$\displaystyle\nabla\times\nabla\times\int_{\mathbb{R}}{dt^{\prime}\int_{\partial\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dV^{\prime}}}$

thus, proving the equation (3).

###  C.3 Derivation of equation (3)

Using the mathematical identity
$\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t)=\nabla^{\prime
2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t)$, valid for
Green’s function
$G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t)$, one can
rewrite the left hand side integral in equation (3) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{\prime
2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

In Appendix B, we have derived two novel vector calculus identities.
Subtracting vector identity (B.107) from vector identity (B.108) yields:

$\int_{V}{\boldsymbol{\mathbf{P}}\nabla^{2}\psi
dV}=\int_{V}{\psi\nabla^{2}\boldsymbol{\mathbf{P}}dV}+\oint_{\partial
V}{\boldsymbol{\mathbf{P}}\left(\nabla\psi\cdot
d\boldsymbol{\mathbf{S}}\right)}-\oint_{\partial
V}{\psi\left(d\boldsymbol{\mathbf{S}}\cdot\nabla\right)\boldsymbol{\mathbf{P}}}$
(C.3.139)

Using vector identity (C.3.139) we can rewrite equation (C.3) as:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime
2}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dV^{\prime}}}+$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\oint_{\partial\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\left(\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\cdot
d\boldsymbol{\mathbf{S}}^{\prime}\right)}}-$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\oint_{\partial\mathbb{R}^{3}}{G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\left(d\boldsymbol{\mathbf{S}}^{\prime}\cdot\nabla^{\prime}\right)\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}}}$

Because surface $\partial\mathbb{R}^{3}$ is an infinite surface the magnitude
of position vector $\boldsymbol{\mathbf{r}}^{\prime}\in\partial\mathbb{R}^{3}$
has an infinite magnitude,
$\left|\boldsymbol{\mathbf{r}}^{\prime}\right|\rightarrow\infty$. In that
case, both right hand side surface integrals over surface
$\partial\mathbb{R}^{3}$ vanish in equation (C.3). Hence, equation (C.3)
becomes:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle=$
$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\nabla^{\prime
2}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{4\pi\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}dV^{\prime}}}$

The operator $\nabla^{\prime}$ does not affect vector
$\boldsymbol{\mathbf{v}}_{s}(t^{\prime})$ which is a function of variable
$t^{\prime}$. Thus, we can rewrite the Laplacian in the equation above as:

$\nabla^{\prime
2}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{4\pi\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\nabla^{\prime
2}\frac{1}{4\pi\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}=-\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)$
(C.3.142)

Inserting equation (C.3.142) into right hand side of equation (C.3) yields:

$\displaystyle\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\nabla^{2}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}=$
$\displaystyle-\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{\epsilon}\boldsymbol{\mathbf{v}}_{s}(t^{\prime})\delta\left(\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right)G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

which proves equation (3).

###  C.4 Derivation of equation (60)

Equation (60) can be derived from equation (67) by taking the divergence of
both sides of this equation to obtain:

$\nabla\cdot\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)+\nabla\cdot\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=\nabla\cdot\frac{q_{s}}{4\pi\epsilon}\frac{\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)}{\left|\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right|^{3}}=\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
(C.4.144)

To find $\nabla\cdot\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)$ we can
write operator $\nabla$ under the right hand side integral of equation (66)
and then apply identity $\nabla
G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t)=-\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t)$
to obtain:

$\nabla\cdot\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]\cdot\nabla^{\prime}G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$
(C.4.145)

From here, using standard vector identity
$\nabla\cdot(\psi\boldsymbol{\mathbf{P}})=\nabla\psi\cdot\boldsymbol{\mathbf{P}}+\psi\nabla\cdot\boldsymbol{\mathbf{P}}$
and divergence theorem it is obtained that:

$\displaystyle\nabla\cdot\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=$
$\displaystyle=$ $\displaystyle-\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\oint_{\partial\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})\cdot
d\boldsymbol{\mathbf{S}}^{\prime}}}$
$\displaystyle+\frac{1}{c^{2}}\frac{\partial}{\partial
t}\int_{\mathbb{R}}{dt^{\prime}\int_{\mathbb{R}^{3}}{\frac{q_{s}}{4\pi\epsilon}\left[\nabla^{\prime}\cdot\nabla^{\prime}\times\nabla^{\prime}\times\frac{\boldsymbol{\mathbf{v}}_{s}(t^{\prime})}{\left|\boldsymbol{\mathbf{r}}^{\prime}-\boldsymbol{\mathbf{r}}_{s}(t^{\prime})\right|}\right]G(\boldsymbol{\mathbf{r}},t;\boldsymbol{\mathbf{r}}^{\prime},t^{\prime})dV^{\prime}}}$

Clearly, the surface integral on the right hand side of equation (C.4)
vanishes as $\left|\boldsymbol{\mathbf{r}}^{\prime}\right|\rightarrow 0$.
Furthermore, because $\nabla\cdot\nabla\times\boldsymbol{\mathbf{P}}=0$ the
second right hand side term vanishes as well. Hence, we can write:

$\nabla\cdot\boldsymbol{\mathbf{K}}(\boldsymbol{\mathbf{r}},t)=0$ (C.4.147)

Inserting equation (C.4.147) into equation (C.4.144) yields:

$\nabla\cdot\boldsymbol{\mathbf{N}}(\boldsymbol{\mathbf{r}},t)=\frac{q_{s}}{\epsilon}\delta\left(\boldsymbol{\mathbf{r}}-\boldsymbol{\mathbf{r}}_{s}(t)\right)$
(C.4.148)

thus, proving the equation (60).

