arXiv subjects

D. N. Nishnianidze

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

32 records · Page 2Linked to original sources

arXiv2007

Two-dimensional Schrödinger Hamiltonians with Effective Mass in SUSY Approach

The general solution of SUSY intertwining relations of first order for two-dimensional Schrödinger operators with position-dependent (effective) mass is built in terms of four arbitrary functions. The procedure of separation of variables for the constructed potentials is demonstrated in general form. The generalization for intertwining of second order is also considered. The general solution for a particular form of intertwining operator is found, its properties - symmetry, irreducibility, separation of variables - are investigated.

hep-th↗

arXiv2007

Exact Solvability of Two-Dimensional Real Singular Morse Potential

The supersymmetric approach in the form of second order intertwining relations is used to prove the exact solvability of two-dimensional Schrodinger equation with generalized two-dimensional Morse potential for $a_0=-1/2$. This two-parametric model is not amenable to conventional separation of variables, but it is completely integrable: the symmetry operator of fourth order in momenta exists. All bound state energies are found explicitly, and all corresponding wave functions are built analytically. By means of shape invariance property, the result is extended to the hierarchy of Morse models with arbitrary integer and half-integer values $a_k=-(k+1)/2.$

hep-th↗

arXiv2007

Pseudo-Hermiticity of an Exactly Solvable Two-Dimensional Model

We study a two-dimensional exactly solvable non-Hermitian $PT-$non-symmetric quantum model with real spectrum, which is not amenable to separation of variables, by supersymmetrical methods. Here we focus attention on the property of pseudo-Hermiticity, biorthogonal expansion and pseudo-metric operator. To our knowledge this is the first time that pseudo-Hermiticity is realized explicitly for a nontrivial two-dimensional case. It is shown that the Hamiltonian of the model is not diagonalizable.

hep-th↗

arXiv2006

New Two-Dimensional Integrable Quantum Models from SUSY Intertwining

Supersymmetrical intertwining relations of second order in the derivatives are investigated for the case of supercharges with deformed hyperbolic metric $g_{ik}=diag(1,-a^2)$. Several classes of particular solutions of these relations are found. The corresponding Hamiltonians do not allow the conventional separation of variables, but they commute with symmetry operators of fourth order in momenta. For some of these models the specific SUSY procedure of separation of variables is applied.

hep-th↗

arXiv2005

Exactly Solvable Two-Dimensional Complex Model with Real Spectrum

Supersymmetrical intertwining relations of second order in derivatives allow to construct a two-dimensional quantum model with complex potential, for which {\it all} energy levels and bound state wave functions are obtained analytically. This model {\it is not amenable} to separation of variables, and it can be considered as a specific complexified version of generalized two-dimensional Morse model with additional $\sinh^{-2}$ term. The energy spectrum of the model is proved to be purely real. To our knowledge, this is a rather rare example of a nontrivial exactly solvable model in two dimensions. The symmetry operator is found, the biorthogonal basis is described, and the pseudo-Hermiticity of the model is demonstrated. The obtained wave functions are found to be common eigenfunctions both of the Hamiltonian and of the symmetry operator.

hep-th↗

arXiv2005

Double Shape Invariance of Two-Dimensional Singular Morse Model

A second shape invariance property of the two-dimensional generalized Morse potential is discovered. Though the potential is not amenable to conventional separation of variables, the above property allows to build purely algebraically part of the spectrum and corresponding wave functions, starting from {\it one} definite state, which can be obtained by the method of $SUSY$-separation of variables, proposed recently.

hep-th↗

arXiv2004

Higher Order Matrix SUSY Transformations in Two-Dimensional Quantum Mechanics

The iteration procedure of supersymmetric transformations for the two-dimensional Schroedinger operator is implemented by means of the matrix form of factorization in terms of matrix 2x2 supercharges. Two different types of iterations are investigated in detail. The particular case of diagonal initial Hamiltonian is considered, and the existence of solutions is demonstrated. Explicit examples illustrate the construction.

