9525575
Factorization of nonlinear supersymmetry in one-dimensional quantum mechanics. II: Proofs of theorems on reducibility
In this paper, we continue to study factorization of supersymmetric (SUSY) transformations in one-dimensional Quantum Mechanics into chains of elementary Darboux transformations with nonsingular coefficients. We define a class of potentials that are invariant under the Darboux-Crum transformations and prove a number of lemmas and theorems substantiating the formulated conjectures on reducibility of differential operators for spectral equivalence transformations. Analysis of the general case is performed with all the necessary proofs. Bibliography: 27 titles.
1279344
Resolutions of Identity for Some Non-Hermitian Hamiltonians. II. Proofs
This part is a continuation of the Part I where we built resolutions of identity for certain non-Hermitian Hamiltonians constructed of biorthogonal sets of their eigen- and associated functions for the spectral problem defined on entire axis. Non-Hermitian Hamiltonians under consideration are taken with continuous spectrum and the following cases are examined: an exceptional point of arbitrary multiplicity situated on a boundary of continuous spectrum and an exceptional point situated inside of continuous spectrum. In the present work the rigorous proofs are given for the resolutions of identity in both cases.
119315901
Spectral design for matrix Hamiltonians: different methods of constructing of a matrix intertwining operator*
We study relations describing the intertwining of n ? n matrix, in general non-Hermitian, one-dimensional Hamiltonians by matrix linear differential operators with nondegenerate coefficients of in the highest degree. Some methods for constructing matrix intertwining operators of the first order of general form are proposed and their interrelations are examined. We construct, as an example, a 2 ? 2 matrix Hamiltonian of general form intertwined by an operator of the first order with the Hamiltonian with zero matrix potential. It is shown that one can add, for the final 2 ? 2 matrix Hamiltonian with respect to the initial matrix Hamiltonian, with the help of an intertwining operator of first order, up to two bound states for different energy values, or up to two bound states described by vector-eigenfunctions for the same energy value, or up to two bound states described by a vector-eigenfunction and an associated vector-function for the same energy value.
119648611
Polynomial supersymmetry for matrix Hamiltonians
Abstract We study intertwining relations for matrix non-Hermitian Hamiltonians by matrix differential operators of arbitrary order. It is established that for any intertwining operator of minimal order there is operator that intertwines the same Hamiltonians in the opposite direction and such that the products of these operators are identical polynomials of the corresponding Hamiltonians. The related polynomial algebra of supersymmetry is constructed. The problems of minimization and reducibility of a matrix intertwining operator are considered and the criteria of minimizability and reducibility are presented. It is shown that there are absolutely irreducible matrix intertwining operators, in contrast to the scalar case.
119142504
Non-linear supersymmetry for non-Hermitian, non-diagonalizable Hamiltonians: II. Rigorous results
Abstract We continue our investigation of the non-linear SUSY for complex potentials started in part I [A.A. Andrianov, F. Cannata, A.V. Sokolov, math-ph/0610024 ] and prove the theorems characterizing its structure in the case of non-diagonalizable Hamiltonians. This part provides the mathematical basis of previous studies. The classes of potentials invariant under SUSY transformations for non-diagonalizable Hamiltonians are specified and the asymptotics of formal eigenfunctions and associated functions are derived. Several results on the normalizability of associated functions at infinities are rigorously proved. Finally the Index Theorem on relation between Jordan structures of intertwined Hamiltonians depending of the behavior of elements of canonical basis of supercharge kernel at infinity is proven.
123155745
Upgrade of the KEDR tagging system
An upgrade of a special system to tag scattered electrons from two-photon processes, based on triple-GEM detectors, is described. The system is intended for the experiments at the VEPP-4M storage ring. Tests of the detector prototype with two gas mixtures confirm the feasibility of the chosen upgrade layout.