ fights with supports from a supply unit
$N$. A battle may consist of several stages but we consider the problem of
finding optimal fire allocation for $B$ in the first stage only. Optimal fire
allocation is a set of three non-negative numbers whose sum equals to one,
such that the remaining force of $B$ is maximal at any instants. In order to
tackle this problem, we introduce the notion of threatening rates which are
computed for $A,R,N$ at the beginning of the battle. Numerical illustrations
are presented to justify the theoretical findings.

Keywords: Nonlinear Lanchester Model, Network Centric Warfare, Optimal fire
allocation.
AMS Subject Classification 2010: 34A34, 65L05.

##  1 Introduction

In 1916, Lanchester [1] introduced a mathematical model for a battle in the
form of a system of differential equations. This model has been extended and
generalized in various ways, such as guerilla model by Deitchman [2], guerilla
model with intelligence by Schaffer [3] and Schreiber [4], counter terrorism
model by Kaplan, Kress and Szechtman (KKS) [5],[6]. There are several problems
involving these models, among of which are the problem of optimal fire
allocation with number of troops being objective function. This problem has
been investigated in various scenarios by Taylor [7], Lin and Mackay [8]. In
these works, the role of military supply, however, has not been studied
thoroughly. In a combat, the victory of either party is not only decided by
the armed forces but also by their supply units. In many historical battles,
firepower was not only aimed at the direct rivals but also their supply units,
see [9].
In 2017, Kim and his colleagues [15] considered a Lanchester’s model where
Blue force $B$ fights against Red force $R$ supported by a supply unit $N$ and
called this a NCW model. This model can be denoted by $\left(B\textit{ vs
}(R,N)\right)$. Kim considered fire allocations in the form of piecewise
constant functions and derived optimal fire allocation so that number of $B$’s
troop is always at its possible maximum. In this paper, we extend Kim’s model
by considering a model of $\left(B\textit{ vs }\\{(R,N),A\\}\right)$ where $A$
stands for an independent force. Let us refer this model as mixed NCW. For
this model, we also consider the problem of finding optimal fire allocation of
$B$ so that its remaining troop at any instant is maximal. By a different
approach with the help of ”threatening rates”, we managed to derive the
optimal fire allocation for $B$. In Lanchester’s model using system of
differential equations, the decreasing rate of troops of a force is computed
by attrition rate of its rival force multiplied by the rival’s number of
troops. In our model, attrition rate of $R$ is assumed to be a linear function
of the amount of $N$’s troop and this supply unit can also be attrited by $B$.
The resulting model is of non-linear Lanchester type. Let us recall that in
classical non-linear Lanchester’s model, the supply units have not been taken
into account but only the armed forces.
In any battle of the type ”one against many”, the strategy will be intuitively
derived as follows: focus all firepower to the entity possessing the ”greatest
threat”. In order to quantify ”threats” posed by the entities $R,\,N,\,A$ to
$B$, we introduce the notion of ”threatening rates” which are computed for
each entity $R,\,N,\,A$ by involving entities number of troops, their
attrition rates against $B$ and $B$’s attrition rates. The fire allocation of
$B$ is assumed to be constant for a certain period of time, which we call
”stage”. The first stage is the period from the beginning of the battle to the
instant when one entity is extirpated - its number of troops reaches zero.
During this stage, the fire allocation is kept constant and the battle moves
to a new stage if one (or more) entity is eliminated. This choice of fire
allocation is meaningful since in planning phase of a battle, this choice
simplifies the logistic operations. Moreover, capturing the states of the
battlefield and altering the fire allocation accordingly is not an easy task.
By using the threatening rates, we justify the intuitive strategy mentioned
above. Thus, the optimal fire allocation for $B$ is focusing all its firepower
to the entity possessing the greatest threatening rate. Several numerical
examples are included to illustrate the theoretical findings.
The rest of the paper is organized as follows. Section 2 is devoted to
inroduce our model and to investigate the optimization problem for this model.
Numerical experiments are presented in Section 3 to illustrate the theoretical
results. Conclusion and some possible further developments are discussed in
the last section.

##  2 Main results

###  2.1 Non-linear Lanchester model of mixed NCW type

Let us consider a battle where $B$ fights against $((R,N),A)$. Our model is
called non-linear Lanchester of NCW type. NCW stands for ”Network Centric
Warfare”, which is a novel notion of modern warfare. For a more detailed
explanation of this notion, the reader is referred to [10, 11, 12, 13]. A
simple diagram for the model is represented in Figure 1.

Figure 1: Diagram of $\left(B\textit{ vs }\\{(R,N),A\\}\right)$ model.

Before formulating the model, let us denote:

  * •

by ${{\beta}_{R}}:$ attrition rate of $B$ against $R$.

  * •

by ${{\beta}_{A}}:$ attrition rate of $B$ against $A$.

  * •

by ${{\beta}_{N}}:$ attrition rate of $B$ against $N$.

  * •

${{f}_{\alpha}}(N):$ attrition function of $N$ complementing $R^{i}$ to $B$.

  * •

$\Pi=({{\pi}_{1}},{{\pi}_{2}},{{\pi}_{3}}):$ fire allocation of $B$ against
$R$, $N$, $A$, respectively.

  * •

$\alpha_{c}^{N}:$ fully-connected attrition rate of $R$ against $B$.

  * •

$\alpha_{d}^{N}:$ disconnected attrition rate of $R$ against $B$,
$\left(\alpha_{d}^{N}\leq\alpha_{c}^{N}\right).$

  * •

${{\gamma}_{A}}:$ attrition rate of $A$ against $B$.

  * •

$R_{0},\,N_{0},\,A_{0}$ are the initial numbers of troops of $R,\,N,\,A$,
respectively.

The fire allocation of $B$ is sought in the set

$\mathcal{P}=\\{\Pi=\left(\pi_{1},\pi_{2},\pi_{3}\right)|\pi_{1}+\pi_{2}+\pi_{3}=1,\,\pi_{i}\in\left[0;1\right],\,\pi_{i}^{\prime}s\text{
are constants },\,i=1,2,3\\}.$ (1)

The attrition function of $R$ against $B$ is assumed to be a linear function
of $N$:

${{f}_{\alpha}}\left(N\right)=\alpha_{d}^{N}+\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right)\frac{N}{{{N}_{0}}},$
(2)

When $N={{N}_{0}}$, ${{f}_{\alpha}}\left(N\right)=\alpha_{c}^{N},$ in other
words, $R$ and $N$ are fully connected. When $N$ is totally eliminated by $B,$
$N=0$, $R$ and $N$ are totally disconnected and
${{f}_{\alpha}}\left(N\right)=\alpha_{d}^{N}.$
Our model can be described by a system of differential equations defined as
follows:

$\left\\{\begin{aligned}
\frac{dB}{dt}&=-\left[\alpha_{d}^{N}+\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right)\frac{N}{N_{0}}\right]R-\gamma_{A}A,\\\
\frac{dR}{dt}&=-\pi_{1}\beta_{R}B,\\\ \frac{dN}{dt}&=-\pi_{2}\beta_{N}B,\\\
\frac{dA}{dt}&=-\pi_{3}\beta_{A}B.\end{aligned}\right.$ (3)

###  2.2 Optimal fire allocation

For the model described by (3), we consider the first stage of the conflict
and study the problem of maximizing the remaining troops of $B$ at any instant
$t$ with respect to fire allocations belonging to the set (1). Let us
introduce the notion of “threatening rates”, which are denoted by
${{b}_{1}},\,{{b}_{2}},\,{{b}_{3}}$, and computed as follows:

$\left\\{\begin{aligned} {{b}_{1}}&=\alpha_{c}^{N}{{\beta}_{R}},\\\
{{b}_{2}}&=\frac{{{\beta}_{N}}\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right){{R}_{0}}}{{{N}_{0}}},\\\
{{b}_{3}}&={{\gamma}_{A}}{{\beta}_{A}}.\end{aligned}\right.$ (4)

These quantities represent the level of harm which $R,\,N,\,A$ can possibly
inflict on $B$. By invoking these rates, the optimal fire allocation of $B$ is
derived in the following theorem.

######  Theorem 1.

Among all the fire allocations of the set (1) for the first stage, the optimal
one for $B$ is as follows:

$\Pi^{*}=\left\\{\begin{aligned} \left(1,0,0\right)\text{ if
}\,&\left({{b}_{1}}\geq{{b}_{2}}\right)\wedge\left({{b}_{1}}\geq{{b}_{3}}\right),\\\
\left(0,1,0\right)\text{ if
}\,&\left({{b}_{2}}\geq{{b}_{1}}\geq{{b}_{3}}\right)\vee\left({{b}_{2}}\geq{{b}_{3}}\geq{{b}_{1}}\right),\\\
\left(0,0,1\right)\text{ if
}\,&\left({{b}_{3}}\geq{{b}_{1}}\geq{{b}_{2}}\right)\vee\left({{b}_{3}}\geq{{b}_{2}}\geq{{b}_{1}}\right).\end{aligned}\right.$

###### Proof.

Let
$X\left(t\right)=\int\limits_{0}^{t}{B\left(s\right)ds\Rightarrow}{X}^{\prime}\left(t\right)=B\left(t\right)$.
We have then

${X}^{\prime\prime}\left(t\right)={B}^{\prime}\left(t\right)=-\left[\alpha_{d}^{N}+\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right)\frac{N}{{{N}_{0}}}\right]R-{{\gamma}_{A}}A.$
(5)

Integrating both sides of the second equation of (3) yields

$\int\limits_{0}^{t}{dR=-\int\limits_{0}^{t}{{{\pi}_{1}}{{\beta}_{R}}B\left(s\right)ds}}\Rightarrow
R\left(t\right)-R\left(0\right)=-{{\pi}_{1}}{{\beta}_{R}}X\left(t\right).$

From this we obtain

$R=-{{\pi}_{1}}{{\beta}_{R}}X\left(t\right)+{{R}_{0}}.$ (6)

By analogous computations, we get

$\displaystyle N$ $\displaystyle=$
$\displaystyle-{{\pi}_{2}}{{\beta}_{N}}X\left(t\right)+{{N}_{0}},$ (7)
$\displaystyle A$ $\displaystyle=$
$\displaystyle-{{\pi}_{3}}{{\beta}_{A}}X\left(t\right)+{{A}_{0}}.$ (8)

where $R_{0},N_{0},A_{0}$ are the corresponding numbers of troops of $R,N,A$
at the beginning of the combat. Substituting $R,N,A$ in (6),(7),(8) into (5)
yields:

${X}^{\prime\prime}\left(t\right)=-{{C}_{1}}{{X}^{2}}\left(t\right)+{{C}_{2}}X\left(t\right)-{{C}_{3}},$
(9)

where:

$\displaystyle{{C}_{1}}$ $\displaystyle=$
$\displaystyle\frac{{{\pi}_{1}}.{{\pi}_{2}}.{{\beta}_{R}}.{{\beta}_{N}}.\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right)}{{{N}_{0}}},$
$\displaystyle{{C}_{2}}$ $\displaystyle=$
$\displaystyle\frac{{{\pi}_{2}}{{\beta}_{N}}\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right){{R}_{0}}+{{\pi}_{1}}{{\beta}_{R}}\alpha_{c}^{N}{{N}_{0}}}{{{N}_{0}}}+{{\gamma}_{A}}{{\pi}_{3}}{{\beta}_{A}},$
$\displaystyle{{C}_{3}}$ $\displaystyle=$
$\displaystyle\alpha_{c}^{N}{{R}_{0}}+{{\gamma}_{A}}{{A}_{0}}.$

Multiplying both sides of (9) by $d{X}^{\prime}\left(t\right)$ and
integrating, we obtain

$\displaystyle{X}^{\prime}\left(t\right)$ $\displaystyle=$ $\displaystyle
B\left(t\right)$ $\displaystyle=$
$\displaystyle\sqrt{-\frac{2}{3}{{C}_{1}}{{X}^{3}}\left(t\right)+{{C}_{2}}{{X}^{2}}\left(t\right)-2{{C}_{3}}X\left(t\right)+{{C}_{4}}}.$

It can be seen that ${{C}_{1}},{{C}_{2}}$ are nonnegative and they depend
solely on the fire allocation $\Pi$, $C_{3}$ is independent of $\Pi$ and
${{C}_{4}}$ is an integral constant. In the stage under consideration, at any
time $t=t_{0}$, $X(t_{0})$ is a positive constant. Therefore, it is observed
that the smaller $C_{1}$ and the greater $C_{2}$ make $B(t_{0})$ greater. We
then need to solve the following multi-objective optimization problem:

$\min\limits_{\Pi\ in\mathcal{P}}\\{(C_{1};-C_{2})\\}.$

Let us denote

$\displaystyle a$ $\displaystyle=$
$\displaystyle\frac{{{\beta}_{R}}{{\beta}_{N}}\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right)}{{{N}_{0}}},$
$\displaystyle{{b}_{1}}$ $\displaystyle=$
$\displaystyle\alpha_{c}^{N}{{\beta}_{R}},$ $\displaystyle{{b}_{2}}$
$\displaystyle=$
$\displaystyle\frac{{{\beta}_{N}}\left(\alpha_{c}^{N}-\alpha_{d}^{N}\right){{R}_{0}}}{{{N}_{0}}},$
$\displaystyle{{b}_{3}}$ $\displaystyle=$
$\displaystyle{{\gamma}_{A}}{{\beta}_{A}},$ $\displaystyle x$ $\displaystyle=$
$\displaystyle\pi_{1},y=\pi_{2},z=\pi_{3}.$

The problem now becomes:

$\min\\{\left(axy,-(b_{1}x+b_{2}y+b_{3}z)\right)\\}\text{ s.t.
}\left\\{\begin{aligned} &0\leq x,y,z\leq 1,\\\ &x+y+z=1.\\\
\end{aligned}\right.$ (10)

To tackle this problem, we will use weighting method (see [14, Section 3.1]).
We thus set

$F_{\lambda}\left(x,y,z\right)=\lambda\left(axy\right)-\left(1-\lambda\right)\left({{b}_{1}}x+{{b}_{2}}y+{{b}_{3}}z\right),$

and obtain:

$\min F_{\lambda}(x,y,z)\text{ s.t. }\left\\{\begin{aligned} &0\leq x,y,z\leq
1,\\\ &x+y+z=1,\\\ &0\leq\lambda\leq 1.\end{aligned}\right.$ (11)

Substituting $x=1-y-z$, we now get the problem:

$\displaystyle\min\\{\lambda(1-y-z)ay-(1-\lambda)({{b}_{1}}+({{b}_{2}}-{{b}_{1}})y+({{b}_{3}}-{{b}_{1}})z)\\}$
(12) $\displaystyle\text{s.t. }\left\\{\begin{aligned} &y,z\geq 0,\\\ &y+z\leq
1,\\\ &0\leq\lambda\leq 1.\end{aligned}\right.$

Let us consider the following five cases:

  1. 1.

