Sergei Petrovich Maydanyuk (Institute for Nuclear Research NAS Ukraine, Kyiv, Ukraine)

New methods of deformation of potentials and their spectral characteristics with discrete spectra

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:
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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