2,556 research outputs found
SUSY transformations with complex factorization constants. Application to spectral singularities
Supersymmetric (SUSY) transformation operators corresponding to complex
factorization constants are analyzed as operators acting in the Hilbert space
of functions square integrable on the positive semiaxis. Obtained results are
applied to Hamiltonians possessing spectral singularities which are
non-Hermitian SUSY partners of selfadjoint operators. A new regularization
procedure for the resolution of the identity operator in terms of continuous
biorthonormal set of the non-Hermitian Hamiltonian eigenfunctions is proposed.
It is also shown that the continuous spectrum eigenfunction has zero binorm (in
the sense of distributions) at the singular point.Comment: Thanks to A. Sokolov a number of inaccuracies are correcte
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
Supersymmetric transformations for coupled channels with threshold differences
The asymptotic behaviour of the superpotential of general SUSY
transformations for a coupled-channel Hamiltonian with different thresholds is
analyzed. It is shown that asymptotically the superpotential can tend to a
diagonal matrix with an arbitrary number of positive and negative entries
depending on the choice of the factorization solution. The transformation of
the Jost matrix is generalized to "non-conservative" SUSY transformations
introduced in Sparenberg et al (2006 J. Phys. A: Math. Gen. 39 L639). Applied
to the zero initial potential the method permits to construct superpartners
with a nontrivially coupled Jost-matrix. Illustrations are given for two- and
three-channel cases.Comment: 17 pages, 3 explicit examples and figures adde
Supersymmetry of the Nonstationary Schr\"odinger equation and Time-Dependent Exactly Solvable Quantum Models
New exactly solvable quantum models are obtained with the help of the
supersymmetric extencion of the nonstationary Schr/"odinger equation.Comment: Talk at the 8th International Conference "Symmetry Methods in
Physics". Dubna, Russia, 28 July - 2 August, 199
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 known propagator, 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
particle (i.e. to transparent potentials).Comment: 31 page
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