If $b_{2}>b_{3}>b_{1},$ since $\lambda\left(1-y-z\right)ay\geq 0,$ it follows:

$\displaystyle\min F_{\lambda}$ $\displaystyle\geq$
$\displaystyle-\left(1-\lambda\right)\left({{b}_{1}}+\left({{b}_{2}}-{{b}_{1}}\right)y+\left({{b}_{3}}-{{b}_{1}}\right)z\right)$
$\displaystyle\geq$
$\displaystyle-\left(1-\lambda\right)\left({{b}_{1}}+\left({{b}_{2}}-{{b}_{1}}\right)\left(y+z\right)\right)$
$\displaystyle\geq$
$\displaystyle-\left(1-\lambda\right){{b}_{2}}=F_{\lambda}\left(0,1,0\right).$

  2. 2.

If ${{b}_{2}}>{{b}_{1}}>{{b}_{3}},$ the problem becomes:

$\min\left\\{\lambda\left(1-y-z\right)ay+\left(1-\lambda\right)\left({{b}_{1}}-{{b}_{3}}\right)z-\left(1-\lambda\right)\left({{b}_{1}}+\left({{b}_{2}}-{{b}_{1}}\right)y\right)\right\\}.$

Therefore:

$\min
F_{\lambda}\geq-\left(1-\lambda\right){{b}_{2}}=F_{\lambda}\left(0,1,0\right).$

  3. 3.

If ${{b}_{3}}>{{b}_{2}}>{{b}_{1}}$ we obtain:

$\displaystyle\min F_{\lambda}$ $\displaystyle\geq$
$\displaystyle-\left(1-\lambda\right)\left({{b}_{1}}+\left({{b}_{3}}-{{b}_{1}}\right)\left(y+z\right)\right)$
$\displaystyle\geq$
$\displaystyle-\left(1-\lambda\right){{b}_{3}}=F_{\lambda}\left(0,0,1\right).$

  4. 4.

If ${{b}_{3}}>{{b}_{1}}>{{b}_{2}}$ the problem is now:

$\min\left\\{\lambda\left(1-y-z\right)ay+\left(1-\lambda\right)\left({{b}_{1}}-{{b}_{2}}\right)y-\left(1-\lambda\right)\left(b+\left({{b}_{3}}-b\right)z\right)\right\\}.$

We yield that:

$\min
F_{\lambda}\geq-\left(1-\lambda\right){{b}_{3}}=F_{\lambda}\left(0,0,1\right).$

  5. 5.

If $b_{2}<b_{1},\,b_{3}<b_{1}$, the problem turns out to be:

$\min\left\\{\lambda\left(1-y-z\right)ay+\left(1-\lambda\right)\left(\left({{b}_{1}}-{{b}_{2}}\right)y+\left({{b}_{1}}-{{b}_{3}}\right)z\right)-\left(1-\lambda\right){{b}_{1}}\right\\}.$

And, it follows that $\min
F_{\lambda}\geq-\left(1-\lambda\right)b=F_{\lambda}\left(1,0,0\right).$

∎

In principle, the battle can be divided into three stages. In the first stage,
due to the computed threatening rates ${{b}_{1}},\,{{b}_{2}},\,{{b}_{3}}$ the
Blue force $B$ will focus all its firepower to one of the entities $R,N,A.$
When one of the entities is eliminated, the second stage begins. When one of
the remaining two forces is extinguisged, the third stage follows. However, if
$R$ and $A$ are out of the picture by the end of the second stage, $B$ will be
no longer attrited and the battle finishes. As stated above, for the first
stage, our results apply. For the second stage, one may use the results by Kim
et. al. [15] or by Lin and Mackay [8]. Thus, by the first case in our proof,
in order that $B\left(t\right)$ is always maximal, for the first stage $B$
should focus its firepower to $N$. After the conclusion of $N$, the second
stage begins, where the results of Lin and Mackay apply, thus $B$ concentrates
its firepower on $R$ or $A$.

The second case in our proof is explained analogously.

By the third case in our proof, $B$ will focus its firepower on $A$. The
second stage in this case has been considered in [15]. The fourth case is
analyzed similarly as the third case. Strategy in the fifth case can be
explained as follows: for the first stage, $B$ concentrates on $R.$ $B$ then
turn its firepower to $A$ for the second stage. And the battle ends when $A$
is totally annihilated.

##  3 Numerical experiments

###  3.1 Case 1

Let us consider equation (3) with coefficients given by

$\alpha_{c}^{N}$ | $\alpha_{d}^{N}$ | $\gamma_{A}$ | $\beta_{R}$ | $\beta_{N}$ | $\beta_{A}$
---|---|---|---|---|---
$0.4$ | $0.15$ | $0.2$ | $0.5$ | $0.3$ | $0.2$
Table 1: Parameters for Case 1.

together with the following initial conditions :
${{B}_{0}}=170;{{R}_{0}}=120;\,\,{{N}_{0}}=20;\,{{A}_{0}}=50$. Threatening
rates are thus computed as:
${{b}_{1}}=0.2;\,\,\,\,{{b}_{2}}=0.45;\,\,\,\,{{b}_{3}}=0.04.$ It is obvious
that ${{b}_{2}}>{{b}_{1}}>{{b}_{3}}$, so the optimal fire allocation for the
first stage is given by $\left(0,1,0\right)$. By using results of Lin and
Mackay, the optimal strategy for the whole battle is given by:

${{\Pi}^{*}}=\left(0,1,0\right)\to\left(1,0,0\right)\to\left(0,0,1\right).$

This strategy should be interpreted as follows: for the first stage, $B$ will
focus all its firepower to $N$. For the second stage, $B$ will concentrate on
$R$ since $b_{1}>b_{2}$. In order to contrast with the optimal strategy, we
use ${{\Pi}_{1}}=\left(1,0,0\right)\to\left(0,0,1\right).$ This strategy is
explained as: $B$ will focus all its firepower on $R$ in the first stage; for
the second stage, it will concentrate on $A.$ The simulation results show that
$B$ still win the battle with this strategy. However its amount of troops at
any instant is always smaller than itself using the optimal one. Amounts of
troops of $B$ using both stragtegies are represented in Figure 2.

Figure 2: Results for Case 1.

###  3.2 Case 2

We now will investigate model (3) with the following parameters

$\alpha_{c}^{N}$ | $\alpha_{d}^{N}$ | $\gamma_{A}$ | $\beta_{R}$ | $\beta_{N}$ | $\beta_{A}$
---|---|---|---|---|---
$0.4$ | $0.15$ | $0.2$ | $0.5$ | $0.2$ | $0.2$
Table 2: Parameters for Case 2

together with these initial conditions:
${{B}_{0}}=170,\,{{R}_{0}}=120,\,{{N}_{0}}=50,\,{{A}_{0}}=50.$ Threatening
rates are thus computed as:
${{b}_{1}}=0.2;\,\,\,\,{{b}_{2}}=0.12;\,\,\,\,{{b}_{2}}=0.04.$ Since
${{b}_{2}}>{{b}_{1}}>{{b}_{3}}$, the optimal fire allocation for the first
stage is ${{\Pi}^{*}}=\left(1,0,0\right).$ Among three strategies we chose to
compare, only ${{\Pi}_{3}}=\left(0.7,0.2,0.1\right)\to\left(0,0,1\right)$
leads to $B$’s victory. However, the number of troops of $B$ is again lower
than one resulting from the optimal strategy. The other two strategies even
result in $B$’s failure. Processes and endings of the simulated battles are
depicted in Figure 3

Figure 3: Results for Case 2.

###  3.3 Case 3

Now we consider the model with parameters given below:

$\alpha_{c}^{N}$ | $\alpha_{d}^{N}$ | $\gamma_{A}$ | $\beta_{R}$ | $\beta_{N}$ | $\beta_{A}$
---|---|---|---|---|---
$0.4$ | $0.2$ | $0.6$ | $0.5$ | $0.2$ | $0.5$
Table 3: Parameters for Case 3

together with initial conditions:
${{B}_{0}}=170;{{R}_{0}}=120;\,\,{{N}_{0}}=60;\,{{A}_{0}}=50.$ The threatening
rates are ${{b}_{1}}=0.2;\,{{b}_{2}}=0.08;\,{{b}_{3}}=0.3$. The optimal fire
allocation is therefore ${{\Pi}^{*}}=\left(0,0,1\right)\to\left(1,0,0\right).$
The two contrasting strategies are
${{\Pi}_{1}}=\left(1,0,0\right)\to\left(0,0,1\right);\,{{\Pi}_{2}}=\left(0,1,0\right)$.
By ${{\Pi}_{1}}$, $B$ win with more troops lost while ${{\Pi}_{2}}$ leads to
$B$’s failure. Processes of the battles are given in Figure 4.

Figure 4: Results for Case 3.

##  4 Conclusion

A nonlinear Lanchester model of mixed NCW type has been introduced together
with the notion of threatening rates. Invoking these rates, threats exposed by
rivals have been quantified and these quantities help derive the optimal
strategy for Blue force, which decreases the losing of its own troops.
Numerical results support the theoretical findings.

## References

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4861

# On the finiteness of the discrete spectrum of a $3\times 3$ operator matrix

Tulkin H. Rasulov  Faculty of Physics and Mathematics, Bukhara State
University, 11 M. Ikbol str., Bukhara, 200100, Uzbekistan<EMAIL_ADDRESS>
(Date: 01/02/2014)

###### Abstract.

An operator matrix $H$ associated with a lattice system describing three
particles in interactions, without conservation of the number of particles, is
considered. The structure of the essential spectrum of $H$ is described by the
spectra of two families of the generalized Friedrichs models. A symmetric
version of the Weinberg equation for eigenvectors of $H$ is obtained. The
conditions which guarantee the finiteness of the number of discrete
eigenvalues located below the bottom of the three-particle branch of the
essential spectrum of $H$ is found.

###### Key words and phrases:

Operator matrix, bosonic Fock space, annihilation and creation operators,
generalized Friedrichs model, essential and discrete spectra, Weinberg
equation, continuity in the uniform operator topology.

###### 2000 Mathematics Subject Classification:

Primary 81Q10; Secondary 35P20, 47N50.

##  1\. Introduction

One of important problems in the spectral theory of Schrödinger operators and
Hamiltonians (operator matrices) in a Fock space is to study the number of
eigenvalues (bound states) located outside the essential spectrum. The first
mathematical result on the finiteness of the discrete spectrum of Schrödinger
operators for general interactions was obtained by Uchiyama in [20]. Under
natural assumptions on the potential, the essential spectrum of the continuous
Schrödinger operator $H_{\rm c}$ of a system of three pair-wise interacting
particles coincides with the half-axis $[\kappa;\infty),$ $\kappa\leq 0.$ In
independent investigations of Yafaev [21] and Zhislin [24], it was shown that
for $\kappa<0$ and a sufficiently rapid decrease of the interactions in the
coordinate space representation the discrete spectrum of $H_{\rm c}$ is
actually finite. In the case $\kappa=0$ the finiteness of the discrete
spectrum of $H_{\rm c}$ with certain decreasing interactions was established
by Yafaev [22]. Yafaev’s results are based on the investigation of the Faddeev
and Weinberg type system of integral equations for the resolvent.

