arXiv subjects

D. N. Nishnianidze

Publications and source records attributed to D. N. Nishnianidze.

At least 19 recordsLinked to original sources

arXiv2022

A New Class of Solvable Two-dimensional Scalar Potentials for Graphene

In the present paper, a systematic approach is presented for solution of two-dimensional massless Dirac equation with external electrostatic potential applied. This approach is based on the new - asymmetric - form of SUSY-like intertwining relations. It allows to build a wide variety of pairs of SUSY-partner external scalar potentials. If one of them is simple enough to be solvable, its partner is also solvable although it may have a non-trivial dependency on both coordinates. Physically, this kind of problems is related to the description of graphene and some other materials with external potential. Solvability obtained by means of asymmetric form of SUSY intertwining relations allows to extend the class of analytically solvable two-dimensional models.

cond-mat.mes-hall↗

arXiv2020

Generalization of SUSY Intertwining Relations: New Exact Solutions of Fokker-Planck Equation

It is commonly known that the Fokker-Planck equation is exactly solvable only for some particular systems, usually with time-independent drift coefficients. To extend the class of solvable problems, we use the intertwining relations of SUSY Quantum Mechanics but in new - asymmetric - form. It turns out that this form is just useful for solution of Fokker-Planck equation. As usual, intertwining provides a partnership between two different systems both described by Fokker-Planck equation. Due to the use of an asymmetric kind of intertwining relations with a suitable ansatz, we managed to obtain a new class of analytically solvable models. What is important, this approach allows us to deal with the drift coefficients depending on both variables, $x,$ and $t.$ An illustrating example of the proposed construction is given explicitly.

quant-ph↗

arXiv2019

New solutions for graphene with scalar potentials by means of generalized intertwining

The intertwining relations between superpartner Hamiltonians are the main ingredients of well known Supersymmetrical Quantum Mechanics (SUSY QM). In the present paper, the generalized form of intertwining is used for investigation of a massless (zero energy) two-dimensional Dirac equation with scalar external potential. This equation is related to the description of graphene and some other materials in the field of external electrostatic potential. The use of modified intertwining relations allows to find analytically solutions for the wave functions in the field of some external scalar potentials which depend on both space coordinates. A few examples of this construction are given explicitly.

cond-mat.mes-hall↗

arXiv2018

Zero Energy States for a Class of Two-Dimensional Potentials in Graphene

The excitations in graphene and some other materials are described by two-dimensional massless Dirac equation with applied external potential of some kind. Solutions of this zero energy equation are built analytically for a wide class of scalar potentials. In contrast to most publications on analytical solutions of massless two-dimensional Dirac equation, our potentials really depend on both spatial coordinates in some bounded domain. Several examples of such construction are given explicitly.

cond-mat.mes-hall↗

arXiv2017

Mapping of Two-Dimensional Schrödinger Equation under the Point Transformation

For the two-dimensional Schrödinger equation, the general form of the point transformations such that the result can be interpreted as a Schrödinger equation with effective (i.e. position dependent) mass is studied. A wide class of such models with different forms of mass function is obtained in this way. Starting from the solvable two-dimensional model, the variety of solvable partner models with effective mass can be built. Several illustrating examples not amenable to the conventional separation of variables are given.

quant-ph↗

arXiv2016

SUSY Method for the Three-Dimensional Schrödinger Equation with Effective Mass

The three-dimensional Schrödinger equation with a position-dependent (effective) mass is studied in the framework of Supersymmetrical (SUSY) Quantum Mechanics. The general solution of SUSY intertwining relations with first order supercharges is obtained without any preliminary constraints. Several forms of coefficient functions of the supercharges are investigated and analytical expressions for the mass function and partner potentials are found. As usual for SUSY Quantum Mechanics with nonsingular superpotentials, the spectra of intertwined Hamiltonians coincide up to zero modes of supercharges, and the corresponding wave functions are connected by intertwining relations. All models are partially integrable by construction: each of them has at least one second order symmetry operator.

