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A. G. Nikitin

Publications and source records attributed to A. G. Nikitin.

At least 19 recordsLinked to original sources
arXiv2025

Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 2. Systems with dilatation and shift symmetries

3d quantum mechanical systems with position dependent masses (PDM) admitting at least one second order integral of motion and symmetries with respect to dilatation or shift transformations are classified. Twenty-seven such systems are specified and the completeness of the classification results is proved.

math-ph↗
arXiv2024

Integrable and superintegrable quantum mechanical systems with position dependent masses invariant with respect to one parametric Lie groups. 1. Systems with cylindric symmetry

Cylindrically symmetric quantum mechanical systems with position dependent masses (PDM) admitting at least one second order integral of motion are classified. It is proved that there exist 68 such systems which are inequivalent. Among them there are twenty seven superintegrable and twelve maximally superintegrable. The arbitrary elements of the correspondinding Hamiltonians (i.e.,masses and potentials) are presented explicitly.

math-ph↗
arXiv2024

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to two parametric Lie groups

Quantum mechanical systems with position dependent masses (PDM) admitting two parametric Lie symmetry groups are classified. Namely, all PDM systems are specified which, in addition to their invariance w.r.t. a two parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which do not accept any Lie group.

math-ph↗
arXiv2023

Superintegrable quantum mechanical systems with position dependent masses invariant with respect to three parametric Lie groups

Quantum mechanical systems with position dependent masses (PDM) admitting for and more dimensional symmetry algebras are classified. Namely, all PDM systems are specified which, in addition to their invariance w.r.t. a three parametric Lie group, admit at least one second order integral of motion. The presented classification is partially extended to the more generic systems which admit one or two parametric Lie groups.

math-ph↗
arXiv2021

Symmetries of Schroedinger equation with scalar and vector potentials

Using the algebraic approach Lie symmetries of time dependent Schroedinger equations for charged particles interacting with superpositions of scalar and vector potentials are classified. Namely, all the inequivalent equations admitting symmetry transformations with respect to continuous groups of transformations are presented. This classification is completed and includes the specification of symmetries and admissible equivalence relations for such equations. In particular, a simple mapping between the free Schroedinger equation and the repulsive oscillator is found which has a clear group-theoretical sense.

math-ph↗
arXiv2020

Symmetries of the Schroedinger-Pauli equation for neutral particles

With using the algebraic approach Lie symmetries of Schrödinger equations with matrix potentials are classified. Thirty three inequivalent equations of such type together with the related symmetry groups are specified, the admissible equivalence relations are clearly indicated. In particular the Boyer results concerning kinematical invariance groups for arbitrary potentials (C. P. Boyer, Helv. Phys. Acta, {\bf 47}, 450--605 (1974)) are clarified and corrected.

math-ph↗
arXiv2020

Superintegrable systems with position dependent mass

First order integrals of motion for Schrödinger equations with position dependent masses are classified. Seventeen classes of such equations with non-equivalent symmetries are specified. They include integrable, superintegrable and maximally superintegrable systems. Among them is a system invariant with respect to the Lie algebra of Lorentz group and a system whose integrals of motion form algebra so(4). Three of the obtained systems are solved exactly.

math-ph↗
arXiv2020

Symmetries and Supersymmetries of Generalized Schrödinger equations

In this survey the contemporary results concerning supersymmetries in generalized Schrödinger equations are presented. Namely, position dependent mass Schödinger equations are discussed as well as the equations with matrix potentials. An extended number of realistic quantum mechanical problems admitting extended supersymmetries is described, an extended class of matrix potentials is classified.

math-ph↗
arXiv2019

Exact solvability of PDM systems with extended Lie symmetries

It is shown that all PDM Schroedinger equations admitting more than five dimensional Lie symmetry algebras (whose completed list can be found in paper~[{\it J.~Math. Phys.} {\bf 58}, , 083508 (2017)] are exactly solvable. The corresponding exact solutions are presented. The supersymmetric aspects of the exactly solvable systems are discussed.