The problem of finiteness of the number of eigenvalues of the three-particle
discrete Schrödinger operators $H_{\rm d}$ was studied by many authors, see
for example, [1, 7, 13]. The authors of [1] used the Faddeev and Weinberg type
equations and an expansion of the Fredholm determinant to prove finiteness of
the discrete spectrum of $H_{\rm d}$ with pair contact interactions when the
corresponding two-particle discrete Schrödinger operators have no virtual
levels. The Birman-Schwinger principle was used in [7] to prove that the
discrete spectrum of the operator $H_{\rm d}$ describing systems of three
particles (two bosons and a third particle of a different nature) is finite.
In [8], applying the methods developed in [22] to the Hamiltonian $H_{\rm d}$
of a system of three arbitrary particles on a lattice, finiteness of the
discrete spectrum of $H_{\rm d}$ is proved if either only one or none of the
two-particle subsystems has a virtual level. In [13], the finiteness of the
number of eigenvalues of $H_{\rm d}$ with a specific class of potentials is
proved where one of the particles has an infinite mass.

In all of the above mentioned papers devoted to the finiteness of the discrete
spectrum, it was considered systems with a fixed number of quasi-particles. It
is worth to mention that there are important problems in the theory of solid-
state physics [12], quantum field theory [6], statistical physics [10, 11],
fluid mechanics [5], magnetohydrodynamics [9] and quantum mechanics [19] where
the number of quasi-particles is finite but not fixed. Recall that the study
of systems describing $n$ particles in interaction without conservation of the
number of particles can be reduced to the investigation of the spectral
properties of self-adjoint operators acting in the $n$-particle cut subspace
of the Fock space [6, 11, 12, 18]. In [18], geometric and commutator
techniques were developed in order to find the location of the spectrum and to
prove absence of singular continuous spectrum for Hamiltonians without the
conservation of particle number.

In the present paper we consider an operator matrix $H$ associated with the
lattice system describing three particles in interactions without conservation
of the number of particles. This operator acts in a three-particle subspace
${\mathcal{H}}$ of the bosonic Fock space and it is a lattice analogue of the
spin-boson Hamiltonian [11]. We find sufficient conditions for the finiteness
of the discrete spectrum of $H.$ Note that the operator matrix $H$ has been
considered before in [14, 15, 16, 23] where only its essential spectrum was
investigated.

The organization of the present paper is as follows. Section 1 is an
introduction to the whole work. In Section 2, the operator matrix $H$ is
described as a bounded self-adjoint operator in ${\mathcal{H}}$ and the main
results are formulated. In Section 3, we prove some auxiliary lemmas. In
Section 4, we obtain a symmetric version of the Weinberg equation for
eigenvectors of $H.$ Section 5 is devoted to the proof of the main results.

##  2\. The operator matrix and main results

###  2.1. The operator matrix

Let ${\mathbb{C}},$ ${\mathbb{R}}$ and ${\mathbb{Z}}$ be the set of all
complex, real and integer numbers, respectively. We denote by
${\mathbb{T}}^{3}$ the three-dimensional torus (the first Brillouin zone,
i.e., the dual group of ${\mathbb{Z}}^{3}$), the cube $(-\pi,\pi]^{3}$ with
appropriately identified sides is equipped with its Haar measure. The torus
${\mathbb{T}}^{3}$ will always be considered as an Abelian group with respect
to the addition and multiplication by real numbers regarded as operations on
the three-dimensional space ${\mathbb{R}}^{3}$ modulo
$(2\pi{\mathbb{Z}})^{3}.$

Let $L_{2}({\mathbb{T}}^{3})$ be the Hilbert space of square integrable
(complex) functions defined on ${\mathbb{T}}^{3}$ and $L_{2}^{\rm
s}(({\mathbb{T}}^{3})^{2})$ be the Hilbert space of square integrable
(complex) symmetric functions defined on $({\mathbb{T}}^{3})^{2}.$ Denote by
${\mathcal{H}}$ the direct sum of spaces ${\mathcal{H}}_{1}={\mathbb{C}},$
${\mathcal{H}}_{1}=L_{2}({\mathbb{T}}^{3})$ and ${\mathcal{H}}_{2}=L_{2}^{\rm
s}(({\mathbb{T}}^{3})^{2}),$ that is,
${\mathcal{H}}={\mathcal{H}}_{0}\oplus{\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2}.$

Let us consider the operator matrix (Hamiltonian) $H$ acting in the Hilbert
space ${\mathcal{H}}$ as

$H=\left(\begin{array}[]{ccc}H_{00}&H_{01}&0\\\ H_{01}^{*}&H_{11}&H_{12}\\\
0&H_{12}^{*}&H_{22}\\\ \end{array}\right),$

where the entries $H_{ij}:{\mathcal{H}}_{j}\to{\mathcal{H}}_{i},$ $i\leq j,$
$i,j=0,1,2$ are defined by

$H_{00}f_{0}=w_{0}f_{0},\quad
H_{01}f_{1}=\int_{{\mathbb{T}}^{3}}v_{0}(s)f_{1}(s)\,ds,\quad(H_{11}f_{1})(p)=w_{1}(p)f_{1}(p),$
$(H_{12}f_{2})(p)=\int_{{\mathbb{T}}^{3}}v_{1}(s)f_{2}(p,s)\,ds,\quad
H_{22}=H_{22}^{0}-V,\quad(H_{22}^{0}f_{2})(p,q)=w_{2}(p,q)f_{2}(p,q),$
$(Vf_{2})(p,q)=v_{2}(q)\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{2}(p,s)\,ds+v_{2}(p)\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{2}(s,q)\,ds.$

Here $f_{i}\in{\mathcal{H}}_{i},$ $i=0,1,2;$ $w_{0}$ is a fixed real number,
$w_{1}(\cdot)$ and $v_{i}(\cdot),$ $i=0,1,2$ are real-valued continuous
functions on ${\mathbb{T}}^{3},$ the function $w_{2}(\cdot,\cdot)$ is a real-
valued continuous symmetric function on $({\mathbb{T}}^{2})^{2}.$ The operator
$H_{ij}^{*}$ $(i<j)$ denotes the adjoint to $H_{ij}$ and

$(H_{01}^{*}f_{0})(p)=v_{0}(p)f_{0},\quad(H_{12}^{*}f_{1})(p,q)=\frac{v_{1}(p)f_{1}(q)+v_{1}(q)f_{1}(p)}{2},\quad
f_{i}\in{\mathcal{H}}_{i},\quad i=0,1.$

It follows that under these assumptions $H$ is bounded and self-adjoint.

We recall that the operators $H_{01}$ and $H_{12}$ (resp. $H_{01}^{*}$ and
$H_{12}^{*}$) are called annihilation (resp. creation) operators,
respectively. In the present paper we consider the case where the number of
annihilations and creations of the particles of the system is equal to 1, that
is, $H_{ij}\equiv 0$ for all $|i-j|>1.$

It is known that the three-particle discrete Schrödinger operator
$\widehat{H}$ in the momentum representation acts on the Hilbert space
$L_{2}(({\mathbb{T}}^{3})^{3}).$ Introducing the total quasi-momentum
$K\in{\mathbb{T}}^{3}$ and choosing relative coordinate system, we decompose
$\widehat{H}$ into the von Neumann direct integral (see for example [1, 7, 8,
13])

$\widehat{H}=\int_{{\mathbb{T}}^{3}}\widehat{H}(K)\,dK,$

where the bounded self-adjoint operator $\widehat{H}(K),$
$K\in{\mathbb{T}}^{3},$ acts on the Hilbert space $L_{2}(\Gamma_{K}).$ Here
$\Gamma_{K}\subset({\mathbb{T}}^{3})^{2}$ being some manifold.

Notice that the operator matrix $H$ satisfies the main spectral properties of
the three-particle discrete Schrödinger operator $\widehat{H}(0),$ where the
role of two-particle discrete Schrödinger operators is played by the family of
the generalized Friedrichs models [3, 4]. For this reason the Hilbert space
${\mathcal{H}}$ is called the three-particle cut subspace of the bosonic Fock
space ${\mathcal{F}}_{\rm s}(L_{2}({\mathbb{T}}^{3}))$ over
$L_{2}({\mathbb{T}}^{3})$ and the operator matrix $H$ is associated to a
system describing three particles in interaction without conservation of the
number of particles. The operator $H_{22}$ is associated to a system of three
quantum particles on a lattice.

To formulate the main results of the paper we introduce the operators $H_{1}$
and $H_{2}$ acting in the Hilbert spaces ${\mathcal{H}}$ and
${\mathcal{H}}_{2},$ respectively, as

$H_{1}:=\left(\begin{array}[]{ccc}H_{00}&H_{01}&0\\\
H_{01}^{*}&H_{11}&H_{12}\\\ 0&H_{12}^{*}&H_{22}^{0}\\\
\end{array}\right),\quad H_{2}:=H_{22},$

and the family of bounded self-adjoint operators (generalized Friedrichs
models) $h(p),$ $p\in{\mathbb{T}}^{3},$ acting in
${\mathcal{H}}_{0}\oplus{\mathcal{H}}_{1}$ as

$h(p)=\left(\begin{array}[]{cc}h_{00}(p)&h_{01}\\\ h_{01}^{*}&h_{11}(p)\\\
\end{array}\right),$

where

$h_{00}(p)f_{0}=w_{1}(p)f_{0},\quad
h_{01}f_{1}=\frac{1}{\sqrt{2}}\int_{{\mathbb{T}}^{3}}v_{1}(s)f_{1}(s)\,ds,$
$h_{11}(p)=h_{11}^{0}(p)-v,\quad(h_{11}^{0}(p)f_{1})(q)=w_{2}(p,q)f_{1}(q),\quad(vf_{1})(q)=v_{2}(q)\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{1}(s)\,ds.$

We recall that the operator $h(p)$ is also called molecular-resonance model
and it is associated with the Hamiltonian of the system consisting of at most
two particles on the three-dimensional lattice, interacting via both a
nonlocal potential and creation and annihilation operators.

In [16] it was shown that for any $p\in{\mathbb{T}}^{3}$ the operator $h(p)$
has at most three eigenvalues.

The spectrum, the essential spectrum, the discrete and point spectrum of a
bounded self-adjoint operator will be denoted by $\sigma(\cdot),$ $\sigma_{\rm
ess}(\cdot),$ $\sigma_{\rm disc}(\cdot)$ and $\sigma_{\rm p}(\cdot)$
respectively.

Set

$m:=\min\limits_{p,q\in{\mathbb{T}}^{3}}w_{2}(p,q),\quad
M:=\max\limits_{p,q\in{\mathbb{T}}^{3}}w_{2}(p,q).$

The following theorem describes the location of the essential spectrum of the
operator $H$ by the spectrum of the family $h(p)$ of the generalized
Friedrichs models [16].

###### Theorem 2.1

For the essential spectrum of $H$ the following equality holds:

$\sigma_{\rm
ess}(H)=\sigma\cup[m;M],\quad\sigma:=\bigcup\limits_{p\in{{\mathbb{T}}^{3}}}\sigma_{\rm
disc}(h(p)).$ (2.1)

Moreover, the set $\sigma_{\rm ess}(H)$ is a union of at most four bounded
closed intervals.

The subsets $\sigma$ and $[m;M]$ are called two-particle and three-particle
branches of the essential spectrum of $H,$ respectively.

###  2.2. Main assumptions

From now on we always assume that $\\{\alpha,\beta\\}=\\{1,2\\}$ and
$\alpha\neq\beta.$ Denote $\bar{\pi}:=(\pi,\pi,\pi).$

###### Assumption 2.2

The function $v_{\alpha}(\cdot)$ is $2\bar{\pi}$ periodic and
$v_{\beta}(\cdot)$ satisfies the condition

$\int_{{\mathbb{T}}^{3}}v_{\beta}(s)g(s)\,ds=0$ (2.2)

for any $2\bar{\pi}$ periodic function $g\in L_{2}({\mathbb{T}}^{3}).$

###### Assumption 2.3

(i) The function $w_{2}(\cdot,\cdot)$ is $2\bar{\pi}$ periodic on each
variable $p$ and $q,$ that is,
$w_{2}(p+2\bar{\pi},q)=w_{2}(p,q+2\bar{\pi})=w_{2}(p,q)$ for all
$p,q\in{\mathbb{T}}^{3};$
(ii) The function $w_{2}(\cdot,\cdot)$ has a unique non-degenerate minimum at
the point $(p_{0},p_{0})\in({\mathbb{T}}^{3})^{2}.$ All third order partial
derivatives of the functions $w_{1}(\cdot)$ and $w_{2}(\cdot,\cdot)$ are
continuous on ${\mathbb{T}}^{3}$ and $({\mathbb{T}}^{3})^{2},$ respectively.