## References

  * [1] J.C. Maxwell, A treatise on electricity and magnetism, Vol I, Claredon Press, Oxford, 1873.
  * [2] J.C. Maxwell, A treatise on electricity and magnetism, Vol II, Claredon Press, Oxford, 1881.
  * [3] A. Liénard, Champ électrique et Magnétique produit par une charge électrique concentrée en un point et animée d’un mouvement quelconque, L’éclairage électrique 16 (1898), 16–21.
  * [4] E. Wiechert, Elektrodynamische elementargesetze, Ann. Phys. 4(1901), 667–689.
  * [5] H.A. Lorentz, La théorie électromagnétique de Maxwell et son application aux corps mouvants, Arch. Néerl. 25(1892), 363–552.
  * [6] O. Darringol, The Electron Theories of Larmor and Lorentz: A Comparative Study, Hist. Stud. Nat. Sci. 24(1994), 265–336.
  * [7] A. Einstein, On the electrodynamics of moving bodies, Ann. Phys. 17 (1905), 891–921.
  * [8] L. Page, Derivation of the fundamental relations of electrodynamics from those of electrostatics, Am. J. Sci., 34 (1912), 57–68.
  * [9] D. H. Frisch, L. Willets, Development of the Maxwell-Lorentz equations from special relativity and Gauss’s Law, Am. J. Phys. 24 (1956), 574–579.
  * [10] R. S. Elliott, Relativity and Electricity, IEEE Spectr. 3 (1966), 140–152.
  * [11] D.H. Kobe, Generalization of Coulomb’s law to Maxwell’s equations using special relativity, Am. J. Phys. 54 (1986), 631–636.
  * [12] P. Lorrain, D.R. Corson, F. Lorrain, Electromagnetic fields and waves: including electric circuits, third ed., W.H. Freeman and Company, New York, 1988, 291–293.
  * [13] J.H. Field, Derivation of the Lorentz force law, the magnetic field concept and the Faraday–Lenz and magnetic Gauss laws using an invariant formulation of the Lorentz transformation, Phys. Scr. 73 (2006), 639–651.
  * [14] A.K. Singal, A first principles derivation of the electromagnetic fields of a point charge in arbitrary motion, Am. J. Phys. 79 (2011), 1036–1041.
  * [15] H.A. Lorentz, Electromagnetic phenomena in a system moving with any velocity smaller than that of light, Proc. R. Neth. Acad. Arts Sci. 6 (1904), 809–831.
  * [16] E.J. Rothwell, M.J. Cloud, Electromagnetics, third ed., CRC Press, New York, 2018.
  * [17] J.D. Jackson, Classical Electrodynamics, Wiley, New York, 1975.
  * [18] M. Schwartz, Principles of Electrodynamics, Dover Publications, New York, 1987.
  * [19] N. Rosen, Field of a particle in uniform motion and uniform acceleration, Ann. Phys. 17 (1962), 269–275.
  * [20] P. Franklin, The meaning of rotation in the special relativity theory, Proc. Natl. Acad. Sci. 8 (1922), 265–268.
  * [21] W. Hauser, On the Fundamental Equations of Electromagnetism, Am. J. Phys. 38 (1970), 80–84.
  * [22] E. Kapuścik , Generalized Helmholtz theorem and gauge invariance of classical field theories, Lett. Nuovo Cimento. 42 (1985), 263–266.
  * [23] A. M. Davis, A generalized Helmholtz theorem for time-varying vector fields, Am. J. Phys. 74 (2006), 72–76.
  * [24] R. Heras, The Helmholtz theorem and retarded fields, Eur. J. Phys. 37 (2016), 72–76.
  * [25] H. Flanders, Differentiation under the integral sign, Am. Math. Mon. 80 (1973), 615–627.


# Position-dependent mass effects on a bilayer graphene catenoid bridge

J. E. G. Silva  Centro de Ciências e Tecnologia
Universidade Federal do Cariri
57072-270, Juazeiro do Norte, Ceará, Brazil<EMAIL_ADDRESS>J.
Furtado  Centro de Ciências e Tecnologia
Universidade Federal do Cariri
57072-270, Juazeiro do Norte, Ceará, Brazil<EMAIL_ADDRESS>A. C. A.
Ramos  Centro de Ciências e Tecnologia
Universidade Federal do Cariri
57072-270, Juazeiro do Norte, Ceará, Brazil<EMAIL_ADDRESS>
###### Abstract

We study the electronic properties of a position-dependent effective mass
electron on a bilayer graphene catenoid bridge. We propose a position-
dependent mass (PDM) as a function of both gaussian and mean curvature. The
hamiltonian exhibits parity and time-reversal steaming from the bridge
symmetry. The effective potential contains the da Costa, centrifugal and PDM
terms which are concentrated around the catenoid bridge. For zero angular
momentum states, the PDM term provides a transition between a reflectionless
to a double-well potential. As a result, the bound states undergo a transition
from a single state around the bridge throat into two states each one located
at rings around the bridge. Above some critical value of the PDM coupling
constant, the degeneracy is restored due to double-well tunneling resonance.

##  I Introduction

Two dimensional structures, as the graphene geim ; novoselov ; katsnelson ,
nanotubes nanotube  and the phosphorene phosphorene  open a new venue to study
the electron properties at low dimensional physics. The geometry of the
graphene layer plays a pivotal role on the electronic structure. The curvature
at the tip of a conical layer produces topological phase furtado , whereas
helical strips induces chiral properties dandoloff1 ; atanasovhelicoid ;
atanasov . Fluctuations of the geometry produces the so-called pseudomagnetic
fields ribbons , whose effects can be seen at ripples contijo  and corrugated
layers corrugated  .

The electron Hamiltonian on the surface can be obtained from the $3D$
Hamiltonain by considering a small surface width $\epsilon$, writing the
Hamiltonian in the tangent and the normal coordinates and then, taking the
limit $\epsilon\rightarrow 0$ hjensen . Starting with the $3D$ Schrödinger
Hamiltonian and appying this thin-layer squeezing method, one obtains a
geometric potential, known as the da Costa potential costa ; costa1 ;
matsutani ; luiz . The geometric da Costa potential depends on the squared of
the gaussian and the mean curvatures and yields to an attractive potential.
This method can also be extended to include external fields ferrari , spin in
a Pauli equation wang  and the Dirac equation on surfaces BJ .

The geometry of the graphene layer can be used to develop new electronic
devices. In Ref. wormhole ; picak  a bridge connecting a bilayer graphene was
devised using a nanotube. In order to obtain a smooth bridge, the
Ref.Dandoloff  proposed a catenoid surface to describe the bilayer and the
bridge using only one surface. This can be achieved due to the catenoid
curvature which is concentrated around the bridge and vanishes asymptotically
spivak . The catenoid is a minimal surface, which is known to provide a stable
graphitic structure terrones .

In Ref.euclides , we explored the effects of the geometry and of external
electric and magnetic fields upon the graphene catenoid bridge. The da Costa
potential provides a reflectionless attractive potential, whereas the symmetry
with respect to the $z$ axis yields to a centrifugal repulsive term euclides .
The geometric potential exhibits a parity and time reversal symmetry which
holds under the action of the magnetic field but it is broken by the external
electric field euclides . The magnetic field produces a double-well potential
which leads to bound states located at symmetric rings around the catendoid
throat. The electric field creates a difference between the asymptotic values
of the effective potential on the upper to the lower layer. This effect
suggests that the catenoid bilayer bridge could be used as a diode euclides .

Besides the geometric potential, it is expected that the surface curvature may
also produce position-dependent mass effects. Indeed, the curvature breaks the
homogeneity of the lattice which compose the layer, thus modifying the
electron effective mass. The PDM can be produced by $p-n$ junctions driven by
curvaturepnbilayer , as well as phonon interactionssinner . Although the
position-dependent mass Hamiltonians in flat surfaces are widely studied pdm1
; pdm2 ; pdm3 ; pdm4 , only recently an extension of the da Costa method
including position-dependent mass effects on curved surfaces was proposed
moraes . The inclusion of an effective mass of form $m^{*}\propto
d^{-\alpha}$, where $d$ is the nanotube diameter, on a corrugated nanotube
lead to significant modifications of the transport properties moraes . In a
cylinder, the gaussian curvature vanishes whereas the mean curvature is a
constant $H=d^{-1}$. Thus, the position-dependent mass considered in
Ref.moraes  is proportional to the a power of the corrugated mean curvature.

