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Trace formulas for the schroedinger operator

โœ Scribed by Peter D. Lax

Publisher
John Wiley and Sons
Year
1994
Tongue
English
Weight
251 KB
Volume
47
Category
Article
ISSN
0010-3640

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โœฆ Synopsis

In this note we give an exceedingly simple, although not quite rigorous, derivation of the well-known trace formula expressing a periodic potential in a onedimensional Schroedinger operator in terms of its periodic, antiperiodic, and Dirichlet respectively Neumann spectrum. The items listed in the bibliography deal with results of this kind. Then we use our method to suggest a trace formula for the potential in a two-dimensional Schroedinger operator.

The idea is to deform the potential. Thus we deal with one-parameter families L(t) of selfadjoint operators that have a common domain, and which depend differentiably on r. We take the case that L(t) has a discrete spectrum {A,l}, tending

๐Ÿ“œ SIMILAR VOLUMES

Notes on the Schroedinger operator
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โœ F Gesztesy; H Holden; B Simon; Z Zhao ๐Ÿ“‚ Article ๐Ÿ“… 1996 ๐Ÿ› Elsevier Science ๐ŸŒ English โš– 902 KB

We prove multidimensional analogs of the trace formula obtained previously for one-dimensional Schro dinger operators. For example, let V be a continuous function on [0, 1] & /R & . For A/[1, ..., &], let &2 A be the Laplace operator on [0, 1] & with mixed Dirichlet Neumann boundary conditions .(x)=