Under the Assumption 2.2 and the part (i) of Assumption 2.3 the discrete
spectrum of $h(p)$ coincides (see Lemma 3.1 below) with the union of discrete
spectra of the operators

$h_{1}(p):=\left(\begin{array}[]{cc}h_{00}(p)&h_{01}\\\
h_{01}^{*}&h_{11}^{0}(p)\\\ \end{array}\right)\quad\mbox{and}\quad
h_{2}(p):=h_{11}(p).$

It follows from the definition of the operator $h_{\alpha}(p)$ that its
structure is simpler than that of $h(p).$ Using the Weyl theorem one can
easily show that

$\sigma_{\rm ess}(h(p))=\sigma_{\rm ess}(h_{1}(p))=\sigma_{\rm
ess}(h_{2}(p))=[m(p);M(p)],$

where the numbers $m(p)$ and $M(p)$ are defined by

$m(p):=\min_{q\in{\mathbb{T}}^{3}}w_{2}(p,q),\quad
M(p):=\max_{q\in{\mathbb{T}}^{3}}w_{2}(p,q).$

For any fixed $p\in{\mathbb{T}}^{3},$ we define the analytic functions in
${\mathbb{C}}\setminus[m(p);M(p)]$ by

$\Delta_{1}(p\,;z):=w_{1}(p)-z-\frac{1}{2}\int_{{\mathbb{T}}^{3}}\frac{v_{1}^{2}(s)\,ds}{w_{2}(p,s)-z},\quad\Delta_{2}(p\,;z):=1-\int_{{\mathbb{T}}^{3}}\frac{v_{2}^{2}(s)\,ds}{w_{2}(p,s)-z},$

which are Fredholm determinants associated with the operators $h_{1}(p)$ and
$h_{2}(p),$ respectively.

Since the function $w_{2}(\cdot,\cdot)$ has a unique non-degenerate minimum at
$(p_{0},p_{0})\in({\mathbb{T}}^{3})^{2}$ and the function $v_{\alpha}(\cdot)$
is a continuous on ${\mathbb{T}}^{3},$ for any $p\in{\mathbb{T}}^{3}$ the
integral

$\int_{{\mathbb{T}}^{3}}\frac{v_{\alpha}^{2}(s)\,ds}{w_{2}(p,s)-m}$

is positive and finite. Then the Lebesgue dominated convergence theorem yields
$\Delta_{\alpha}(p_{0}\,;m)=\lim\limits_{p\to p_{0}}\Delta_{\alpha}(p\,;m),$
and hence the function $\Delta_{\alpha}(\cdot\,;m)$ is a continuous on
${\mathbb{T}}^{3}.$

Note that using the fact [2, 3]

$\sigma_{\rm
ess}(H_{\alpha})=\sigma_{\alpha}\cup[m;M],\quad\sigma_{\alpha}:=\bigcup\limits_{p\in{{\mathbb{T}}^{3}}}\sigma_{\rm
disc}(h_{\alpha}(p))$

together with Assumption 2.2 and part (i) of Assumption 2.3 the equality (2.1)
can be written as

$\sigma_{\rm ess}(H)=\sigma_{\rm ess}(H_{1})\cup\sigma_{\rm ess}(H_{2}).$
(2.3)

It was shown in [2, 3] that if
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ then
$\sigma_{\alpha}\cap(-\infty;m]\neq\emptyset.$ Assuming
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ we introduce the
following numbers:

$E_{\rm
min}^{(\alpha)}:=\min\left\\{\sigma_{\alpha}\cap(-\infty;m]\right\\},\quad
E_{\rm max}^{(\alpha)}:=\max\left\\{\sigma_{\alpha}\cap(-\infty;m]\right\\}.$

The following theorem [2, 16] describes the structure of the part of the
essential spectrum of $H_{\alpha}$ located in $(-\infty;M].$

###### Theorem 2.4

Let part (ii) of Assumption 2.3 be fulfilled. Then the following assertions
hold.
(i) If $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0,$ then

$(-\infty;M]\cap\sigma_{\rm ess}(H_{\alpha})=[m;M].$

(ii) If $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0,$ then

$(-\infty;M]\cap\sigma_{\rm ess}(H_{\alpha})=[E_{\rm min}^{(\alpha)};M],\quad
E_{\rm min}^{(\alpha)}<m.$

(iii) If $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ then

$(-\infty;M]\cap\sigma_{\rm ess}(H_{\alpha})=[E_{\rm min}^{(\alpha)};E_{\rm
max}^{(\alpha)}]\cup[m;M],\quad E_{\rm max}^{(\alpha)}<m.$

We notice that if Assumption 2.2 and part (i) of Assumption 2.3 hold, then
Theorem 2.4 together with the equality (2.3) describes the structure of the
part of the essential spectrum of $H$ located in $(-\infty;M].$

If $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ then from
$E_{\rm min}^{(\alpha)},E_{\rm max}^{(\alpha)}\in\sigma_{\alpha}$ it follows
that there exist positive integers $n_{\alpha},$ $k_{\alpha}$ and points
$\\{p_{\alpha i}\\}_{i=1}^{n_{\alpha}},\\{q_{\alpha
j}\\}_{j=1}^{k_{\alpha}}\subset{\mathbb{T}}^{3}$ such that

$\\{p\in{\mathbb{T}}^{3}:\Delta_{\alpha}(p\,;E_{\rm
min}^{(\alpha)})=0\\}=\\{p_{\alpha 1},\ldots,p_{\alpha n_{\alpha}}\\},$
$\\{p\in{\mathbb{T}}^{3}:\Delta_{\alpha}(p\,;E_{\rm
max}^{(\alpha)})=0\\}=\\{q_{\alpha 1},\ldots,q_{\alpha k_{\alpha}}\\}.$

###### Assumption 2.5

There exist positive numbers $C,$ $\delta$ and $\beta_{\alpha i}\in(0;2],$
$i=1,\ldots,n_{\alpha}$ such that

$|\Delta_{\alpha}(p\,;E_{\rm min}^{(\alpha)})|\geq C|p-p_{\alpha
i}|^{\beta_{\alpha i}},\quad p\in U_{\delta}(p_{\alpha i}),\quad
i=1,\ldots,n_{\alpha},$

and the inequality $\Delta_{\alpha}(p\,;E_{\rm min}^{(\alpha)})>0$ holds for
all $p\in{\mathbb{T}}^{3}\setminus\\{p_{\alpha 1},\ldots,p_{\alpha
n_{\alpha}}\\}.$

###### Assumption 2.6

There exist positive numbers $K,$ $\rho$ and $\gamma_{\alpha j}\in(0;2],$
$j=1,\ldots,k_{\alpha}$ such that

$|\Delta_{\alpha}(p\,;E_{\rm max}^{(\alpha)})|\geq K|p-q_{\alpha
j}|^{\gamma_{\alpha j}},\quad p\in U_{\rho}(q_{\alpha j}),\quad
j=1,\ldots,k_{\alpha},$

and the inequality $\Delta_{\alpha}(p\,;E_{\rm max}^{(\alpha)})<0$ holds for
all $p\in{\mathbb{T}}^{3}\setminus\\{q_{\alpha 1},\ldots,q_{\alpha
k_{\alpha}}\\}.$

###  2.3. Statement of the main results

Here we formulate main results of the paper.

###### Theorem 2.7

Let part (i) of Assumption 2.3 be fulfilled.
(i) If Assumption 2.2 holds with $\alpha=1$ and in addition, the functions
$v_{0}(\cdot),$ $w_{1}(\cdot)$ are $2\bar{\pi}$ periodic, then $\sigma_{\rm
disc}(H_{1})\subset\sigma_{\rm p}(H).$
(i) If Assumption 2.2 holds with $\alpha=2,$ then $\sigma_{\rm
disc}(H_{2})\subset\sigma_{\rm p}(H).$

###### Theorem 2.8

Let Assumptions 2.2 and 2.3 be fulfilled. Assume
($\alpha$.1) $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)>0;$
($\alpha$.2) $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0$ and
Assumption 2.5 holds;
($\alpha$.3) $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
Assumptions $\ref{A3},\ref{A4}$ hold.

If for some $i,j\in\\{1,2,3\\}$ the conditions $(1.i)$ and $(2.j)$ hold, then
the operator matrix $H$ has a finite number of discrete eigenvalues lying on
the left of $m.$

###### Remark 2.9

The class of functions $w_{1}(\cdot),$ $v_{i}(\cdot),$ $i=1,2$ and
$w_{2}(\cdot,\cdot)$ satisfying the conditions in Theorem 2.8 is nonempty
$($see Lemma $\ref{Example}).$

###### Remark 2.10

Note that comparing Theorems 2.7 and 2.8 we have that if the condition
$(\alpha.j)$ in Theorem 2.8 holds for some $j\in\\{1,2,3\\},$ then the
operator $H_{\alpha}$ has a finite number of discrete eigenvalues lying on the
left of $m.$ If
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)=\Delta_{\alpha}(p_{0}\,;m)=0$
and $v_{\alpha}(p_{0})\neq 0,$ then $\min\sigma_{\rm ess}(H_{\alpha})=m$ and
it was shown in [3] for $\alpha=1$ and in [2] for $\alpha=2$ that the operator
$H_{\alpha}$ has infinitely many eigenvalues lying on the left of $m.$ Hence,
in this case by Theorem 2.7 the operator $H$ also has infinitely many
eigenvalues lying on the left of $m.$

##  3\. Some auxiliary statements

The following lemma describes the relation between the eigenvalues of the
operators $h(p)$ and $h_{\alpha}(p).$

###### Lemma 3.1

Let Assumption 2.2 and part (i) of Assumption 2.3 be fulfilled. For any fixed
$p\in{\mathbb{T}}^{3}$ the number $z(p)\in{\mathbb{C}}\setminus[m(p);M(p)]$ is
an eigenvalue for $h(p)$ if and only if $z(p)$ is an eigenvalue for at least
one of the operators $h_{1}(p)$ and $h_{2}(p).$

###### Proof.

Let $p\in{\mathbb{T}}^{3}$ be fixed. Suppose
$(f_{0},f_{1})\in{\mathcal{H}}_{0}\oplus{\mathcal{H}}_{1}$ is an eigenvector
of the operator $h(p)$ associated with the eigenvalue
$z(p)\in{\mathbb{C}}\setminus[m(p);M(p)]$. Then $f_{0}$ and $f_{1}$ satisfy
the following system of equations:

$\begin{gathered}(w_{1}(p)-z(p))f_{0}+\frac{1}{\sqrt{2}}\int_{{\mathbb{T}}^{3}}v_{1}(s)f_{1}(s)\,ds=0,$$\\\
\frac{1}{\sqrt{2}}v_{1}(q)f_{0}+(w_{2}(p,q)-z(p))f_{1}(q)-v_{2}(q)\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{1}(s)\,ds=0.\end{gathered}$
(3.1)

Since for any $q\in{\mathbb{T}}^{3}$ the relation $w_{2}(p,q)-z(p)\neq 0$
holds, from the second equation in the system (3.1) for $f_{1}$ we have

$f_{1}(q)=\frac{C_{f_{1}}v_{2}(q)}{w_{2}(p,q)-z(p)}-\frac{1}{\sqrt{2}}\frac{v_{1}(q)f_{0}}{w_{2}(p,q)-z(p)},$
(3.2)

where

$C_{f_{1}}=\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{1}(s)\,ds.$ (3.3)

Substituting the expression (3.2) for $f_{1}$ into the first equation of the
system (3.1) and the equality (3.3), we conclude that the system of equations
(3.1) has a nontrivial solution if and only if the system of equations

$\displaystyle\Delta_{1}(p\,;z(p))f_{0}+\frac{1}{\sqrt{2}}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)v_{2}(s)\,ds}{w_{2}(p,s)-z(p)}C_{f_{1}}=0,$
$\displaystyle\frac{1}{\sqrt{2}}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)v_{2}(s)\,ds}{w_{2}(p,s)-z(p)}f_{0}+\Delta_{2}(p\,;z(p))C_{f_{1}}=0$

has a nontrivial solution $(f_{0},C_{f_{1}})\in{\mathbb{C}}^{2},$ i.e. if the
condition

$\Delta_{1}(p\,;z(p))\Delta_{2}(p\,;z(p))-\frac{1}{2}\Bigg{(}\,\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)v_{2}(s)\,ds}{w_{2}(p,s)-z(p)}\,\Bigg{)}^{2}=0$

is satisfied.