In this work we study the effects of the position-dependent mass upon the
electron on a catenoid bridge. In section II, we propose an isotropic
position-dependent effective mass as a function of the gaussian and the mean
curvatures. Then, we analyze qualitatively the features of the effective
potential, such as its behaviour with respect to the parity, time-reversal
symmetry and hermiticity. In section III, we obtain the bound states and the
corresponding energy spectrum. Final remarks and perspectives are outlined in
section IV.

##  II Position-dependent mass electron on a catenoid surface

In this section we introduce the geometry and the dynamics of the electron on
the doublelayer catenoid bridge considering the effects of a curvature-
dependent mass. As shown in fig.1, the double layer graphene bridge is
realized as a smooth minimal surface (least area) joining the two planes. Near
the bridge throat, depicted in fig.1, the symmetry about the $z$ axis and the
minimal radius $R$ are shown.

After squeezing the electron wave-function on a surface, the spinless
stationary Schrödinger equation has the form

$-\frac{\hbar^{2}}{2m^{*}}\nabla^{2}\Psi+V_{dc}\Psi=E\Psi,$ (1)

where
$\nabla^{2}\Psi=\frac{1}{\sqrt{g}}\partial_{a}(\sqrt{g}g^{ab}\partial_{b}\Psi)$
is the Laplacian operator on the surface, $g^{ab}$ is the induced metric of
the surface and $V_{dC}=-\frac{\hbar^{2}}{2m^{*}}(H^{2}-K)$ is a potential
induced by the surface curvature, known as the da Costa potential costa . The
geometric potential depends both on the the mean curvature $H$ and the
Gaussian curvature $K$ costa .

Recently, the electronic properties of the electron on the bilayer catenoid
bridge was investigated assuming a constant effective mass $m^{*}$ euclides .
Since the effective mass is also modified by the break of the lattice
homogeneity driven by the curvature, we consider the modified position-
dependent mass Schrödinger equation in the form moraes

$\displaystyle(\mathcal{K}-V_{dC})\Psi=E\Psi,$ (2)

where the position-dependent mass kinetic operator $\mathcal{K}$ is defined as

$\mathcal{K}\Psi=-\frac{\hbar^{2}}{2}\left[\frac{1}{m^{*}}\nabla^{2}\Psi+\frac{1}{3}\nabla^{2}\left(\frac{1}{m^{*}}\right)\Psi+\partial^{j}\left(\frac{1}{m^{*}}\right)(\partial_{i}\Psi)\right],$
(3)

and $m^{*}=m^{*}(x)$. Note that for a constant effective mass $m^{*}$ the
Eq.(1) is obtained.

Adopting a coordinate system formed from the the meridian
$u=u(z)=R\sinh\left(z/R\right)$ and the parallel $\phi$, where
$u\in(-\infty,\infty)$ and the parallel $\phi\in[0,2\pi)$, a point on the
catenoid surface can be written as

$\vec{r}=\sqrt{R^{2}+u^{2}}(\cos\phi\hat{i}+\sin\phi\hat{j})+R\sinh^{-1}(u/R)\hat{k},$
(4)

where $R$ the radius of the catenoid bridge, as shown in figure 1.

Figure 1: Catenoid surface and coordinate system

In this coordinate system, the induced metric on the catenoid is $g_{uu}=1$
and $g_{\phi\phi}=R^{2}+u^{2}$, and the da Costa potential

$V_{dC}=-\frac{\hbar^{2}}{2m^{*}}\frac{R^{2}}{(R^{2}+u^{2})^{2}}.$ (5)

It is worthwhile to mention that the da Costa potential $V_{dc}$ exhibits a
parity-symmetrical potential well with respect to $u=0$. In addition, the da
Costa potential vanishes asymptotically, reflecting the asymptotic flat
geometry of the catenoid. Note that the curvature leads to an attractive
potential which tends to trap the electron around the origin.

The axial symmetry leads to the periodic behavior of the wave function in the
form

$\Psi(u,\phi)=\Phi(u)e^{i\nu\phi},$ (6)

where $\nu$ is the orbital quantum number. Substituting the Eq.(6) into Eq.(2)
yields to

$\displaystyle-$
$\displaystyle\frac{\hbar^{2}}{2m^{*}}\left[\partial_{u}^{2}\Phi+\frac{u}{R^{2}+u^{2}}\partial_{u}\Phi-\frac{\nu^{2}}{R^{2}+u^{2}}\Phi\right]$
(7)
$\displaystyle-\frac{\hbar^{2}}{6}\left[\partial_{u}^{2}\left(\frac{1}{m^{*}}\right)+\frac{u}{R^{2}+u^{2}}\partial_{u}\left(\frac{1}{m^{*}}\right)+\frac{1}{R^{2}+u^{2}}\partial_{\phi}^{2}\left(\frac{1}{m^{*}}\right)\right]\Phi$
$\displaystyle-\frac{\hbar^{2}}{2}\left[\partial_{u}\left(\frac{1}{m^{*}}\right)(\partial_{u}\Phi)+\frac{i\nu}{R^{2}+u^{2}}\partial_{\phi}\left(\frac{1}{m^{*}}\Phi\right)\right]$
$\displaystyle-\frac{\hbar^{2}}{2m^{*}}\frac{R^{2}}{(R^{2}+u^{2})^{2}}\Phi=E\Phi.$

The Eq.(7) depends on the position-dependence of the effective mass function
$m^{*}$. In general, the effective mass may be anisotropic, as a result of a
non-symmetric energy band moraes ; anisotropicmass . In Ref.moraes , the
authors explored the electronic properties of position-dependent Schrödinger
equation (2) on a nanotube. For that purpose, they considered $m^{*}\propto
d^{-\alpha}$, where $d$ is the nanotube diameter moraes .