hep-th↗

arXiv2016

Solution of Second Order Supersymmetrical Intertwining Relations in Minkowski Plane

Supersymmetrical (SUSY) intertwining relations are generalized to the case of quantum Hamiltonians in Minkowski space. For intertwining operators (supercharges) of second order in derivatives the intertwined Hamiltonians correspond to completely integrable systems with the symmetry operators of fourth order in momenta. In terms of components, the itertwining relations correspond to the system of nonlinear differential equations which are solvable with the simplest - constant - ansatzes for the "metric" matrix in second order part of the supercharges. The corresponding potentials are built explicitly both for diagonalizable and nondiagonalizable form of "metric" matrices, and their properties are discussed.

hep-th↗

arXiv2016

Some Properties of the Shape Invariant Two-Dimensional Scarf II Model

Two-dimensional Scarf~II quantum model is considered in the framework of Supersymmetrical Quantum Mechanics (SUSY QM). Previously obtained results for this integrable system are systematized, and some new properties are derived. In particular, it is shown that the model is exactly or quasi-exactly solvable in different regions of parameter of the system. The degeneracy of the spectrum is detected for some specific values of parameters. The action of the symmetry operators of fourth order in momenta is calculated for the arbitrary wave functions, obtained by means of double shape invariance.

quant-ph↗

arXiv2015

New Implicitly Solvable Potential Produced by Second Order Shape Invariance

The procedure proposed recently by J.Bougie, A.Gangopadhyaya and J.V.Mallow to study the general form of shape invariant potentials in one-dimensional Supersymmetric Quantum Mechanics (SUSY QM) is generalized to the case of Higher Order SUSY QM with supercharges of second order in momentum. A new shape invariant potential is constructed by this method. It is singular at the origin, it grows at infinity, and its spectrum depends on the choice of connection conditions in the singular point. The corresponding Schrödinger equation is solved explicitly: the wave functions are constructed analytically, and the energy spectrum is defined implicitly via the transcendental equation which involves Confluent Hypergeometric functions.

quant-ph↗

arXiv2012

Three-Dimensional Shape Invariant Non-Separable Model With Equidistant Spectrum

A class of three-dimensional models which satisfy supersymmetric intertwining relations with the simplest - oscillator-like - variant of shape invariance is constructed. It is proved that the models are not amenable to conventional separation of variables for the complex potentials, but their spectra are real and equidistant (like for isotropic harmonic oscillator). The special case of such potential with quadratic interaction is solved completely. The Hamiltonian of the system is non-diagonalizable, and its wave functions and associated functions are built analytically. The symmetry properties of the model and degeneracy of energy levels are studied.

hep-th↗

arXiv2012

General Solution of the Two-Dimensional Intertwining Relations for Supercharges with Hyperbolic (Lorentz) Metrics

The supersymmetrical intertwining relations are the most productive part of the supersymmetrical method in two-dimensional Quantum Mechanics. Most interesting are relations with hyperbolic form of derivatives in supercharges. So far, several explicit solutions were found, and they provide nontrivial two-dimensional potentials which were further studied by means of supersymmetrical methods. In the present paper the general solution of intertwining relations with hyperbolic structure of supercharges is obtained. The corresponding potentials are built explicitly, and it is evident that some of them were not known before now.

hep-th↗

arXiv2012

Equidistance of the Complex 2-Dim Anharmonic Oscillator Spectrum: Exact Solution

We study a class of quantum two-dimensional models with complex potentials of specific form. They can be considered as the generalization of a recently studied model with quadratic interaction not amenable to conventional separation of variables. In the present case, the property of shape invariance provides the equidistant form of the spectrum and the algorithm to construct eigenfunctions analytically. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of identity must include also the corresponding associated functions. In the specific case of anharmonic second-plus-fourth order interaction, expressions for the wave functions and associated functions are constructed explicitly for the lowest levels, and the recursive algorithm to produce higher level wave functions is given.