By part (i) of Assumption 2.3 for any fixed $p\in{\mathbb{T}}^{3}$ the
function $(w_{2}(p,\cdot)-z(p))^{-1}\in L_{2}({\mathbb{T}}^{3})$ is
$2\bar{\pi}$ periodic. Applying Assumption 2.2 we obtain

$\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)v_{2}(s)\,ds}{w_{2}(p,s)-z(p)}=0.$

If we set $v_{2}(q)\equiv 0$ in the operator $h(p),$ then $h(p)=h_{1}(p);$ in
this case the number $z(p)\in{\mathbb{C}}\setminus[m(p);M(p)]$ is an
eigenvalue of $h_{1}(p)$ if and only if $\Delta_{1}(p\,;z(p))=0.$ Similarly
one can show that the number $z(p)\in{\mathbb{C}}\setminus[m(p);M(p)]$ is an
eigenvalue of $h_{2}(p)$ if and only if $\Delta_{2}(p\,;z(p))=0.$ The lemma is
proved. ∎

###### Lemma 3.2

Let $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)>0.$ Then there
exists a positive number $C_{1}$ such that the inequality
$\Delta_{\alpha}(p\,;z)\geq C_{1}$ holds for all $p\in{\mathbb{T}}^{3}$ and
$z\leq m.$

###### Proof.

Since for any $p\in{\mathbb{T}}^{3}$ the function $\Delta_{\alpha}(p\,;\cdot)$
is monotonically decreasing in $(-\infty;m],$ we have

$\Delta_{\alpha}(p\,;z)\geq\Delta_{\alpha}(p\,;m)\geq\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)>0$

for all $p\in{\mathbb{T}}^{3}$ and $z\leq m.$ Now setting
$C_{1}:=\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)$ we complete
the proof of lemma. ∎

For some $\delta>0$ we set

$U_{\delta}(p_{0}):=\\{p\in{\mathbb{T}}^{3}:|p-p_{0}|<\delta\\}.$

###### Lemma 3.3

If Assumption 2.5 resp. 2.6 holds, then for any $\delta>0$ there exist the
positive numbers $C_{1}(\delta)$ and $C_{2}(\delta)$ such that
(i) $\Delta_{\alpha}(p\,;E_{\rm min}^{(\alpha)})\geq C_{1}(\delta)$ for any
$p\in{\mathbb{T}}^{3}\setminus\bigcup\limits_{i=1}^{n_{\alpha}}U_{\delta}(p_{\alpha
i});$
resp.
(ii) $|\Delta_{\alpha}(p\,;E_{\rm max}^{(\alpha)})|\geq C_{2}(\delta)$ for any
$p\in{\mathbb{T}}^{3}\setminus\bigcup\limits_{j=1}^{k_{\alpha}}U_{\delta}(q_{\alpha
j}).$

###### Proof.

Let Assumption 2.5 be fulfilled. Then the inequality
$\Delta_{\alpha}(p\,;E_{\rm min}^{(\alpha)})>0$ holds for any
${\mathbb{T}}^{3}\setminus\\{p_{\alpha 1},\ldots,p_{\alpha n_{\alpha}}\\}.$
Since for any $\delta>0$ the set
${\mathbb{T}}^{3}\setminus\bigcup\limits_{i=1}^{n_{\alpha}}U_{\delta}(p_{\alpha
i})$ is compact and $\Delta_{\alpha}(\cdot\,;E_{\rm min}^{(\alpha)})$ is the
positive continuous function on this set, there exists the number
$C_{1}(\delta)>0$ such that the assertion (i) of lemma holds. Proof of
assertion (ii) is similar. ∎

###### Lemma 3.4

Let part (ii) of Assumption 2.3 be fulfilled. Then there exist positive
numbers $C_{1},C_{2},C_{3}$ and $\delta$ such that the following inequalities
hold:
(i) $C_{1}(|p-p_{0}|^{2}+|q-p_{0}|^{2})\leq w_{2}(p,q)-m\leq
C_{2}(|p-p_{0}|^{2}+|q-p_{0}|^{2}),$ $p,q\in U_{\delta}(p_{0});$
(ii) $w_{2}(p,q)-m\geq C_{3},$ $(p,q)\not\in U_{\delta}(p_{0})\times
U_{\delta}(p_{0}).$

###### Proof.

By part (ii) of Assumption 2.3 the all third order partial derivatives of
$w_{2}(\cdot,\cdot)$ are continuous on $({\mathbb{T}}^{3})^{2}$ and it has a
unique non-degenerate minimum at the point
$(p_{0},p_{0})\in({\mathbb{T}}^{3})^{2}.$ Then by the Hadamard lemma [25]
there exists a $\delta$-neighborhood of the point $p_{0}\in{\mathbb{T}}^{3}$
such that the following decomposition holds:

$\displaystyle w_{2}(p,q)$
$\displaystyle=m+\frac{1}{2}\left((W_{1}(p-p_{0}),p-p_{0})+2(W_{2}(p-p_{0}),q-p_{0})+(W_{1}(q-p_{0}),q-p_{0})\right)$
$\displaystyle\quad+\sum\limits_{|s|+|l|=3}H_{sl}(p,q)\prod_{i=1}^{3}(p^{(i)}-p_{0}^{(i)})^{s_{i}}(q^{(i)}-p_{0}^{(i)})^{l_{i}},\quad
p,q\in U_{\delta}(p_{0}),$

where

$W_{1}:=\left(\frac{\partial^{2}w_{2}(p_{0},p_{0})}{\partial p^{(i)}\partial
p^{(j)}}\right)_{i,j=1}^{3},\quad
W_{2}:=\left(\frac{\partial^{2}w_{2}(p_{0},p_{0})}{\partial p^{(i)}\partial
q^{(j)}}\right)_{i,j=1}^{3},$ $s=(s_{1},s_{2},s_{3}),\quad
l=(l_{1},l_{2},l_{3}),\quad|s|=s_{1}+s_{2}+s_{3},\quad
s_{i},l_{i}\in\\{0,1,2,3\\},\quad i=1,2,3,$

and $H_{sl}(\cdot,\cdot)$ with $|s|+|l|=3$ are continuous functions in
$U_{\delta}(p_{0})\times U_{\delta}(p_{0}).$ Therefore, there exist positive
numbers $C_{1},C_{2},C_{3}$ such that (i) and (ii) hold true. ∎

##  4\. The Weinberg type system of integral equations

In this section we derive an analogue of the Weinberg type system of integral
equations for the eigenvectors, corresponding to the eigenvalues of $H,$ lying
on the left of $m.$

Let $\tau_{\rm ess}(H)$ be the lower bound of the essential spectrum of $H.$
It is clear that $\Delta_{\alpha}(p\,;z)>0$ for all $p\in{\mathbb{T}}^{3}$ and
$z\in(-\infty;\tau_{\rm ess}(H));$ if
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ then
$\Delta_{\alpha}(p\,;z)<0$ for all $p\in{\mathbb{T}}^{3}$ and $z\in(E_{\rm
max}^{(\alpha)};m).$ So ${\rm sign}(\Delta_{\alpha}(p\,;z))$ depends on the
location of $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ and does not depend
on $p\in{\mathbb{T}}^{3}.$ For $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$
we set $\xi_{\alpha}(z):={\rm sign}(\Delta_{\alpha}(p\,;z)).$

Let for any $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ the operator $W(z)$
act in the Hilbert space ${\mathcal{H}}$ as a $3\times 3$ operator matrix with
entries $W_{ij}(z):{\mathcal{H}}_{j}\to{\mathcal{H}}_{i},$ $i,j=0,1,2$ defined
by

$\begin{gathered}W_{00}(z)g_{0}=(1+z-w_{0})g_{0},\quad
W_{01}(z)g_{1}=-\int_{{\mathbb{T}}^{3}}\frac{v_{0}(s)g_{1}(s)\,ds}{\sqrt{\xi_{1}(z)\Delta_{1}(s\,;z)}},\\\
W_{02}(z)\equiv
0,\quad(W_{10}(z)g_{0})(p)=-\frac{\xi_{1}(z)v_{0}(p)g_{0}}{\sqrt{\xi_{1}(z)\Delta_{1}(p\,;z)}},\\\
(W_{11}(z)g_{1})(p)=\frac{\xi_{1}(z)v_{1}(p)}{2\sqrt{\xi_{1}(z)\Delta_{1}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)g_{1}(s)\,ds}{\sqrt{\xi_{1}(z)\Delta_{1}(s\,;z)}(w_{2}(p,s)-z)},\\\
(W_{12}(z)g_{2})(p)=-\frac{\xi_{1}(z)v_{2}(p)}{\sqrt{\xi_{1}(z)\Delta_{1}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)v_{2}(t)g_{2}(s,t)\,ds\,dt}{\sqrt{\xi_{2}(z)\Delta_{2}(s\,;z)}(w_{2}(p,s)-z)},\\\
(W_{20}(z)g_{0})(p,q)=-\frac{v_{1}(p)(W_{10}(z)g_{0})(q)+v_{1}(q)(W_{10}(z)g_{0})(p)}{2(w_{2}(p,q)-z)},\end{gathered}$
$\displaystyle(W_{21}(z)g_{1})(p,q)$
$\displaystyle=-\frac{\xi_{2}(z)v_{1}(p)v_{2}(q)}{2(w_{2}(p,q)-z)\sqrt{\xi_{2}(z)\Delta_{2}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)g_{1}(s)\,ds}{\sqrt{\xi_{1}(z)\Delta_{1}(s\,;z)}(w_{2}(p,s)-z)}$
$\displaystyle\quad-\frac{\xi_{2}(z)v_{1}(q)v_{2}(p)}{2(w_{2}(p,q)-z)\sqrt{\xi_{2}(z)\Delta_{2}(q\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)g_{1}(s)\,ds}{\sqrt{\xi_{1}(z)\Delta_{1}(s\,;z)}(w_{2}(q,s)-z)}$
$\displaystyle\quad-\frac{v_{1}(p)(W_{11}(z)g_{1})(q)+v_{1}(q)(W_{11}(z)g_{1})(p)}{2(w_{2}(p,q)-z)},$
$\displaystyle(W_{22}(z)g_{2})(p,q)$
$\displaystyle=\frac{\xi_{2}(z)v_{2}(p)v_{2}(q)}{(w_{2}(p,q)-z)\sqrt{\xi_{2}(z)\Delta_{2}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)v_{2}(t)g_{2}(s,t)\,ds\,dt}{\sqrt{\xi_{2}(z)\Delta_{2}(s\,;z)}(w_{2}(p,s)-z)}$
$\displaystyle\quad+\frac{\xi_{2}(z)v_{2}(p)v_{2}(q)}{(w_{2}(p,q)-z)\sqrt{\xi_{2}(z)\Delta_{2}(q\,;z)}}\int_{{\mathbb{T}}^{3}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)v_{2}(t)g_{2}(s,t)\,ds\,dt}{\sqrt{\xi_{2}(z)\Delta_{2}(s\,;z)}(w_{2}(q,s)-z)}$
$\displaystyle\quad-\frac{v_{1}(p)(W_{12}(z)g_{2})(q)+v_{1}(q)(W_{12}(z)g_{2})(p)}{2(w_{2}(p,q)-z)},$

where $g_{i}\in{\mathcal{H}}_{i},$ $i=0,1,2.$

We have the following lemma.

###### Lemma 4.1

Let Assumption 2.2 and part (i) of Assumption 2.3 be fulfilled. If
$f\in{\mathcal{H}}$ is an eigenvector corresponding to the eigenvalue
$z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ of $H,$ then $f$ satisfies the
Weinberg equation $W(z)f=f.$

###### Proof.

Let $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ be an eigenvalue of the
operator $H$ and $f=(f_{0},f_{1},f_{2})\in{\mathcal{H}}$ be the corresponding
eigenvector. Then $f_{0},$ $f_{1}$ and $f_{2}$ satisfy the system of equations

$\begin{gathered}(H_{00}-z)f_{0}+H_{01}f_{1}=0,\\\
(H_{10}f_{0})(p)+((H_{11}-z)f_{1})(p)+(H_{12}f_{2})(p)=0,\\\
(H_{21}f_{1})(p,q)+((H_{22}^{0}-z)f_{2})(p,q)-(Vf_{2})(p,q)=0.\end{gathered}$
(4.1)

Since $z<m,$ from the third equation of the system (4.1) for $f_{2}$ we have

$f_{2}(p,q)=\frac{v_{2}(q)\overline{f}_{2}(p)+v_{2}(p)\overline{f}_{2}(q)}{w_{2}(p,q)-z}-\frac{v_{1}(q)f_{1}(p)+v_{1}(p)f_{1}(q)}{2(w_{2}(p,q)-z)},$
(4.2)

where

$\overline{f}_{2}(p)=\int_{{\mathbb{T}}^{3}}v_{2}(s)f_{2}(p,s)\,ds.$ (4.3)

Substituting the expression (4.2) for $f_{2}$ into the second equation in the
system (4.1) and the equality (4.3) and using Assumptions 2.2 and 2.3, we
obtain

$\begin{gathered}f_{0}=(1+z-w_{0})f_{0}-\int_{{\mathbb{T}}^{3}}v_{0}(s)f_{1}(s)\,ds=0,\\\
\Delta_{1}(p\,;z)f_{1}(p)=-v_{0}(p)f_{0}+\frac{v_{1}(p)}{2}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)f_{1}(s)\,ds}{w_{2}(p,s)-z}-v_{2}(p)\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)\overline{f}_{2}(s)\,ds}{w_{2}(p,s)-z},\\\
\Delta_{2}(p\,;z)\overline{f}_{2}(p)=-\frac{v_{1}(p)}{2}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)f_{1}(s)\,ds}{w_{2}(p,s)-z}+v_{2}(p)\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)\overline{f}_{2}(s\,)\,ds}{w_{2}(p,s)-z}.\end{gathered}$
(4.4)

It is clear that the inequality $\xi_{\alpha}(z)\Delta_{\alpha}(p\,;z)>0$
holds for all $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ and
$p\in{\mathbb{T}}^{3}.$ Therefore, the system of equations (4.4) has a
nontrivial solution if and only if the following system of equations:

$\displaystyle f_{0}$ $\displaystyle=W_{00}(z)f_{0}+W_{01}(z)f_{1}=0,$
$\displaystyle f_{1}(p)$
$\displaystyle=(W_{10}(z)f_{0})(p)+(W_{11}(z)f_{1})(p)$
$\displaystyle\quad-\frac{\xi_{1}(z)v_{2}(p)}{\sqrt{\xi_{1}(z)\Delta_{1}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)\overline{f}_{2}(s)\,ds}{\sqrt{\xi_{2}(z)\Delta_{2}(s\,;z)}(w_{2}(p,s)-z)},$
$\displaystyle\overline{f}_{2}(p)$
$\displaystyle=-\frac{\xi_{2}(z)v_{1}(p)}{2\sqrt{\xi_{2}(z)\Delta_{2}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)f_{1}(s)\,ds}{\sqrt{\xi_{1}(z)\Delta_{1}(s\,;z)}(w_{2}(p,s)-z)}$
$\displaystyle\quad+\frac{\xi_{2}(z)v_{2}(p)}{\sqrt{\xi_{2}(z)\Delta_{2}(p\,;z)}}\int_{{\mathbb{T}}^{3}}\frac{v_{2}(s)\overline{f}_{2}(s)\,ds}{\sqrt{\xi_{2}(z)\Delta_{2}(s\,;z)}(w_{2}(p,s)-z)}$

has a nontrivial solution.

