Boris F. Samsonov

Darboux transformation, factorization, and supersymmetry in one-dimensional quantum mechanics

We introduce an N-order Darboux transformation operator as a particular case of general transformation operators. It is shown that this operator can always be represented as a product of N first-order Darboux transformation operators. The relationship between this transformation and the factorization method is investigated. Supercharge operators are introduced. They are differential operators of order N. It is shown that these operators and super-Hamiltonian form a superalgebra of order N. For N=2, we have a quadratic superalgebra analogous to the Sklyanin quadratic algebras. The relationship between the transformation introduced and the inverse scattering problem in quantum mechanics is established. An elementary N-parametric potential that has exactly N predetermined discrete spectrum levels is constructed. The paper concludes with some examples of new exactly soluble potentials.

Darboux transformation of the Schrodinger equation

The recent developments in the theory of the generation of potentials for which the Schrodinger equation has an exact solution are discussed. The generalization of the Darboux transformation to the nonstationary Schrodinger equation is studied in detail. The supersymmetric generalization of the nonstationary Schrodinger equation is formulated. Versions corresponding to exact and spontaneously broken supersymmetry are discussed. New, exactly solvable nonstationary potentials are obtained as examples. The stationary Darboux transformation is viewed as a special case of the new transformation. Families of isospectral potentials with the spectra of the harmonic oscillator and the hydrogen-like atom are obtained. The effectiveness of these methods for describing the coherent states of the transformed Hamiltonians is demonstrated.

New possibilities for supersymmetry breakdown in quantum mechanics and second order irreducible Darboux transformations

Abstract New types of irreducible second-order Darboux transformations for the one dimensional Schrodinger equation are described. The main feature of such transformations is that the transformation functions have eigenvalues greater then the ground state energy of the initial Hamiltonian. These transformations lead to a second-derivative supersymmetric models with unusual properties of supersymmetry breakdown.

Projective Hilbert space structures at exceptional points

A non-Hermitian complex symmetric 2 × 2-matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseuxexpanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behaviour in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase-jump behaviour are analysed, and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT symmetrically extended quantum mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.

New features in supersymmetry breakdown in quantum mechanics

The supersymmetric quantum mechanical model based on higher-derivative supercharge operators possessing unbroken supersymmetry and discrete energies below the vacuum state energy is described. As an example harmonic oscillator potential is considered.

Naimark-Dilated PT-Symmetric Brachistochrone

The quantum mechanical brachistochrone system with a PT-symmetric Hamiltonian is Naimark-dilated and reinterpreted as a subsystem of a Hermitian system in a higher-dimensional Hilbert space. This opens a way to a direct experimental implementation of the recently hypothesized PT-symmetric ultrafast brachistochrone regime of Bender et al. [Phys. Rev. Lett. 98, 040403 (2007)] in an entangled two-spin system.

Intertwining technique for the one-dimensional stationary Dirac equation

The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac Hamiltonians, quadratic supersymmetry, closed extension of transformation operators, chains of transformations, and finally particular cases of pseudoscalar and scalar potentials. The method is widely illustrated by numerous examples.

Nonlocal supersymmetric deformations of periodic potentials

Irreducible second-order Darboux transformations are applied to the periodic Schrodinger operators. It is shown that for the pairs of factorization energies inside the same forbidden band they can create new nonsingular potentials with periodicity defects and bound states embedded in the spectral gaps. The method is applied to the Lame and periodic piece-wise transparent potentials. An interesting phenomenon of translational Darboux invariance reveals nonlocal aspects of the supersymmetric deformations.

Supersymmetry of a nonstationary Schrödinger equation

Abstract The supersymmetry of a one-dimensional time-dependent Schrodinger equation is established. It is intimately connected with the time-dependent Darboux transformation. With the help of this transformation new exactly solvable time-dependent potentials are generated.

Chains of Darboux transformations for the matrix Schrödinger equation

Chains of Darboux transformations for the matrix Schr¨ odinger equation are considered. A matrix generalization of the well-known for the scalar equation Crum-Krein formulae for the resulting action of such chains is given.