Andrey Pupasov

Singular matrix Darboux transformations in the inverse-scattering method

Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented determinant formulas for the resulting action of the chain. A determinant representation of the Kohlhoff-von Geramb solution to the Marchenko equation is given.

SUSY transformation of the Green function and a trace formula

An integral relation is established between the Green functions corresponding to two Hamiltonians which are supersymmetric (SUSY) partners and in general may possess both discrete and continuous spectra. It is shown that when the continuous spectrum is present, the trace of the difference of the Green functions for SUSY partners is a finite quantity which may or may not be equal to zero despite the divergence of the traces of each Green function. Our findings are illustrated by using the free particle example considered both on the whole real line and on a half line.

Exact propagators for complex SUSY partners of real potentials

A method for calculating exact propagators for those complex potentials with a real spectrum which are SUSY partners of real potentials is presented. It is illustrated by examples of propagators for some complex SUSY partners of the harmonic oscillator and zero potentials.

Exact propagators for SUSY partners

Pairs of SUSY partner Hamiltonians are studied which are interrelated by usual (linear) or polynomial supersymmetry. Assuming the model of one of the Hamiltonians as exactly solvable with a known propagator, the expressions for propagators of partner models are derived. The corresponding general results are applied to ‘a particle in a box’, the harmonic oscillator and a free

Coupling between scattering channels with SUSY transformations for equal thresholds

Supersymmetric (SUSY) transformations of the multichannel Schr¨ odinger equation with equal thresholds and arbitrary partial waves in all channels are studied. The structures of the transformation function and the superpotential are analysed. Relations between Jost and scattering matrices of superpartner potentials are obtained. In particular, we show that a special type of SUSY transformation allows us to introduce a coupling between scattering channels starting from a potential with an uncoupled scattering matrix. The possibility for this coupling to be trivial is discussed. We show that the transformation introduces bound and virtual states with a definite degeneracy at the factorization energy. A detailed study of the potential and scattering matrices is given for the 2 × 2 case. The possibility of inverting coupledchannel scattering data by such a SUSY transformation is demonstrated by several examples (s–s, s–p and s–d partial waves).

Reconstructing the nucleon-nucleon potential by a new coupled-channel inversion method.

A second-order supersymmetric transformation is presented, for the two-channel Schrödinger equation with equal thresholds. It adds a Breit-Wigner term to the mixing parameter, without modifying the eigenphase shifts, and modifies the potential matrix analytically. The iteration of a few such transformations allows a precise fit of realistic mixing parameters in terms of a Padé expansion of both the scattering matrix and the effective-range function. The method is applied to build an exactly solvable potential for the neutron-proton (3)S1-(3)D1 case.

Eigenphase preserving two-channel SUSY transformations

We propose a new kind of supersymmetric (SUSY) transformation in the case of the two-channel scattering problem with equal thresholds for partial waves of the same parity. This two-fold transformation is based on two imaginary factorization energies with opposite signs and with mutually conjugated factorization solutions. We call it an eigenphase preserving SUSY transformation as it relates two Hamiltonians, the scattering matrices of which have identical eigenphase shifts. In contrast to known phase-equivalent transformations, the mixing parameter is modified by the eigenphase preserving transformation.

EXACTLY-SOLVABLE COUPLED-CHANNEL MODELS FROM SUPERSYMMETRIC QUANTUM MECHANICS

Starting from a system of N radial Schrodinger equations with a vanishing potential and finite threshold differences between the channels, a coupled N × N exactly-solvable potential model is obtained with the help of a single non-conservative supersymmetric transformation. The obtained potential matrix, which subsumes a result obtained in the literature, has a compact analytical form, as well as its Jost matrix. It depends on N(N + 1)/2 unconstrained parameters and on one upper-bounded parameter, the factorization energy. For N = 2, previous results are reviewed, in particular regarding the number of bound states and resonances of the potential. A schematic inverse problem with one resonance is considered.

Spectral properties of non-conservative multichannel SUSY partners of the zero potential

Spectral properties of a coupled N × N potential model obtained with the help of a single non-conservative supersymmetric (SUSY) transformation starting from a system of N radial Schrodinger equations with the zero potential and finite threshold differences between the channels are studied. The structure of the system of polynomial equations which determine the zeros of the Jost-matrix determinant is analyzed. In particular, we show that the Jost-matrix determinant has N2N−1 zeros which may all correspond to virtual states. The number of bound states satisfies 0 ≤ nb ≤ N. The maximal number of resonances is nr = (N − 1)2N−2. A perturbation technique for a small coupling approximation is developed. A detailed study of the inverse spectral problem is given for the 2 × 2 case.

Exact Propagators for Soliton Potentials

Using the method of Darboux transformations (or equivalently supersymmet- ric quantum mechanics) we obtain an explicit expression for the propagator for the one- dimensional Schrodinger equation with a multi-soliton potential.