Substituting the last expressions for $f_{1}$ and $\overline{f}_{2}$ into the
formula (4.2) and using the equality (4.3), we obtain the Weinberg equation
$W(z)f=f.$ ∎

Set

$\Sigma:=\overline{[\tau_{\rm ess}(H)-1;m]\setminus\sigma_{\rm ess}(H)}.$

###### Lemma 4.2

Let assumptions in Theorem 2.8 be fulfilled. Then the operator $W(z)$ is
compact for $z\in\Sigma$ and the operator-valued function $W(z)$ is continuous
in the uniform operator topology for $z\in\Sigma.$

###### Proof.

First for the convenience using Theorem 2.7 we describe the structure of the
set $\Sigma:$
(i) if $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0,$ then
$\Sigma=[m-1;m];$
(ii) if $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0,$ then
$\Sigma=[E_{\rm min}-1;E_{\rm min}],$ where $E_{\rm min}=\min\\{E_{\rm
min}^{(1)},E_{\rm min}^{(2)}\\}$ and $E_{\rm min}<m;$
(iii) if $\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0$ and
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\beta}(p\,;m)\geq 0,$ then
$\Sigma=[E_{\rm min}^{(\alpha)}-1;E_{\rm min}^{(\alpha)}]$ with $E_{\rm
min}^{(\alpha)}<m;$
(iv) if $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\beta}(p\,;m)\geq 0,$ then
$\Sigma=[E_{\rm min}^{(\alpha)}-1;E_{\rm min}^{(\alpha)}]\cup[E_{\rm
max}^{(\alpha)};m]$ with $E_{\rm max}^{(\alpha)}<m;$
(v) if $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\beta}(p\,;m)<0$ and
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\beta}(p\,;m)\geq 0,$ then

$\Sigma=\left\\{\begin{array}[]{ll}\qquad\quad\,\,[E_{\rm min}-1;E_{\rm
min}],\qquad\quad\,\,\,\quad\mbox{if}\,\,E_{\rm max}^{(\alpha)}\geq E_{\rm
min}^{(\beta)},\\\ [E_{\rm min}^{(\alpha)}-1;E_{\rm
min}^{(\alpha)}]\cup[E_{\rm max}^{(\alpha)};E_{\rm
min}^{(\beta)}],\,\,\quad\mbox{if}\,\,E_{\rm max}^{(\alpha)}<E_{\rm
min}^{(\beta)}\end{array}\right.$

with $E_{\rm max}^{(\alpha)},E_{\rm min}^{(\beta)}<m;$
(vi) if $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ then
$\Sigma=[\tau_{\rm ess}(H)-1;m]\setminus\\{(E_{\rm min}^{(1)};E_{\rm
max}^{(1)})\cup(E_{\rm min}^{(2)};E_{\rm max}^{(2)})\\}$ with $E_{\rm
max}^{(\alpha)}<m.$

We will prove the statement of the lemma for the case (vi) with $E_{\rm
min}:=E_{\rm min}^{(1)}=E_{\rm min}^{(2)}$ and $E_{\rm max}:=E_{\rm
max}^{(1)}=E_{\rm max}^{(2)}.$ Other cases can be proven in a similar.

Let $\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
Assumptions 2.5, 2.6 be fulfilled. For $z\in(-\infty;m)\setminus\sigma_{\rm
ess}(H)$ denote by $W(p,q,s,t;z)$ the kernel of the operator $W_{22}(z).$

We have the following inequalities:

$w_{2}(p,q)-z\geq m-E_{\rm min}>0\quad\mbox{for all}\quad
p,q\in{\mathbb{T}}^{3},\quad z\leq E_{\rm min};$ $w_{2}(p,q)-z\geq(m-E_{\rm
max})/2>0\quad\mbox{for all}\quad p,q\in{\mathbb{T}}^{3},\quad z\in[E_{\rm
max};(m+E_{\rm max})/2].$

Then by Assumptions 2.5, 2.6 and Lemma 3.3 the function
$|W(\cdot,\cdot,\cdot,\cdot;z)|$ can be estimated by

$\displaystyle C_{1}$
$\displaystyle\times\left(1\\!+\\!\sum_{i=1}^{n_{2}}\frac{\chi_{\delta}(s-p_{2i})}{|s-p_{2i}|^{\beta_{2i}/2}}\right)$
$\displaystyle\times\left(1\\!+\\!\sum_{i=1}^{n_{1}}\frac{\chi_{\delta}(p-p_{1i})}{|p-p_{1i}|^{\beta_{1i}/2}}\\!+\\!\sum_{i=1}^{n_{1}}\frac{\chi_{\delta}(q-p_{1i})}{|q-p_{1i}|^{\beta_{1i}/2}}\\!+\\!\sum_{i=1}^{n_{2}}\frac{\chi_{\delta}(p-p_{2i})}{|p-p_{2i}|^{\beta_{2i}/2}}\\!+\\!\sum_{i=1}^{n_{2}}\frac{\chi_{\delta}(q-p_{2i})}{|q-p_{2i}|^{\beta_{2i}/2}}\right)$

for $z\leq E_{\rm min}$ and by

$\displaystyle C_{2}$
$\displaystyle\times\left(1\\!+\\!\sum_{j=1}^{k_{2}}\frac{\chi_{\rho}(s-q_{2j})}{|s-q_{2j}|^{\gamma_{2j}/2}}\right)$
$\displaystyle\times\left(1\\!+\\!\sum_{j=1}^{k_{1}}\frac{\chi_{\rho}(p-q_{1j})}{|p-q_{1j}|^{\gamma_{1j}/2}}\\!+\\!\sum_{j=1}^{k_{1}}\frac{\chi_{\rho}(q-q_{1j})}{|q-q_{1j}|^{\gamma_{1j}/2}}\\!+\\!\sum_{j=1}^{k_{2}}\frac{\chi_{\rho}(p-q_{2j})}{|p-q_{2j}|^{\gamma_{2j}/2}}\\!+\\!\sum_{j=1}^{k_{2}}\frac{\chi_{\rho}(q-q_{2j})}{|q-q_{2j}|^{\gamma_{2j}/2}}\right)$

for $z\in[E_{\rm max},(m+E_{\rm max})/2],$ where $\chi_{\delta}(\cdot)$ is the
characteristic function of $U_{\delta}(\overline{0}).$

Since $\xi_{\alpha}(z)=1$ for any $z\in(E_{\rm max};m)$ and
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0,$ for any
$z\in(E_{\rm max};m)$ we have
$\max\limits_{p\in{\mathbb{T}}^{3}}(\xi_{\alpha}(z)\Delta_{\alpha}(p\,;m))>0.$
Therefore, Lemmas 3.2 and 3.4 imply that the function
$|W(\cdot,\cdot,\cdot,\cdot;z)|$ can be estimated by

$\displaystyle
C_{3}\left(1+\frac{\chi_{\delta}(p-p_{0})\chi_{\delta}(q-p_{0})}{|p-p_{0}|^{2}+|q-p_{0}|^{2}}\right)\left(1+\frac{\chi_{\delta}(p-p_{0})\chi_{\delta}(s-p_{0})}{|p-p_{0}|^{2}+|s-p_{0}|^{2}}+\frac{\chi_{\delta}(q-p_{0})\chi_{\delta}(s-p_{0})}{|q-p_{0}|^{2}+|s-p_{0}|^{2}}\right)$

for $z\in[(m+E_{\rm max})/2;m].$

The latter three functions are square integrable on $({\mathbb{T}}^{3})^{4}$
and hence the operator $W_{22}(z)$ is Hilbert Schmidt for any
$z\in(-\infty;E_{\rm min}]\cup[E_{\rm max};m].$

A similar argument shows that the operators $W_{11}(z),$ $W_{12}(z)$ and
$W_{21}(z)$ are also Hilbert Schmidt for any $z\in\Sigma.$

For any $z\in(-\infty;m)\setminus\sigma_{\rm ess}(H)$ the kernel function of
$W_{ij}(z),$ $i,j=1,2$ is continuous on its domain. Therefore the continuity
of the operator-valued functions $W_{ij}(z),$ $i,j=1,2$ in the uniform
operator topology for $z\in\Sigma$ follows from Lebesgue’s dominated
convergence theorem.

Since for all $z\in\Sigma$ the operators $W_{00}(z),$ $W_{01}(z),$ $W_{10}(z)$
and $W_{20}(z)$ are of rank 1 and continuous in the uniform operator topology
for $z\in\Sigma,$ one concludes that $W(z)$ is compact for $z\in\Sigma$ and
the operator-valued function $W(z)$ is continuous in the uniform operator
topology for $z\in\Sigma.$ ∎

##  5\. Proof of the main results

In this section we prove Theorems 2.7 and 2.8.

###### Proof of Theorem 2.7.

Let $\alpha=1.$ If $z_{1}\in{\mathbb{C}}\setminus\sigma_{\rm ess}(H_{1})$ is
an eigenvalue of the operator $H_{1}$ and
$f=(f_{0},f_{1},f_{2})\in{\mathcal{H}}$ is the corresponding eigenvector, then
$f_{0},$ $f_{1}$ and $f_{2}$ are satisfy the following system of equations:

$\begin{gathered}(w_{0}-z_{1})f_{0}+\int_{{\mathbb{T}}^{3}}v_{0}(s)f_{1}(s)\,ds=0,\\\
v_{0}(p)f_{0}+(w_{1}(p)-z_{1})f_{1}(p)+\int_{{\mathbb{T}}^{3}}v_{1}(s)f_{2}(p,s)\,ds=0,\\\
\frac{1}{2}(v_{1}(p)f_{1}(q)+v_{1}(q)f_{1}(p))+(w_{2}(p,q)-z_{1})f_{2}(p,q)=0.\end{gathered}$
(5.1)

Since $z_{1}\not\in[m;M],$ from the third equation of the system (5.1) for
$f_{2}$ we have

$f_{2}(p,q)=-\frac{v_{1}(p)f_{1}(q)+v_{1}(q)f_{1}(p)}{2(w_{2}(p,q)-z_{1})}.$
(5.2)

Substituting the expression (5.2) for $f_{2}$ into the second equation of the
system (5.1), we obtain

$\Delta_{1}(p\,;z_{1})f_{1}(p)=\frac{v_{1}(p)}{2}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)f_{1}(s)\,ds}{w_{2}(p,s)-z_{1}}-v_{0}(p)f_{0}.$

Since $z_{1}\not\in\sigma_{\rm ess}(H_{1})$ the inequality
$\Delta_{1}(p\,;z_{1})\neq 0$ holds for all $p\in{\mathbb{T}}^{3}.$ From the
last equation we have

$f_{1}(p)=\frac{v_{1}(p)}{2\Delta_{1}(p\,;z_{1})}\int_{{\mathbb{T}}^{3}}\frac{v_{1}(s)f_{1}(s)\,ds}{w_{2}(p,s)-z_{1}}-\frac{v_{0}(p)f_{0}}{\Delta_{1}(p\,;z_{1})}.$

The functions $v_{0}(\cdot),$ $v_{1}(\cdot),$ $w_{1}(\cdot)$ and
$w_{2}(\cdot,q),$ $q\in{\mathbb{T}}^{3}$ are $2\bar{\pi}$ periodic and hence
the function $f_{1}(\cdot)$ is also $2\bar{\pi}$ periodic. Therefore, for any
fixed $p\in{\mathbb{T}}^{3}$ the function $f_{2}(p,\cdot)$ defined by (5.2),
is $2\bar{\pi}$ periodic. Hence this function satisfies the condition (2.2),
that is, $Vf_{2}=0.$ So the number $z_{1}\in\sigma_{\rm disc}(H_{1})$ is an
eigenvalue of $H$ with the same eigenvector
$f=(f_{0},f_{1},f_{2})\in{\mathcal{H}}.$ Therefore, $\sigma_{\rm
disc}(H_{1})\subset\sigma_{\rm p}(H).$