math-ph↗
arXiv2017

Group classification of (1+3)-dimensional Schrödinger equations with position dependent mass

Kinematical invariance groups of the 3d Schrödinger equations with position dependent masses (PDM) and arbitrary potentials are classified. It is shown that there exist 94 classes of such equations defined up to the generic equivalence group, and 70 classes defined up to the equivalence groupoid. The maximally extended kinematical invariance algebras of such equations appears to be eight dimensional. The specific symmetries connected with the presence of the ambiguity parameters are discussed and an extended class of systems which keep their forms for arbitrary or particular changes of these parameters is specified. The exact solution of the selected PDM Schrödinger equation is presented. This equation describes a deformed 3d isotropic harmonic oscillator and possesses extended continuous symmetries and hidden supersymmetries with two different superpotentials as well.

math-ph↗
arXiv2017

The maximal 'kinematical' invariance group for an arbitrary potential revised

Group classification of one particle Schrödinger equations with arbitrary potentials (C. P. Boyer, Helv. Phys. Acta {\bf 47}, p. 450, 1974) is revised. The corrected completed list of non-equivalent potentials and the corresponding symmetries is presented together with exact identification of symmetry algebras and admissible equivalence transformations.

math-ph↗
arXiv2017

Group classification of Schrödinger equations with position dependent mass

Maximal kinematical invariance groups of $2d$ Schrödinger equation with a position dependent mass and arbitrary potential are classified. It is demonstrated that there exist seven classes of such equations possessing non-equivalent continuous symmetry group. Three of these classes include arbitrary functions while the remaining ones are defined up to arbitrary parameters. In particular, for the case of a constant mass the class missing in the Boyer classification (Boyer C P 1974 Helv. Phys. Acta{\bf 47}, 450) is indicated. A constructive test of (non)equivalence of a PDM system to a constant mass system is proposed.

math-ph↗
arXiv2016

Higher order symmetries for linear and nonlinear Schroedinger equations

We study arbitrary order symmetry operators for the linear Schrödinger equations with arbitrary number of spatial variables. We deduce determining equations for coefficient functions of such operators and consider in detail some cases when these equations can be explicitly solved. In addition, the complete group classification of the nonlinear Schrödinger equation is presented.

math-ph↗
arXiv2015

Superintegrable and shape invariant systems with position dependent mass

Second order integrals of motion for 3d quantum mechanical systems with position dependent masses (PDM) are classified. Namely, all PDM systems are specified which, in addition to their rotation invariance, admit at least one second order integral of motion. All such systems appear to be also shape invariant and exactly solvable. Moreover, some of them possess the property of double shape invariance and can be solved using two different superpotentials. Among them there are systems with double shape invariance which present nice bridges between the Coulomb and isotropic oscillator systems. A simple algorithm for calculation the discrete spectrum and the corresponding state vectors for the considered PDM systems is presented and applied to solve five of the found systems.

math-ph↗
arXiv2014

Non-standard Dirac equations for non-standard spinors

Generalized Dirac equation with operator mass term is presented. Its solutions are non standard (ELKO) spinors which are eigenvectors of the charge conjugation and dual helicity operators. It is demonstrated that in spite of their non covariant nature ELKO can serve as a carrier space of a representation of Poincaré group. However, the corresponding boost generators are not manifestly covariant and generate non-local momentum dependent transformations which are presented explicitly. These results present a new look on group-theoretical grounds of ELKO theories.

math-ph↗
arXiv2014

Laplace-Runge-Lenz vector with spin in any dimension

Superintegrable d - dimensional quantum mechanical systems with spin, which admit a generalized Laplace-Runge-Lenz vector are presented. The systems with spins 0, 1/2 and 1 are considered in detail. All these systems are exactly solvable for arbitrary d, and their solutions are presented explicitly.

math-ph↗