M. V. Ioffe

Higher derivative supersymmetry and the Witten index

We propose a higher-order derivative generation of supersymmetric quantum mechanics. It is formally based on the standard superalgebra but supercharges involve differential operators of order n. As a result, their anticommutator entails a polynomial of a Hamiltonian. The Witten index does not characterize spontaneous SUSY breaking in such models. The construction naturally arises after truncation of the order n parasupersymmetric quantum mechanics which in turn is built by glueing of n ordinary supersymmetric systems.

The factorization method and quantum systems with equivalent energy spectra

Abstract An algorithm for constructing a chain of quantum hamiltonians with interconnected energy spectra is proposed for an arbitrary number of space dimensions. Some physical models of that kind are discussed. The supersymmetric nature of the relation between connected quantum systems is pointed out.

Polynomial SUSY in quantum mechanics and second derivative Darboux transformations

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.Abstract We give a classification of second-order polynomial SUSY quantum mechanics in one and two dimensions. 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.

Systems with higher-order shape invariance: spectral and algebraic properties

Abstract We study systems of two intertwining relations of first or second order for the same (up to a constant shift) partner Schrodinger operators. It is shown that the corresponding Hamiltonians possess a higher order shape invariance which is equivalent to the ladder equation. We analyze with particular attention irreducible second order Darboux transformations which together with the first order act as building blocks. For the third order shape-invariance irreducible Darboux transformations entail only one sequence of equidistant levels while for the reducible case the structure consists of up to three infinite sequences of equidistant levels and, in some cases, singlets or doublets of isolated levels.

Supersymmetric origin of equivalent quantum systems

Abstract The previously established equivalence of certain multidimensional quantum hamiltonians is shown to be a consequence of the supersymmetry in quantum mechanics. Thereby the supersymmetric quantum mechanics can serve as a regular source of equivalent quantum systems in arbitrary space dimensions.

POLYNOMIAL SUPERSYMMETRY AND DYNAMICAL SYMMETRIES IN QUANTUM MECHANICS

A polynomial generalization of supersymmetry in quantum mechanics in one and two dimensions is proposed. Polynomial superalgebras in one dimension are classified. In two dimensions, a detailed analysis is made for supercharges of second order with respect to derivatives and it is shown that in all cases the binomial superalgebra leads to hidden dynamical symmetry generated by the central charge.

Nonlinear supersymmetric quantum mechanics: concepts and realizations

Nonlinear SUSY approach to preparation of quantum systems with pre-planned spectral properties is reviewed. Possible multidimensional extensions of Nonlinear SUSY are described. The full classification of ladder-reducible and irreducible chains of SUSY algebras in one-dimensional QM is given. Emergence of hidden symmetries and spectrum generating algebras is elucidated in the context of Nonlinear SUSY in one- and two-dimensional QM.

New methods for the two-dimensional Schrödinger equation: SUSY-separation of variables and shape invariance

Two new methods for the investigation of two-dimensional quantum systems, whose Hamiltonians are not amenable to separation of variables, are proposed. The first one—SUSY-separation of variables—is based on the intertwining relations of higher-order SUSY quantum mechanics (HSUSY QM) with supercharges allowing separation of variables. The second one is a generalization of shape invariance. While in one dimension shape invariance allows us to solve algebraically a class of (exactly solvable) quantum problems, its generalization to higher dimensions has not been explored yet. 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 us 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.

A SUSY approach for investigation of two-dimensional quantum mechanical systems

Different ways to incorporate two-dimensional systems which are not amenable to separation of variables treatment into the framework of supersymmetrical quantum mechanics (SUSY QM) are analysed. In particular, the direct generalization of one-dimensional Witten SUSY QM is based on supercharges of first order in momenta and allows one to connect the eigenvalues and eigenfunctions of two scalar and one matrix Schrodinger operators. The use of second-order supercharges leads to polynomial supersymmetry and relates a pair of scalar Hamiltonians, giving a set of partner systems with almost coinciding spectra. This class of systems can be studied by means of a new method of SUSY separation of variables, where supercharges allow separation of variables, but Hamiltonians do not. The method of shape invariance is generalized to two-dimensional models in order to construct purely algebraically a chain of eigenstates and eigenvalues for generalized Morse potential models in two dimensions.

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

Abstract 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.