Here we propose that the curvature breaks the homogeneity of the system, and
the consequence of such breaking leads to a modification in the effective mass
of the particle moving along the surface of the system. Therefore, is natural
to assume that the effective mass is related to the surface curvature. The
surface curvature information is encoded in the second fundamental form
$b_{ij}$ which satisfies spivak

$\frac{\partial^{2}\vec{r}}{\partial q^{1}\partial
q^{j}}=b_{ij}\hat{\mathbf{n}}+\Gamma^{k}_{ij}\frac{\partial\vec{r}}{\partial
q^{k}},$ (8)

where $\hat{\mathbf{n}}$ in the unit normal vector to the surface and
$\Gamma^{k}_{ij}$ are the Christoffel symbols. Since the catenoid geometry is
isotropic, we adopt a position-dependent mass in the form,

$m^{*}(x)=m^{*}f(H,K),$ (9)

being $m^{*}$ the effective mass of the electron in a flat graphene sheet and
$f(H,K)$ is a particular function of the scalar invariants of the second
fundamental form, the mean curvature $H=Tr(b_{ij})/2$ and gaussian curvature
$K=det(b_{ij})$. The presence of the both curvatures $H$ and $K$ not only
leads to the position dependence of the effective mass $m^{*}$ but also
guarantees that intrinsic and extrinsic deformations of the surface modifies
the electron effective mass. It is worthwhile to mention that the position
dependent mass considered in Ref.moraes  satisfies Eq.(9) since the nanotube
has $H=d^{-1}$ and $K=0$, and so $m^{*}(x)=m^{*}H^{\alpha}$.

In this work we consider an isotropic ansatz of form $m^{*}(x)=M(u)$ in the
form

$M(u)=m^{*}[1+\lambda(H^{2}-K)^{\alpha}],$ (10)

being $\lambda$ a parameter given by the system, called coupling factor. The
function $f(H,K)=1+\lambda(H^{2}-K)^{\alpha}$ is dimensionless and it is build
from the invariant term $(H^{2}-K)$, known in the elasticity theory as the
Wilmore energy wilmore ; Helfrich . For $\alpha=1$, the coupling factor
$\lambda$ has dimension of $length^{2}$ whereas the curvature term $(H^{2}-K)$
has dimension of $length^{-2}$. Note that for the case of the catenoid (10)
gives us

$M(u)=m^{*}r(u),$ (11)

where the function $r(u)$ is defined by

$r(u)=1+\frac{\lambda R^{2}}{(u^{2}+R^{2})^{2}}.$ (12)

Note that $M(u)\rightarrow m^{*}$ when $u\rightarrow\pm\infty$, in agreement
with the fact that the catenoid is asymptotically flat when
$u\rightarrow\infty$. When $u\rightarrow 0$ then $M(u)\rightarrow
m^{*}(1+\lambda/R^{2})$. Thus, $\sqrt{|\lambda|}$ measures a length of
influence of the position-dependent mass effects.

The coupling constant $\lambda$ can assume positive or negative values. For
$\lambda>0$, the effective mass is augmented near the bridge throat and
assumes the flat value $m^{*}$ asymptotically. For $\lambda<0$, the curvature
reduces the effective electron mass near the bridge. In order to prevent a
vanishing effective mass, the coupling constant $\lambda$ should satisfy
$\sqrt{-\lambda}\neq R$. For $\sqrt{-\lambda}<R$, the effective mass is
everywhere positive. On the other hand, for $\sqrt{-\lambda}>R$ the effective
mass is negative around the bridge throat interval $-R<u<R$ and positive for
$|u|>R$. The effects of the coupling constant $\lambda$ on the bound states
will be explore in details in the next section.

Using the curvature-dependent mass ansatz (11), the stationary Schrödinger
equation along the meridian (7) reads

$\displaystyle-$
$\displaystyle\frac{h^{2}}{2m^{*}}\left[\partial_{u}^{2}\Phi+\frac{u}{R^{2}+u^{2}}\partial_{u}\Phi-\frac{\nu^{2}}{R^{2}+u^{2}}\Phi\right]$
(13)
$\displaystyle-\frac{\hbar^{2}}{6}\left[\partial_{u}^{2}\left(\frac{1}{m^{*}}\right)+\frac{u}{R^{2}+u^{2}}\partial_{u}\left(\frac{1}{m^{*}}\right)\right]\Phi$
$\displaystyle-\frac{\hbar^{2}}{2}\partial_{u}\left(\frac{1}{m^{*}}\right)\partial_{u}\Phi-\frac{\hbar^{2}}{2m^{*}}\frac{R^{2}}{(R^{2}+u^{2})^{2}}\Phi=E\Phi$

Note that the Eq.13 exhibits parity and time-reversal invariance, as a result
of the catenoid geometric symmetries, as we can see from the acting of parity
$\mathcal{P}\hat{u}(z)\mathcal{P}=\hat{u}(-z)=-\hat{u}(z)$ and time reversal
$\mathcal{T}\hat{u}(z)\mathcal{T}=\hat{u}(z)$ operators upon $u(z)$.
Nonetheless, the first order derivative terms renders the Hamiltonian non-
Hermitian with respect to the meridian momentum
$\hat{P}_{u}:=-i\hbar\partial_{u}$. The non-Hermiticity of the free electron
Hamiltonian is not a problem, since the space-time reflection symmetry is
preserved, the spectrum of the eigenvalues of the Hamiltonian is completely
real Bender ; Bender2 . Besides, there is an Hermitean equivalent Hamiltonian
that can be achieved by a simple changing of variables. Considering the change
on the wave function

$\Phi(u)=\frac{1}{2}e^{\log\left(R^{4}+\lambda
R^{2}+2R^{2}u^{2}+u^{4}\right)-\frac{5}{2}\left(\log\left(R^{2}+u^{2}\right)\right)}y(u),$
(14)

leads to an one dimensional Hermitian Schrödinger-like equation

$-\frac{\hbar^{2}}{2M(u)}\frac{d^{2}y}{du^{2}}+V_{eff}(u)y=Ey,$ (15)

whose effective potential is given by

$V_{eff}(u)=V_{c}(u)+V_{G}(u)+V_{\lambda}(u),$ (16)

where $V_{c}(u)$ is the centrifugal potential, $V_{G}(u)$ is the geometric
potential and $V_{\lambda}(u)$ is the coupling potential, that are given by:

$V_{c}(u)=\frac{\hbar^{2}}{2M(u)}\left[\frac{\nu^{2}}{(R^{2}+u^{2})}\right]$
(17)
$V_{G}(u)=-\frac{\hbar^{2}}{2M(u)}\left[\frac{(2R^{2}+u^{2})}{4(R^{2}+u^{2})^{2}}\right]$
(18) $V_{\lambda}(u)=\frac{\hbar^{2}}{2M(u)}\left\\{\frac{2\lambda
R^{2}[\lambda R^{2}(R^{2}-2u^{2})+(R^{2}-4u^{2})(R^{2}+u^{2})^{2}]}{3[\lambda
R^{2}(R^{2}+u^{2})+(R^{2}+u^{2})^{3}]^{2}}\right\\}.$ (19)