Let now $z_{2}\in\sigma_{\rm disc}(H_{2})$ and $g_{2}\in{\mathcal{H}}_{2}$ be
the eigenfunction corresponding to the discrete eigenvalue $z_{2}.$ Then
similar analysis shows that $H_{12}g_{2}=0,$ which guarantee that the number
$z_{2}\in\sigma_{\rm disc}(H_{2})$ is an eigenvalue of $H$ and corresponding
eigenvector $g$ has form $g=(0,0,g_{2})\in{\mathcal{H}},$ that is,
$\sigma_{\rm disc}(H_{2})\subset\sigma_{\rm p}(H).$ Theorem 2.7 is proved. ∎

###### Proof of Theorem 2.8.

We prove the finiteness of the number of discrete eigenvalues located on the
left of $m$ for the case when
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0.$ Other cases can
be proven similarly. Suppose that the operator $H$ has an infinite number of
discrete eigenvalues $\left(E_{k}\right)_{k\in{\mathbb{N}}}\subset(E_{\rm
max};m).$ Then three cases are possible

(i) $\lim\limits_{k\to\infty}E_{k}=m;$

(ii) $\lim\limits_{k\to\infty}E_{k}=E_{\rm max};$

(iii) there exist
$\left(E_{k}^{\prime}\right)_{k\in{\mathbb{N}}},\left(E_{k}^{\prime\prime}\right)_{k\in{\mathbb{N}}}\subset\left(E_{k}\right)_{k\in{\mathbb{N}}}$
such that $\lim\limits_{k\to\infty}E_{k}^{\prime}=m$ and

$\lim\limits_{k\to\infty}E_{k}^{\prime\prime}=E_{\rm max}.$

Let us consider the case (iii). For each $k\in{\mathbb{N}}$ we denote by
$\varphi_{k}\in{\mathcal{H}}$ and $\psi_{k}\in{\mathcal{H}}$ the orthonormal
eigenvectors corresponding to the eigenvalues $E_{k}^{\prime}$ and
$E_{k}^{\prime\prime},$ respectively. Then it follows from Lemma 4.1 that
$\varphi_{k}=W(E_{k}^{\prime})\varphi_{k}$ and
$\psi_{k}=W(E_{k}^{\prime\prime})\psi_{k}$ for any $k\in{\mathbb{N}}.$ By
virtue of Lemma 4.2 the operators $W(E_{\rm max}),W(m)$ are compact and
$\|W(z)-W(E_{\rm max})\|\to 0$ and $\|W(z)-W(m)\|\to 0$ as $z\to E_{\rm
max}+0$ and $z\to m-0,$ respectively. Therefore,

$\displaystyle
1=\|\varphi_{k}\|=\|W(E_{k}^{\prime})\varphi_{k}\|\leq\|(W(E_{k}^{\prime})-W(E_{\rm
max}))\varphi_{k}\|+\|W(E_{\rm max})\varphi_{k}\|\to 0,$ $\displaystyle
1=\|\psi_{k}\|=\|W(E_{k}^{\prime\prime})\psi_{k}\|\leq\|(W(E_{k}^{\prime\prime})-W(m))\psi_{k}\|+\|W(m)\psi_{k}\|\to
0$

as $k\to\infty.$ This contradiction implies that the points $z=E_{\rm max}$
and $z=m$ can not be limit points of the set of discrete eigenvalues of $H$
belonging to the interval $(E_{\rm max};m).$ Similar arguments show that other
edges of $\Sigma$ are also cannot be accumulation point for the set of
discrete eigenvalues of $H$ smaller than $m.$ ∎

The following example shows that the class of functions $w_{1}(\cdot),$
$v_{i}(\cdot),$ $i=1,2$ and $w_{2}(\cdot,\cdot)$ satisfying the conditions of
Theorem 2.8 is nonempty.

###### Lemma 5.1

Let

$\widehat{v}_{1}(p):=\\!\sum\limits_{i=1}^{3}c_{i}\cos
p^{(i)},\quad\widehat{v}_{2}(p):=\\!\sum\limits_{i=1}^{3}d_{i}\cos(p^{(i)}/2),\quad
v_{\alpha}(p):=\\!\\!\sqrt{2^{2-\alpha}\mu_{\alpha}}\,\widehat{v}_{\alpha}(p),\quad\alpha=1,2,$
$w_{1}(p)\equiv 1,\quad
w_{2}(p,q)=\varepsilon(p)+\varepsilon(q),\quad\varepsilon(p)=\sum\limits_{i=1}^{3}(1-\cos
p^{(i)}),$

where $\mu_{\alpha}>0;$ $c_{i},d_{i},$ $i=1,2,3$ are arbitrary real numbers.

Set

$\mu_{\alpha}^{(0)}:=\Bigg{(}\int_{{\mathbb{T}}^{3}}\frac{\widehat{v}_{\alpha}^{2}(s)\,ds}{\varepsilon(s)}\Bigg{)}^{-1},\quad\mu_{\alpha}^{(1)}:=\Bigg{(}\int_{{\mathbb{T}}^{3}}\frac{\widehat{v}_{\alpha}^{2}(s)\,ds}{6+\varepsilon(s)}\Bigg{)}^{-1}.$

Then the functions $w_{1}(\cdot),$ $v_{\alpha}(\cdot),$ $\alpha=1,2$ and
$w_{2}(\cdot,\cdot)$ are satisfy Assumptions 2.2, 2.3, 2.5, 2.6. Moreover,

(i) if $0<\mu_{\alpha}<\mu_{\alpha}^{(0)},$ then
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)>0;$

(ii) if $\mu_{\alpha}^{(0)}<\mu_{\alpha}\leq\mu_{\alpha}^{(1)},$ then
$\min\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0$ and
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)\geq 0;$

(iii) if $\mu_{\alpha}>\mu_{\alpha}^{(1)},$ then
$\max\limits_{p\in{\mathbb{T}}^{3}}\Delta_{\alpha}(p\,;m)<0.$

###### Proof.

Let $g\in L_{2}({\mathbb{T}}^{3})$ be as in Assumption 2.2. Then we have

$\int_{{\mathbb{T}}^{3}}v_{2}(s)g(s)\,ds=\int_{{\mathbb{T}}^{3}}v_{2}(s+2\bar{\pi})g(s+2\bar{\pi})\,ds=-\int_{{\mathbb{T}}^{3}}v_{2}(s)g(s)\,ds,$

which yields the equality (2.2), that is, Assumption 2.2 holds with $\alpha=1$
and $\beta=2.$

From the definition of $w_{2}(\cdot,\cdot)$ it follows that this function has
a unique zero non-degenerate minimum at
$(\overline{0},\overline{0})\in({\mathbb{T}}^{3})^{2}$ and it satisfies all
conditions of Assumption 2.3.

The assertions (i)–(iii) directly follow from the definition of the numbers
$\mu_{\alpha}^{(0)}$ and $\mu_{\alpha}^{(1)}.$

Let $\mu_{\alpha}^{(0)}<\mu_{\alpha}\leq\mu_{\alpha}^{(1)}.$ We prove that the
function $\Delta_{\alpha}(\cdot;E_{\rm min}^{(\alpha)})$ has a unique non-
degenerate minimum at $\overline{0}\in{\mathbb{T}}^{3}.$ Simple calculations
show that $\Delta_{\alpha}(p;E_{\rm
min}^{(\alpha)})>\Delta_{\alpha}(\overline{0};$ $E_{\rm min}^{(\alpha)})$ for
all $p\neq\overline{0}.$

Since $E_{\rm min}^{(\alpha)}\in(-\infty,0),$ it is clear that the function
$\Delta_{\alpha}(\cdot\,;E_{\rm min}^{(\alpha)})$ is twice continuously
differentiable in ${\bf T}^{3}.$ Moreover, from the equalities

$\displaystyle\frac{\partial^{2}\Delta_{\alpha}(p\,;E_{\rm
min}^{(\alpha)})}{\partial p^{(i)}\partial p^{(i)}}$
$\displaystyle=\mu_{\alpha}\cos\,p^{(i)}\int\limits_{{\mathbb{T}}^{3}}\frac{\widetilde{v}_{\alpha}^{2}(s)\,ds}{(\varepsilon(p)+\varepsilon(s)-E_{\rm
min}^{(\alpha)})^{2}}$
$\displaystyle\quad-2\mu_{\alpha}(\sin\,p^{(i)})^{2}\int_{{\mathbb{T}}^{3}}\frac{\widetilde{v}_{\alpha}^{2}(s)\,ds}{(\varepsilon(p)+\varepsilon(s)-E_{\rm
min}^{(\alpha)})^{3}},\quad i=1,2,3,$
$\displaystyle\frac{\partial^{2}\Delta_{\alpha}(p\,;E_{\rm
min}^{(\alpha)})}{\partial p^{(i)}\partial p^{(j)}}$
$\displaystyle=-2\mu_{\alpha}\sin\,p^{(i)}\sin\,p^{(j)}\int_{{\mathbb{T}}^{3}}\frac{\widetilde{v}_{\alpha}^{2}(s)\,ds}{(\varepsilon(p)+\varepsilon(s)-E_{\rm
min}^{(\alpha)})^{3}},\quad i\neq j,\quad i,j=1,2,3$

we get

$\frac{\partial^{2}\Delta_{\alpha}(\overline{0}\,;E_{\rm
min}^{(\alpha)})}{\partial p^{(i)}\partial
p^{(i)}}>0,\quad\frac{\partial^{2}\Delta_{\alpha}(\overline{0}\,;E_{\rm
min}^{(\alpha)})}{\partial p^{(i)}\partial p^{(j)}}=0,\quad i\neq j,\quad
i,j=1,2,3.$

Using these facts, one may verify that the matrix of the second order partial
derivatives of the function $\Delta_{\alpha}(\cdot;E_{\rm min}^{(\alpha)})$ at
the point $p=\overline{0}$ is positive definite. Thus the function
$\Delta_{\alpha}(\cdot;E_{\rm min}^{(\alpha)})$ has a non-degenerate minimum
at the point $p=\overline{0}.$ Then the equality
$\Delta_{\alpha}(\overline{0};E_{\rm min}^{(\alpha)})=0$ implies that there
exist the numbers $\delta>0$ and $C>0$ such that

$|\Delta_{\alpha}(p\,;E_{\rm min}^{(\alpha)})|\geq C|p|^{2},\quad p\in
U_{\delta}(\overline{0}),$

that is, Assumption 2.5 holds with $n_{\alpha}=1,$ $p_{\alpha 1}=\overline{0}$
and $\beta_{\alpha 1}=2.$

In the case $\mu_{\alpha}>\mu_{\alpha}^{(1)}$ one can similarly show that
there exist $\rho>0$ and $K>0$ such that

$|\Delta_{\alpha}(p\,;E_{\rm max}^{(\alpha)})|\geq
K|p-\overline{\pi}|^{2},\quad p\in U_{\rho}(\overline{\pi}),$

that is, Assumption 2.6 holds with $k_{\alpha}=1,$ $q_{\alpha
1}=\overline{\pi}$ and $\gamma_{\alpha 1}=2.$ ∎

Acknowledgments. This work was supported by the TOSCA Erasmus Mundus grant.
The author wishes to thank the University of L’Aquila for the invitation and
hospitality.

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# Invariant Meta Learning for Out-of-Distribution Generalization

Penghao Jiang, Ke Xin, Zifeng Wang, Chunxi Li
The Australian National University, Canberra, Australia

###### Abstract

Modern deep learning techniques have illustrated their excellent capabilities
in many areas, but relies on large training data. Optimization-based meta-
learning train a model on a variety tasks, such that it can solve new learning
tasks using only a small number of training samples. However, these methods
assumes that training and test data are identically and independently
distributed. To overcome such limitation, in this paper, we propose invariant
meta learning for out-of-distribution tasks. Specifically, invariant meta
learning find invariant optimal meta-initialization, and fast adapt to out-of-
distribution tasks with regularization penalty. Extensive experiments
demonstrate the effectiveness of our proposed invariant meta learning on out-
ofdistribution few-shot tasks.

##  1 Introduction

††footnotetext: * The first two authors contributed equally as joint first
authorship. The last two authors contributed equally as joint second
authorship.

Modern deep learning techniques have illustrated their excellent capabilities
in many areas like computer vision, natural language processing and
recommendation, etc [11]. However, these methods relies on large training
data. To overcome this limitation, few-shot learning methods such as meta
learning has been proposed [6]. Most popular meta learning approaches is the
optimization-based metalearning [4, 16], which is model-agnostic and can be
applied to various downstream tasks. However, many recent researches have
revealed the vulnerability of machine learning model when exposed to data with
different distributions.

Figure 1: Illustration example of how the distribution shifts between training
data and testing data hamper the performance of model predictions. Figure 2:
Causal framework of dog perdiction task. Due to the spurious correlation, the
model tends to focus on both grass and dog, which lead to failed prediction in
other distributions.