math-ph↗

arXiv2012

Supersymmetrical Separation of Variables for Scarf II Model: Partial Solvability

Recently, a new quantum model - two-dimensional generalization of the Scarf II - was completely solved analytically by SUSY method for the integer values of parameter. Now, the same integrable model, but with arbitrary values of parameter, will be studied by means of supersymmetrical intertwining relations. The Hamiltonian does not allow the conventional separation of variables, but the supercharge operator does allow, leading to the partial solvability of the model. This approach, which can be called as the first variant of SUSY-separation, together with shape invariance of the model, provides analytical calculation of the part of spectrum and corresponding wave functions (quasi-exact-solvability). The model is shown to obey two different variants of shape invariance which can be combined effectively in construction of energy levels and wave functions.

hep-th↗

arXiv2012

Analytical Solution of Two-Dimensional Scarf II Model by Means of SUSY Methods

New two-dimensional quantum model - the generalization of the Scarf II - is completely solved analytically for the integer values of parameter. This model being not amenable to conventional procedure of separation of variables is solved by recently proposed method of supersymmetrical separation. The latter is based on two constituents of SUSY Quantum Mechanics: the intertwining relations with second order supercharges and the property of shape invariance. As a result, all energies of bound states were found, and the analytical expressions for corresponding wave functions were obtained.

hep-th↗

arXiv2011

New Two-Dimensional Quantum Models with Shape Invariance

Two-dimensional quantum models which obey the property of shape invariance are built in the framework of polynomial two-dimensional SUSY Quantum Mechanics. They are obtained using the expressions for known one-dimensional shape invariant potentials. The constructed Hamiltonians are integrable with symmetry operators of fourth order in momenta, and they are not amenable to the conventional separation of variables.

hep-th↗

arXiv2010

New Exactly Solvable Two-Dimensional Quantum Model Not Amenable to Separation of Variables

The supersymmetric intertwining relations with second order supercharges allow to investigate new two-dimensional model which is not amenable to standard separation of variables. The corresponding potential being the two-dimensional generalization of well known one-dimensional Pöschl-Teller model is proven to be exactly solvable for arbitrary integer value of parameter $p:$ all its bound state energy eigenvalues are found analytically, and the algorithm for analytical calculation of all wave functions is given. The shape invariance of the model and its integrability are of essential importance to obtain these results.

hep-th↗

arXiv2010

Exactly Solvable Non-Separable and Non-Diagonalizable 2-Dim Model with Quadratic Complex Interaction

We study a quantum model with non-isotropic two-dimensional oscillator potential but with additional quadratic interaction $x_1x_2$ with imaginary coupling constant. It is shown, that for a specific connection between coupling constant and oscillator frequences, the model {\it is not} amenable to a conventional separation of variables. The property of shape invariance allows to find analytically all eigenfunctions and the spectrum is found to be equidistant. It is shown that the Hamiltonian is non-diagonalizable, and the resolution of the identity must include also the corresponding associated functions. These functions are constructed explicitly, and their properties are investigated. The problem of $R-$separation of variables in two-dimensional systems is discussed.

hep-th↗

arXiv2009

Three-Dimensional Solutions of Supersymmetrical Intertwining Relations and Pairs of Isospectral Hamiltonians

The general solution of SUSY intertwining relations for three-dimensional Schrödinger operators is built using the class of second order supercharges with nondegenerate constant metric. This solution includes several models with arbitrary parameters. We are interested only in quantum systems which are not amenable to separation of variables, i.e. can not be reduced to lower dimensional problems. All constructed Hamiltonians are partially integrable - each of them commutes with a symmetry operator of fourth order in momenta. The same models can be considered also for complex values of parameters leading to a class of non-Hermitian isospectral Hamiltonians.

hep-th↗