Although the equation (15) resembles a Schrödinger equation, the operator on
the left side $\hat{\mathcal{H}}y(u)=r(u)Ey(u)$ is not properly the
Hamiltonian, since the ”eigenvalue” depends on $u$. By performing another wave
function redefinition as $y(u)=\frac{s_{0}}{\sqrt{r(u)}}\chi(u)$, being
$s_{0}$ and arbitrary constant, we obtain the more familiar PDM Schrödinger
equation pdm1 ; pdm2 ; pdm3 ; pdm4

$\displaystyle-\frac{\hbar^{2}}{2}\frac{d}{du}\left[\frac{1}{M(u)}\frac{d\chi}{du}\right]+\widetilde{V}_{eff}(u)\chi=E\chi,$
(20)

and the effective potential is rewritten as

$\displaystyle\widetilde{V}_{eff}(u)=V_{c}(u)+V_{G}(u)+\widetilde{V}_{\lambda}(u),$
(21)

since, due to the redefinition of the wave function $y(u)$, the coupling
potential is rewritten as

$\displaystyle\widetilde{V}_{\lambda}(u)=V_{\lambda}(u)+\frac{\hbar^{2}}{2M(u)}\left(-\frac{1}{2}\frac{r^{\prime\prime}(u)}{r(u)}+\frac{3}{4}\left(\frac{r^{\prime}(u)}{r(u)}\right)^{2}\right).$
(22)

Thus, the electron’s dynamics on a catenoid with Hamiltonian (13) is Hermitian
equivalent to an electron under the action of the effective potential in (21).
Moreover, the effects of curvature, PDM and angular momentum are encoded in
the effective potential. The equivalence between a $\mathcal{P}$ $\mathcal{T}$
symmetric non-hermitean Hamiltonian and a Hermitean Hamiltonian has attracted
much attention in last years Jones ; Andrianov1 ; Andrianov2 . Moreover, since
(20) satisfies the continuity equation $\frac{\partial|\chi|^{2}}{\partial
t}+\frac{\partial J_{u}}{\partial u}=0$, the square of the wave function
$|\chi|^{2}$ represents a true probability density.

Asymptocially, i.e. for $u\rightarrow\pm\infty$, the effective potential in
Eq. 21 vanishes and the PDM function reaches its asymptotic value
$M(U)\rightarrow m^{*}$, so that the respective asymptotic solution of Eq.(20)
are of form

$\chi(u)\approx A\cos(ku+\varphi),$ (23)

where $k^{2}=\frac{2m^{*}\varepsilon}{\hbar^{2}}$. Therefore, far from the
bridge throat where the curvature effects vanish the electron behaves as free
states. In the next section, we analyze the effects of the curvature and PDM
on bound states near the bridge throat.

###  II.1 Qualitative analysis

Before obtain the bound states and their respective spectra, let us discuss
some qualitative features of the effective potential.

In the fig.2, we show the effective potential given by the eq.22, for $R=30$
Å, and $\nu=0$, for some values of $\lambda$. The eq.11 shows that the mass
depends on coupling factor, $\lambda$, and that in addition, $\lambda$ can
assume negative values, but not any negative value, there is a minimum limit
value, which we call a critical coupling factor, $\lambda_{c}$, from which the
mass becomes negative, this critical value is given by $\lambda_{c}=-R^{2}$.
For $R=30$ Å, the critical coupling factor é $\lambda_{c}=-900$ Å2. In the
insert of the figure, we observe a deep well for $\lambda=-800$ Å2, and it
becomes deeper when $\lambda$ assumes values close to the critical value,
$\lambda=\lambda_{c}$, in this case the coupling effect suppresses the
geometric effect. The effective potential decreases its depth when the
$\lambda$, having negative values, is increased until it reaches zero, as
shown in the figure. A single well is observed for $\lambda=0$, this is the
case when the effective mass is the same over the entire catenoid,
$M(u)=m^{*}=0.03m_{0}$. euclides  When $\lambda$ assumes positive values, the
effective potential is drastically altered, generating two wells symmetrical
in relation to the origin of the catenoid, in addition the double well becomes
softer when lambda takes on higher values, this behavior is shown in the
results taken for for $\lambda=10^{3}$ and $10^{5}$ Å2. Under these conditions
the coupling is so intense that the geometric effect is suppressed. Therefore,
the geometric effect is suppressed by the coupling effect in two limits, when
$\lambda$ tends to $\lambda_{c}$ and when $\lambda$ tends to infinity.

Figure 2: The effective potential for $R=30\AA$ and $\nu=0$ for some values of
$\lambda$, obtained by the eq.21. Figure 3: The effective potential for
$R=30\AA$ and $\nu=1$ for some values of $\lambda$, obtained by the eq.21.

The effective potential for $R=30\AA$ and $\nu=1$ is shown in the fig.3. In
this figure, in addition to the competition between the geometric and coupling
potentials, discussed in fig.2, there is the centrifugal potential that
significantly modifies the effective potential. When comparing the figures
inserted in figs.2 and 3, we observe that the centrifugal effect is suppressed
for $\lambda=-800$Å2, because the effective potentials have practically no
changes. However, when $\lambda$, having negatives values, approaches zero,
the centrifugal term becomes more relevant than the effect of coupling and a
barrier, instead of a well, appears at the origin of the catenoid, as can be
seen in fig.3 for $\lambda=-100$ Å2. For $\lambda=0$ the effective mass is
constant, $M(u)=m^{*}$, and the effective potential takes the form of a
barrier euclides . We also observed that when $\lambda$ assumes large positive
values, the coupling effect changes the effective potential to form a double
well symmetrical in relation to the origin of the catenoid, suppressing the
geometric and centrifugal effects. This behavior is shown in the fig.3 for
$\lambda=10^{5}$ Å2, the same is observed in fig.2 for the same value of
$\lambda$. Here, it is worth noting that both the geometric effect and the
centrifugal effect are suppressed by the two-limit coupling effect, when
lambda tends to $\lambda_{c}$ and when lambda tends to infinity, as discussed
in the fig.2.

Figure 4: The effective potential for $R=70\AA$ and $\nu=0$ for some values of
$\lambda$, obtained by the eq.21.

The effective potential for $\nu=0$ and $R=70\AA$ is shown in the fig.4, for
some values of $\lambda$. The increase in the radius of the catenoid throat
increases the domain of the lambda values, considering that the minimum value
of the critical lambda decreases to $-4900\AA^{2}$. The behavior of the
effective potential shown in fig.4 is similar to that of fig.2, that is, the
potential has a deep well in the origin of the catenoid for $\lambda$ values
close to the critical value. The depth of the well decreases when $\lambda$,
having negative values, has its value increased to zero and finally when
$\lambda$ increases, having its positive values, a double well symmetrical in
relation to the origin of the catenoid appears, and the increase in $\lambda$
decreases the depth of the double well. In both cases the centrifugal
potential is not present ($\nu=0$), so the geometric potential competes with
the coupling potential. Although qualitatively the results presented in fig.4
and 2 are similar, the radius of the catenoid throat is different. In this
condition, increasing the radius value, $R$, decreases the geometric
confinement on the effective potential, as studied in our recent work, see
ref. euclides , as the critical lambda, $\lambda_{c}$, depends on the radius,
the increase in the radius also causes the coupling potential to decrease its
influence on the effective potential, but more smoothly, this is more evident
when we compare the effective potentials with $\lambda$ values close to their
respective critical lambdas, see the effective potential for
$\lambda=-800\AA^{2}$ in the figure inserted in fig.2, and for
$\lambda=-4000\AA^{2}$ in the figure inserted in fig.4.