hep-th↗

arXiv2004

SUSY Intertwining Relations of Third Order in Derivatives

The general solution of the intertwining relations between a pair of Schrödinger Hamiltonians by the supercharges of third order in derivatives is obtained. The solution is expressed in terms of one arbitrary function. Some properties of the spectrum of the Hamiltonian are derived, and wave functions for three energy levels are constructed. This construction can be interpreted as addition of three new levels to the spectrum of partner potential: a ground state and a pair of levels between successive excited states. Possible types of factorization of the third order supercharges are analysed, the connection with earlier known results is discussed.

hep-th↗

arXiv2003

Two-dimensional SUSY-pseudo-Hermiticity without separation of variables

We study SUSY-intertwining for non-Hermitian Hamiltonians with special emphasis to the two-dimensional generalized Morse potential, which does not allow for separation of variables. The complexified methods of SUSY-separation of variables and two-dimensional shape invariance are used to construct particular solutions - both for complex conjugated energy pairs and for non-paired complex energies.

hep-th↗

arXiv2002

New Methods for Two-Dimensional Schrödinger Equation: SUSY-separation of Variables and Shape Invariance

Two new methods for investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. 1)The first one - $SUSY-$ separation of variables - is based on the intertwining relations of Higher order SUSY Quantum Mechanics (HSUSY QM) with supercharges allowing for separation of variables. 2)The second one is a generalization of shape invariance. While in one dimension shape invariance allows to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been yet explored. Here we provide a formal framework in HSUSY QM for two-dimensional quantum mechanical systems for which shape invariance holds. Given the knowledge of one eigenvalue and eigenfunction, shape invariance allows to construct a chain of new eigenfunctions and eigenvalues. These methods are applied to a two-dimensional quantum system, and partial explicit solvability is achieved in the sense that only part of the spectrum is found analytically and a limited set of eigenfunctions is constructed explicitly.

hep-th↗

arXiv1999

Classical Integrable 2-dim Models Inspired by SUSY Quantum Mechanics

A class of integrable 2-dim classical systems with integrals of motion of fourth order in momenta is obtained from the quantum analogues with the help of deformed SUSY algebra. With similar technique a new class of potentials connected with Lax method is found which provides the integrability of corresponding 2-dim hamiltonian systems. In addition, some integrable 2-dim systems with potentials expressed in elliptic functions are explored.

solv-int↗

arXiv1997

Matrix Hamiltonians: SUSY approach to hidden symmetries

A new supersymmetric approach to the analysis of dynamical symmetries for matrix quantum systems is presented. Contrary to standard one dimensional quantum mechanics where there is no role for an additional symmetry due to nondegeneracy, matrix hamiltonians allow for non-trivial residual symmetries. This approach is based on a generalization of the intertwining relations familiar in SUSY Quantum Mechanics. The corresponding matrix supercharges, of first or of second order in derivatives, lead to an algebra which incorporates an additional block diagonal differential matrix operator (referred to as a "hidden" symmetry operator) found to commute with the superhamiltonian. We discuss some physical interpretations of such dynamical systems in terms of spin 1/2 particle in a magnetic field or in terms of coupled channel problem. Particular attention is paid to the case of transparent matrix potentials.

quant-ph↗

arXiv1996

Higher Order SUSY in Quantum Mechanics and Integrability of Two-dimensional Hamiltonians

The new method based on the SUSY algebra with supercharges of higher order in derivatives is proposed to search for dynamical symmetry operators in 2-dim quantum and classical systems. These symmetry operators arise when closing the SUSY algebra for a wide set of potentials. In some cases they are of 2-nd order in derivatives. The particular solutions are obtained also for potentials accepting symmetry operators of 4-th order. The investigation of quasiclassical limit of the SUSY algebra yields new classical integrals of motion for a certain type of systems which are polynomials of 4-th order in momenta. The general SUSY-inspired algorithm to construct classical systems with additional integrals of motion is outlined.

solv-int↗

arXiv1994

Polynomial SUSY in Quantum Mechanics and Second Derivative Darboux Transformation

We give the classification of second-order polynomial SUSY Quantum Mechanics in one and two dimensions. The particular attention is paid to the irreducible supercharges which cannot be built by repetition of ordinary Darboux transformations. In two dimensions it is found that the binomial superalgebra leads to the dynamic symmetry generated by a central charge operator.

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