Such massive gap is induced by the violation of a fundamental assumption that
training and test data are identically and independently distributed (a.k.a.
i.i.d. assumption), upon which most of the existing meta learning models are
developed [4, 16]. In many real cases where i.i.d. assumption can hardly be
satisfied, especially those high-stake applications such as healthcare,
military and autonomous driving, instead of generalization within the training
distribution, the ability to generalize under distribution shift is of more
critical significance. As shown in Figure 1, given traning data where dogs are
on the grass, model could not make accurate predictions in testing data where
dogs are in water, cage or street. The reason is that the supurious
correlation between grass and dog in traning data hamper the performance of
model. Due to the spurious correlation, the model tends to focus on both grass
and dog, which lead to failed prediction in other distribution such as dogs
are in water, cage or street as shown in Figure 2. However, recent meta
learning methods could not overcome the distribution shifts between training
and testing data. In this paper, we consider a realistic scenario where tasks
come from different distributions (out-of-distribution, OOD).

In this paper, to overcome the problem mentioned above, we propose Invariant
Meta Learning (IML) for out-of- distribution tasks, a general learning
framework that jointly adjusts gradient magnitudes and directions.
Specifically, invariant meta learning find invariant optimal
metainitialization, and fast adapt to out-of-distribution tasks with
regularization penalty. To summarize, our main contributions are:

  * •

We consider the challenge of out-of-distribution tasks faced by few-shot
learning, we show a natural idea to jointly adjust gradient magnitudes and
directions of all tasks in the meta optimization process;

  * •

We propose Invariant Meta Learning (IML) for out-ofdistribution tasks, a
general learning framework that jointly adjusts gradient magnitudes and
directions;

  * •

We conduct extensive experiments and analysis to demonstrate that our approach
effectively improves the performance and generalization ability under both
indistribution and out-of-distribution few-shot settings, and thus it can be
regarded as a better baseline.

##  2 Method

In this section, we introduce our proposed Invariant Meta Learning (IML) to
address the out-of-distribution problem in few-shot tasks. IML learns
invariant optimal predictors based on optimization based meta learning
framework. To learn invariant optimal meta-initialization in optimization
based meta learning, the main challenge is that OOD problem exacerbates the
inconsistency in both task-gradient magnitudes and directions. To overcome
such problem, IML finds invariant optimal initialization, and adapt to outof-
distribution tasks with regularization penalty.

Model-agnostic meta-learning (MAML) [4] is an approach to optimization-based
meta-learning that is related to our work. For some parametric model
$f_{\theta}$, MAML aims to find a single set of parameters $\theta$ which,
using a few optimization steps, can be successfully adapted to any novel task
sampled from the same distribution. For a particular task instance
$\mathcal{T}_{i}=\left(\mathcal{D}^{tr},\mathcal{D}^{val}\right)$, the
parameters are adapted to task-specific model parameters $\theta_{i}^{\prime}$
by applying some differentiable function, typically an update rule of the
form:

$\theta_{i}^{\prime}=\mathcal{G}\left(\theta,\mathcal{D}^{tr}\right),$ (1)

where $\mathcal{G}$ is typically implemented as a step of gradient descent on
the few-shot training set $\mathcal{D}^{\text{tr
}},\theta_{i}^{\prime}=\theta-$
$\alpha\nabla_{\theta}\mathcal{L}_{\mathcal{T}_{i}}^{tr}\left(f_{\theta}\right)$.
Generally, multiple sequential adaptation steps can be applied. The learning
rate $\alpha$ can also be metalearned concurrently, in which case we refer to
this algorithm as Meta-SGD [13]. During meta-training, the parameters $\theta$
are updated by back-propagating through the adaptation procedure, in order to
reduce errors on the validation set $\mathcal{D}^{{val}}$ :

$\theta\leftarrow\theta-\eta\nabla_{\theta}\sum_{\mathcal{T}_{i}\sim
p(\mathcal{T})}\mathcal{L}_{\mathcal{T}_{i}}^{val}\left(f_{\theta_{i}^{\prime}}\right).$
(2)

The approach includes the main ingredients of optimization-based meta-learning
with neural networks: initialization is done by maintaining an explicit set of
model parameters $\theta$; the adaptation procedure, or “inner loop”, takes
$\theta$ as input and returns $\theta_{i}^{\prime}$ adapted specifically for
task instance $\mathcal{T}_{i}$, by iteratively using gradient descent (Eq.
1); and termination, which is handled simply by choosing a fixed number of
optimization steps in the “inner loop”. MAML updates $\theta$ by
differentiating through the “inner loop” in order to minimize errors of
instance-specific adapted models $f_{\theta_{i}^{\prime}}$ on the
corresponding validation set (Eq. 2). We refer to this process as the “outer
loop” of meta-learning. We use the same stages to describe IML.

Invariant Meta Learning (IML) finds invariant optimal meta-initialization, and
fast adapt to out-of-distribution tasks with regularization penalty. MAML fast
adapt network to new task during the inner loop and learns universal meta-
initialization in outer loop. Similarly, in IML, we update network with the
bi-level update, optimizing classifier in the inner loop and learning feature
representation in the outer loop. For the inner-level optimization, the
parameters $\theta$ of the predictor become $\theta_{i}$ while adapting to the
task $t_{i}\in\mathcal{T}_{tr}$. This correspond to the inner optimization of
MAML, except that each task $t_{i}$ has a corresponding network $\theta_{i}$.
The optimization in the inner loop can be defined as follows:

$\theta_{i}^{\prime}=\theta-\alpha\nabla_{\theta}\mathcal{L}_{\mathcal{T}_{i}}^{tr}\left(f_{\theta}\right)$
(3)

where $\alpha$ is a learning rate of the inner optimization.

With inner optimized network $f_{\theta_{i}^{\prime}}$, we have outer loop
objective function with variance penalty regularizer:

$\mathcal{L}^{{val}}=\sum_{\mathcal{T}_{i}\sim
p\left(\mathcal{T}^{tr}\right)}\sum_{\mathcal{T}_{j}\sim
p\left(\mathcal{T}^{{val}}\right)}\mathcal{L}_{\mathcal{T}_{j}}^{{val}}\left(f_{\theta_{i}^{\prime}}\right)$
(4)
$\theta\leftarrow\theta-\eta\nabla_{\theta}\mathcal{L}^{{val}}-\beta\lambda\operatorname{trace}\left(\operatorname{Var}_{\mathcal{T}^{{val}}}\left(\nabla_{\theta}\mathcal{L}^{{val}}\right)\right)$
(5)

where $\eta,\beta$ are the learning rate of the outer loop optimization,
$t_{j}$ is task $j$ for outer loop optimization for the network
$\theta_{i}^{\prime},\mathcal{L}$ is the loss function for outer loop
optimization. Note that the inner optimized network $f_{\theta_{i}^{\prime}}$
is used to update meta-initialization in outer loop with $t_{j}$ whereas it is
updated from meta-initialization with $t_{i}$ in ther inner loop. IML learn
invariant meta-initialization obtained from the discrepancy among different
training tasks with variance penalty regularizer.

Method | miniImageNet | CUB | SUN
---|---|---|---
5-way 1-shot | 5-way 5-shot | 5-way 1-shot | 5-way 5-shot | 5-way 1-shot | 5-way 5-shot
Meta-Learner LSTM | 24.99 | 29.79 | 36.23 | 44.39 | 30.99 | 44.86
MAML | 45.69 | 60.90 | 48.87 | 63.99 | 57.75 | 71.45
Reptile | 26.59 | 39.87 | 27.21 | 42.35 | 28.30 | 51.62
Matching Network | 47.63 | 56.28 | 53.06 | 62.19 | 55.02 | 62.57
Prototypical Network | 46.15 | 65.56 | 48.21 | 57.80 | 55.70 | 67.32
Relation Network | 47.64 | 63.65 | 52.76 | 64.71 | 58.29 | 72.15
Baseline | 23.84 | 32.09 | 25.14 | 35.35 | 27.44 | 34.54
Baseline++ | 30.15 | 41.19 | 32.48 | 42.43 | 35.56 | 44.42
IML | 48.35 | 67.21 | 54.18 | 65.85 | 59.24 | 74.18
Table 1: Average accuracy (%) comparison to state-of-the-arts with 95% confidence intervals on 5-way classification tasks under the in-distribution FSL setting. Best results are displayed in boldface. Method | miniImageNet$\rightarrow$ CUB | miniImageNet$\rightarrow$ SUN | CUB$\rightarrow$miniImageNet
---|---|---|---
5-way 1-shot | 5-way 5-shot | 5-way 1-shot | 5-way 5-shot | 5-way 1-shot | 5-way 5-shot
Meta-Learner LSTM | 23.77 | 30.58 | 25.52 | 32.14 | 22.58 | 28.18
MAML | 40.29 | 53.01 | 46.07 | 59.08 | 33.36 | 41.58
Reptile | 24.66 | 40.86 | 32.15 | 50.38 | 24.56 | 40.60
Matching Network | 38.34 | 47.64 | 39.58 | 53.20 | 26.23 | 32.90
Prototypical Network | 36.60 | 54.36 | 46.31 | 66.21 | 29.22 | 38.73
Relation Network | 39.33 | 50.64 | 44.55 | 61.45 | 28.64 | 38.01
Baseline | 24.16 | 32.73 | 25.49 | 37.15 | 22.98 | 28.41
Baseline++ | 29.40 | 40.48 | 30.44 | 41.71 | 23.41 | 25.82
IML | 41.27 | 57.34 | 50.42 | 69.15 | 34.26 | 44.17
Table 2: Average accuracy (%) comparison to state-of-the-arts with 95%
confidence intervals on 5-way classification tasks under the in-distribution
FSL setting. Best results are displayed in boldface.

##  3 Experiments

### Datasets.

In this paper, we address the few-shot classification problem under both in-
distribution and out-ofdistribution FSL settings. These settings are conducted
on three benchmark datasets: miniImageNet [23], Caltech- UCSD-Birds 200-2011
(CUB) [25], and SUN Attribute Database (SUN) [15].

### Baselines.

To evaluate the effectiveness of the proposed framework, we consider the
following representative meta learning methods on the few-shot image
classification task: MAML [5], Reptile [14], Matching Network [23],
Prototypical Network [20], Relation Network [21], Baseline and Baseline++ [3].

### Experimental Settings.

We conduct experiments on 5-way 1-shot and 5-way 5 -shot settings, there are
15 query samples per class in each task. We report the average accuracy (%)
and the corresponding $95\%$ confidence interval over the 2000 tasks randomly
sampled from novel classes. To fairly evaluate the original performance of
each method, we use the same 4-layer ConvNet [23] as the backbone for all
methods and do not adopt any data augmentation during training. All methods
are trained via SGD with Adam [10], and the initial learning rate is set to
$e^{-3}$. For each method, models are trained for 40,000 tasks at most, and
the best model on the validation classes is used to evaluate the final
reporting performance in the meta-test phase.

### Evaluation Using the In-Distribution Setting.

Table 1 shows the comparative results under the in-distribution FSL setting on
three benchmark datasets. It is observed that IML outperforms the original
MAML in all in-distribution FSL scenarios. For 1-shot and 5-shot on
miniImageNet $\rightarrow$ miniImageNet, IML achieves about $1\%$ higher
performance than Prototypical Network. However, IML achieves $5\%$ and $10\%$
higher performance for 1-shot and 5-shot on CUB $\rightarrow$ CUB, and $3\%$
and $6\%$ higher performance on SUN $\rightarrow$ SUN. As the latter two
scenarios are conducted on finegrained classification datasets, we attribute
the promising improvement to that the categories in these fine-grained
datasets share more local concepts than those in coarsegrained datasets, and
thus a more discriminative space can be rapidly learned with a few steps of
adaptation. Moreover, IML achieves the best performance among all baselines in
all in-distribution FSL scenarios, which shows that our approach can be
considered as a better baseline option under the in-distribution FSL setting.

### Evaluation Using the Out-of-Distribution Setting.

We also conduct out-of-distribution FSL experiments and report the comparative
results in Table 2. Compared to the results under the in-distribution setting,
it can be observed that all approaches suffer from a larger discrepancy
between the distributions of training and testing tasks, which results in a
performance decline in all scenarios. However, IML still outperforms the
original MAML in all out-of-distribution FSL scenarios, demonstrating that the
bilevel optimization strategy for adaptation and the learning of transferable
latent factors can be utilized to improve simple meta learning approaches.
Also, IML achieves all the best results, indicating that our approach can be
regarded as a promising baseline under the out-of-distribution setting.

##  4 Conclusion

In this paper, we consider the challenge of out-ofdistribution tasks faced by
few-shot learning. We propose Invariant Meta Learning (IML) for out-of-
distribution tasks, a general learning framework that jointly adjusts gradient
magnitudes and directions. Extensive experiments demonstrate that our approach
effectively improves the performance and generalization ability under both in-
distribution and out-of-distribution few-shot settings, and thus it can be
regarded as a better baseline.

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