Figure 5: The effective potential for $R=70\AA$ and $\nu=1$ for some values of
$\lambda$, obtained by the eq.21.

The effective potential for $\nu=1$ and $R=70\AA$ is shown in the fig.5. This
effective potential presents a qualitative behavior similar to the effective
potential shown in fig.3. In both cases, the centrifugal potential is present
($\nu=1$), and the results show that the increase in the radius, $R$,
decreased both the geometric effect and the coupling effect on the effective
potential, see the effective potential for $\lambda=-800\AA^{2}$ in the figure
inserted in fig.3, and for $\lambda=-4000\AA^{2}$ in the figure inserted in
fig.5.

##  III Bound states

In the previous section, we discussed the effective potential generated by the
surface of a catenoid of radius $R$, made of graphene, in which the effective
mass of the electron depends on a coupling factor $\lambda$. In this section
we will look at the electronic states accessible in these potentials.

For this, we solve numerically the eq.20, using the finite difference method
Ramos11 , for the effective potential given by the eq.21, for some values of
$R$ and $\lambda$. In the calculations, we use $m^{*}=0.03m_{0}$, which is the
effective mass the electron on a single layer graphene sheet and $m_{0}$ is
the resting mass of the electron.

Figure 6: The bound states and their probability densities for a catenoid with
radius $R=30\AA$ and $\nu=0$. The solid black line represents the effective
potential for: a) $\lambda=-850\AA^{2}$, c) $\lambda=0\AA^{2}$, e)
$\lambda=10^{3}\AA^{2}$ and g) $\lambda=10^{5}\AA^{2}$. The red dashed and
blue dotted lines correspond to the first and second bound states and their
probability densities, respectively.

The fig.6 shows the bound states for $R=30\AA$ and $\nu=0$ for four values of
$\lambda$. We observe that the electronic state more confined is given by the
effective potential for $\lambda=-850$ Å2 shown in the fig.6 a), the energy of
this state is -26,020.21 meV and the probability density function shows that
the electron has high probability of being found close to the origin of the
catenoid, see the fig.6 b). Making use of the angular symmetry of the catenoid
and taking the width of the half height of the probability density function,
which is $\Delta u=8.6$ Å, we can visualize a probability cloud around the
origin of the catenoid in the form of a ring, which we can call a probability
ring. From what we see, this state is very localized, this is because the
coupling factor chosen is very close to $\lambda_{c}$, in this situation the
electron has a very small effective mass.

The probability ring is wider, $\Delta u=100.8$ Å, for $\lambda=0$ Å2, so in
this condition the electron is less confined, as shown by the value of the
energy level of the bound state which is -18.63 meV, as shown in fig.6 c) and
d).

Two confined energy states appear for $\lambda=10^{3}$ Å2, the first state is
-11.70 meV and the second is -0.24 meV, see fig.6 e). These states are not
Gaussian functions, but states, called hybrids, that arise from the mixture of
the states of the two wells symmetrical in relation to the origin of the
catenoid Studart . The first state (dashed red line) presents a probability
density function in the form of two Gaussians, practically overlapping, with
their maximum separated from $\Delta u=64.8$ Å, one located at $u=-32.4$ Å,
and the other located at $u=32.4$ Å, see the fig.6 f.1). The cloud of
probability associated with these states are in the form of two rings very
close symmetrical in relation to the origin of the catenoid.

The second state (dotted blue line) presents also a probability density
function in the form of two Gaussians, practically overlapping, but with their
maximum separated from $\Delta u=211.2$ Å, one located at $u=-105.6$ Å, and
the other located at $u=105.6$ Å, see the fig.6 f.2). The cloud of probability
associated with these states are also in the form of two rings very close
symmetrical in relation to the origin of the catenoid, but wider which
represents that this state is less confined in relation to the first.

Now, for $\lambda=10^{5}$ Å2 these two hybrid states are practically
degenerate, as they have almost the same energy value, $-6.19$ and $-6.17$
meV, see fig.6 g), the probability density functions of the first hybrid state
(red dotted line) and the second hybrid state (blue dotted line) are shown in
fig.6 h). The probability density function of each of these states takes the
form of two Gaussians, one of the maximums being in $u=-90$ Å, and the other
is in $u=90$ Å. Again, taking into account the angular symmetry of the
catenoid and the probability density function of the first state (red dashed
line), fig.6 h), we can visualize a probability cloud in the form of two rings
located symmetrically in relation to the origin of the catenoid, with the
width of each of these rings being $\Delta u=124.9$ Å. This probability
density function expresses the fact that that an electron initially located in
one of the wells can pass from one side to the other of the catenoid, as it
presents a resonant tunneling. The same discussion holds for the second state
as it has the same probability density function (blue dotted line) Studart .

It is worth mentioning that the effective potential for $\lambda=10^{6}$ Å2
the energy of the bound states are -2.26 and -2.26 meV. This reflects what was
said in the previous section, the effective potential becomes more shallow as
the coupling factor increases, $\lambda$, consequently the energy of the
connected states gets smaller and smaller to the point of ceasing to exist. In
these conditions the effective mass of the electron is so large that the
effective potential can no longer confine it, so an electron coming from
infinity, on the surface of the catenoid, passes through the throat of the
catenoid without realizing its existence.

Figure 7: The bound states and their probability densities for a catenoid with
radius $R=30\AA$ and $\nu=1$. The solid black line represents the effective
potential for: a) $\lambda=-850\AA^{2}$, c) $\lambda=-600\AA^{2}$, e)
$\lambda=1.5\times 10^{4}\AA^{2}$ and g) $\lambda=10^{5}\AA^{2}$. The dashed
red and blue dotted lines correspond to the first and second bound states and
their probability densities, respectively.

The fig.7 shows the bound states for $R=30\AA$ for four values of $\lambda$,
however, we take in account the centrifugal potential, $\nu=1$. We observe
that the states become less localized when compared with the results, see the
fig.6, where the centrifugal potential is absent. When looking at figs.6 a)
and 7 a), we do not observe the influence of the centrifugal potential on the
effective potential, because apparently the potentials are very similar, and
even the probability density functions have, approximately, the same width
($\Delta u=9.2$ Å), however, the energy value of the bound state is changed to
-24,206.83 meV.

