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Name:
Miloslav Znojil
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE015475912
Author(s):
Yuri Berest and Alexander Veselov
Shorttitle:
On the structure of singularities of integrable Schroedinger operators
Source:
Lett. Math. Phys. 52, No. 2, 103-111 (2000)
Classification:
47A68Factorization theory including Wiener-Hopf and spectral factorizations
35J10Schroedinger operator
58K05Critical points of functions and mappings
32S25Surface and hypersurface singularities
13N10Rings of differential operators and their modules
35Q40Equations from quantum mechanics
Primary Classification:
Secondary Classification:
Keywords:
n-dimensional Schroedinger operator, Darboux transformation, Hadamard problem, D-integrability
Review:

The classical nineteenth century Darboux transformation proved
extremely useful as an efficient tool of analysis in quantum
mechanics. Unfortunately, virtually all its applications (e.g., in
the Schroedinger's factorization algorithm, in the context of the
famous inverse scattering transformation or within the Witten's
supersymmetric quantum mechanics) shared a common restriction to
single dimension. In such a context, Berest and Veselov pay
attention to its n-dimensional version. They discover that among
all the partial differential equations, the Schroedinger equation
which is of the second order in derivatives remains exceptional
since the hypersurfaces of singularities of the Darboux-induced
potentials are just hyperplanes. This result could significantly
simplify the classification of the D-integrable models. It has
already inspired also a generalization to the Riemannian case,
with hyperplanes replaced by the totally geodesic hypersurfaces.


Remarks to the editors: