Sergei Petrovich Maydanyuk (Institute for Nuclear Research NAS Ukraine, Kyiv, Ukraine)
New methods of deformation of potentials and their spectral characteristics with discrete spectra
Abstract:
In paper a generalization on definition of superpotential, connecting two one-dimensional potentials V1 and
V2 with discrete energy spectra, where function of factorization is defined concerning unbound state or
"discontinuous" state (i. e. the state for potential with discontinues) at arbitrary energy of factorization (which can
be non-coincident with the lowest energy level), is analyzed.
On the basis of Darboux transformations [1] and with taking into account of such superpotential
generalization, two new methods of double SUSY-transformations with coincident (method A) and with different
(method B) energies of transformation, allowing to deform the given potential V1 and its spectral
characteristics with obtaining exact solutions, are constructed (foundations of these methods are putted in
[2]). One can use these methods for calculation of wave functions and energy spectra at the
following elementary types of the deformation:
- change of slope of wave function of bound state at arbitrary selected level at selected boundary point (with keeping all
levels),
-
displacement of arbitrary selected level (with keeping the other levels).
With a purpose to prove the methods, as the given V1 a rectangular well with finite width and infinitely high walls
is chosen. Here, by the method A we reconstruct all pictures of deformation (without levels displacement), which were
obtained early by methods of inverse problem [3]. Interdependence between parameters of the
deformation for the methods of SUSY QM and the inverse problem has found, an analysis of a behavior of wave functions and
the potential under the deformation has fulfilled, a classification has proposed for zero-points of the potential, nodes of
the deformed wave functions, points, where wave functions are not deformed, an analysis of angles of wave functions leaving
from such points has fulfilled.
Using the method B, new deformations of the rectangular well are found with its transformation into one-, two- or
many-well potential, restricted by the rectangular well width.
A formalism of construction of n-parametric family of isospectral potentials has developed
[4].
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V. G. Bagrov and B. F. Samsonov,
Darboux transformation of the Schrödinger equation,
Physics of Particles and Nuclei
28 (4), 374-397 (1997).
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S. P. Maydanyuk,
Search of a general form of superpotential in hierarchy with discrete energy spectrum,
57 p., hep-th/0512034.
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B. N. Zakhariev, N. A. Kostov and E. B. Plehanov,
Exactly solbable one- and manychannel models
(Quantum intuition lessons),
Physics of elementary particles and atomic nuclei
21 (Iss. 4), 914-962 (1990).
- [4]
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F. Cooper, A. Khare and U. Sukhatme,
Supersymmetry and quantum mechanics,
Physics Reports 251 (5-6), 267-385 (1995);
hep-th/9405029.