The fig.7 c) shows the energy of the bound state, for $\lambda=-600$ Å2, which
is -123.70 meV, and the probability ring, located at the origin, has a width
of $\Delta u=28$ Å, as shown the fig.7 d). We observed that for values of
$\lambda$ closer to $\lambda_{c}$, the electronic state is more confined.

No bound state is observed for $\lambda=0$ Å2, because for $\nu=1$, the
effective potential is a barrier and not a well. This barrier is shaped like a
Gaussian, whose height is 71.43 meV and its width is 35.71 meV euclides .

Due to the presence of the orbital angular momentum, $\nu=1$, the linked
states were only obtained for values of $\lambda$ greater than $10^{4}$ Å2,
then we calculate the effective potential for $\lambda=1.5\times 10^{4}$, for
this configuration, two hybrid states are observed as shown in the fig. 7 e)
and f), the energy of the first (dashed red line) and second (dotted blue
line) states are are -0.72 meV and -0.33 meV, respectively. The probability
cloud, of the first state (dashed red line), is in the form of two rings, one
located at $u=-63.6$ Å and the other located at $u=63.6$ Å, each ring having a
width of $\Delta u=134.8$ Å. While the second (dotted blue line), the
probability rings are located in $u=-66.0$ Å and $u=66.0$ Å, and the width of
each one is $\Delta u=147.6$ Å.

Two hybrid states also appear for $\lambda=10^{5}$, fig.7 g) and h), the
energy of the first (dashed red line) and second (dotted blue line) states are
-1.69 meV and -1.66 meV, respectively. Their probability rings are practically
coincident and they are located at $u=-92.4$ Å and $u=92.4$ Å, with the width
of the rings being $\Delta u=150$ Å. These states are less confined compared
to the states shown, in figs. 6 g) and h), where the centrifugal potential is
absent, $\nu=0$.

Figure 8: The bound states and their probability densities for a catenoid with
radius $R=70\AA$ and $\nu=0$. The solid black line represents the effective
potential for: a) $\lambda=-850\AA^{2}$, c) $\lambda=0\AA^{2}$, e)
$\lambda=10^{3}\AA^{2}$ and g) $\lambda=10^{5}\AA^{2}$. The dashed red and
blue dotted lines correspond to the first and second bound states and their
probability densities, respectively.

The fig.8 shows the bound states and their probability density functions for
$\nu=0$ and $R=70$ Å, for four values of the $\lambda$. According to the
results shown in fig.8, the increase in the radius of the catenoid decreases
and even remove the bound state, as is the case for $\lambda=10^{3}$ Å2. So,
increasing the radius of the catenoid decreases the effect of the geometric
potential on the effective potential of the system. For example, for
$\lambda=-850$ Å2, the energy of the bound state is -5.14 meV, and the width
of the probability density function is $\Delta u=168.4$ Å.

For $\lambda=0$ Å2 the energy of the bound state is -3.42 meV, and the width
of the probability density function is $\Delta u=236.8$ Å. Now for
$\lambda=10^{3}$ Å2, one of the states seen in the fig.6 e) is removed, and
the remaining one has energy of -2.63 meV, with the width of the bound state
being $\Delta u=312.0$ Å, see the fig.8 f).

The same hybrid states that appear for $R=30$ Å and $\lambda=10^{5}$ Å2, for
both $\nu=0$ and $\nu=1$, see the figs.6 and 7, also appear for $R=70$ Å, as
shown in fig.8 g) and h), however they are less localized. The energy of the
first state (dashed red line) is -1.97 meV, and of the second (dotted blue
line) is -1.87 meV. Here we notice a slight breakdown of degeneracy, although
the probability density functions, which are shown in the fig.8 h), are
overlapping. From the probability density functions we can visualize two
probability rings, one located at $u=-140$ Åand the other at $u=140$ Å, being
the width of each of these rings $\Delta u=219.2$ Å. These hybrid states
become even less localized to take into account the orbital angular momentum,
$\nu=1$, for $R=70$ Å, as shown in the fig.9 g) and h), in which the energy of
the first state (dashed red line) is -0.19 meV and the second state (dotted
blue line) is -0.13 meV. The probability cloud of the first state (dashed red
line) is in the form of two rings, one located at $u=-153.6$ Å and the other
located at $u=153.6$ Å, each having a width of $\Delta u=299.2$ Å. For the
second state (dotted blue line), the cloud of probability is also shaped like
two rings, one located at $u=-158.4$ Å and the other located at $u=158.4$ Å,
each having a width of $\Delta u=313.6$ Å.

Figure 9: The bound states and their probability densities for a catenoid with
radius $R=70\AA$ and $\nu=1$. The solid black line represents the effective
potential for: a) $\lambda=-4850\AA^{2}$, c) $\lambda=-3000\AA^{2}$, e)
$\lambda=10^{5}\AA^{2}$ and g) $\lambda=10^{6}\AA^{2}$. The dashed red and
blue dotted lines correspond to the first and second bound states and their
probability densities, respectively.

No state linked to $\lambda=-850$ Å2 is found when in addition to increasing
the radius of the catenoid to 70 Å, we take into account the centrifugal
potential, $\nu=1$.

The energy of the bound state for $\nu=1$, $R=70$ Å, and $\lambda=-4850$Å2 is
-152,812.01 meV, and the probability density function has a width equals to
$\Delta u=9.2$ Å. Both the energy of the bound state and its probability
density function are shown in the figures 9 a) and b). As we can see, this
state is strongly linked because the value of $\lambda$ used is close to
$\lambda_{c}=-4900$ Å2. The absence of the centrifugal potential ($\nu=0$)
alters the energy of the bound state, to -154,638.10 meV, even for a coupling
potential so close to the critical $\lambda_{c}$.

Increasing $\lambda$ to -3000 Å2 the energy of the bound state is altered to
-6.49 meV, as well as, its probability density function, the width is $\Delta
u=76$Å, as shown the figs.9 c) and d).

Finally, for $\lambda=10^{6}$ Å2, the energy of the first state (dashed red
line) is -0.274 meV and the second state (dotted blue line) is -0.273 meV. The
probability cloud of the first state (dashed red line) and the second state
(dotted blue line) is overlapping. The two probability clouds are shaped like
two rings, one located at $u=-248.0$ Å and the other located at $u=248.0$ Å,
each having a width of $\Delta u=$ 395.2 Å. The question of these states
becoming less and less localized when the coupling factor increases, has been
discussed previously in the text.

##  IV Final Remarks and perspectives

We investigated the electronic states of a position-dependent mass (PDM)
electron confined on the surface of a graphene catenoid bridge. In addition to
the usual geometric potential, we considered a PDM mass as a function of the
mean and the gaussian curvatures.

The coupling between the curvature and the electron mass is controlled by a
coupling constant $\lambda$, which can take any value in the interval
$\lambda_{c}<\lambda<\infty$, where the critical value $\lambda_{c}$ is given
by $\lambda_{c}=-R^{2}$ and $R$ is the radius of the caternoid throat.

The effective potential contains the usual geometric da Costa potential, the
centrifugal term and the PDM corrections. Asymptotically, the effective
potential vanishes and free states are allowed. Near the bridge throat, the
effective potential is strongly dependent on the PDM parameter $\lambda$. For
$\lambda_{c}<\lambda<0$, the potential exhibits an attractive volcano-like
shape around the origin, regardless the value of the angular momentum.

For $0\leq\lambda<\infty$ the is rather dependent on the angular momentum. For
$\nu=0$, $\lambda=0$ leads to a reflectionless potential euclides  and as
$\lambda$ increases the potential takes the form of a double-well potential
near the bridge throat. For $\nu=1$, the centrifugal term enhances the barrier
near the origin.

The PDM parameter also modifies the number and behaviour of the bound states.
For $\lambda_{c}<\lambda<0$, only one bound state located around the origin
was found. As $\lambda$ increases keeping the volcano-shape potential the
width of the bound state also increases. For $\lambda>0$, the PDM parameter
breaks the bound state degeneracy leading to two states. Nonetheless, for
greater values of $\lambda$, the double-well symmetry restore the degeneracy.
In fact, there are two hybrid states, and these states are degenerate Studart
. The probability density function of these states indicates that resonant
tunneling is possible. Our results also indicate that if the coupling factor
continues to increase, that double potential well becomes very shallow and the
bound states are suppressed.

As future developments we point out the analysis of the transport features of
this graphene bridge, as well as the interaction of the confined electrons
with electromagnetic fields.

## Acknowledgments

J.E.G.Silva thanks the Conselho Nacional de Desenvolvimento Científico e
Tecnológico (CNPq), grants no 312356/2017-0 for financial support.

## References

  * (1) A. K. Geim, K. S. Novoselov, Nature Materials 6, 183 (2007).
  * (2) A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, A. K. Geim, Rev. Mod. Phys. 81, 109 (2009).
  * (3) M. Katsnelson, Graphene: Carbon in two dimensions, Cambridge University Press, Cambridge, (2012).
  * (4) S. Berber, Y.K. Kwon and D. Tomanek, Phys. Rev. Lett. 84, 4613 (2000).
  * (5) A. Carvalho, M. Wang, X. Zhu, A. S. Rodin, H. Su, A. H. Castro Neto, Nat. Rev. Mat. 1, 11 (2016).
  * (6) C. Furtado , F. Moraes , A.M. de M. Carvalho, Phys. Lett. A 372, 5368, (2008).
  * (7) R. Dandoloff, T. T. Truong, Phys. Lett. A 325, 233 (2004).
  * (8) V. Atanasov, R. Dandoloff and A. Saxena Phys. Rev. B 79, 033404 (2009).
  * (9) V. Atanasov, A. Saxena, Phys. Rev. B 92, 035440 (2015).
  * (10) F. Guinea, A. K. Geim, M. I. Katsnelson, and K. S. Novoselov, Phys. Rev. B 81, 035408 (2010).
  * (11) F. de Juan, A. Cortijo, M. A. H. Vozmediano, Phys. Rev. B 76, 165409, (2007).
  * (12) V. Atanasov, A. Saxena, Phys. Rev. B 81, 205409 (2010).
  * (13) H. Jensen, H. Koppe, Ann. Phys. 63, 586 (1971).
  * (14) R. C. T. da Costa, Phys. Rev. A 23, 1982 (1981).
  * (15) R. C. T. da Costa, Phys. Rev. A 25, 2893 (1982).
  * (16) S. Matsutani, J. Phys. Soc. Japan 61 (1992) 3825.
  * (17) Luiz C.B. da Silva, Cristiano C. Bastos, Fábio G. Ribeiro, Ann. of Phys. 379 (2017) 13.
  * (18) G. Ferrari, G. Cuoghi, Phys. Rev. Lett. 100, 230403 (2008).
  * (19) Y. L. Wang, L. Du, C. T. Xu, X. J. Liu and H. S. Zong Phys. Rev. A90, 042117 (2014).
  * (20) Mark Burgess, B. Jensen, Phys. Rev. A 48, 3, 1861 (1993).
  * (21) J. González, J. Herrero, Nucl. Phys. B 825, 426 (2010).
  * (22) R. Pincak, J. Smotlacha, Quantum Matter 5, 114 (2016).
  * (23) R. Dandoloff, Phys. Lett. A, 373 (2009).
  * (24) R. Dandoloff, A. Saxena, B. Jensen, Phys. Rev. A 81, 014102 (2010).
  * (25) M. Spivak, A comprehensive introduction to differential geometry, Publish or Perish, Houston, (1999).
  * (26) H. Terrones, A. L. Mackay, Chem. Phys. Let. 207, 45, (1993).
  * (27) J.E.G. Silva et al., Phys. Lett. A 384, 126458 (2020).
  * (28) Y. N. Joglekar, A. Saxena, Phys. Rev. B 80 (2009) 153405.
  * (29) A. Sinner and K. Ziegler, Annals Phys. 400, 262-278 (2019).
  * (30) M. G. Burt, Journal of Physics: Condensed Matter 4 (32) 6651 (1992) .
  * (31) S. Y. Ren, Y.-C. Chang, Annals Phys. 325 (5) (2010) 937 – 947.
  * (32) J.M. Lévi-Le-Blond, Phys. Rev. A 52, 1845 (1995).
  * (33) Gang Chen, Zi-dong Chen, Phy. Lett. A 331, 312 (2004).
  * (34) F. Serafim et al, Physica E 108, 139 (2019).
  * (35) Pedro H. Souza et al, Annalen Der Physik, 2018, 530, 1800112.
  * (36) T. J. Willmore, J. London Math. Soc. 3, 307 (1971).
  * (37) Helfrich W., Z. Naturforsch. C 28 693 (1973).
  * (38) C. M. Bender, Rept. Prog. Phys. 70, 947 (2007)
  * (39) C. M. Bender and S. Boettcher, Phys. Rev. Lett. 80, 5243 (1998)
  * (40) H. F. Jones and J. Mateo, Phys. Rev. D 73, 085002 (2006)
  * (41) A. A. Andrianov, Phys. Rev. D 76, 025003 (2007)
  * (42) A. A. Andrianov, Annals Phys. 140, 82 (1982).
  * (43) A. C. A. Ramos, G. A. Farias and N. S. Almeida, Physica E 43, 1878 (2011).
  * (44) M. Novaes and N. Studart, Mecânica Quântica Básica - São Paulo: Editora Livraria da Física (Série MNPEF), (2016).