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Perturbation theory for the Schrödinger operator with a periodic potential. (English) Zbl 0883.35002

Lecture Notes in Mathematics. 1663. Berlin: Springer. vii, 352 p. (1997).
The main aim of this book is to construct perturbation formulae for Bloch eigenvalues and their spectral projections in a high energy region on a rich set of quasimomenta. The construction of these formulae is connected with the investigation of a complicated picture of the crystal diffraction. Another problem considered here is a semibounded crystal problem, i.e., the Schrödinger operator which has a zero potential in a half space and a periodic potential in the other half space. The interaction of a plane wave with a semicrystal is studied. First, the asymptotic expansion of the reflection coefficients in a high energy region is obtained, this expansion is valid for a rich set of momenta of the incident plane wave. Second, the connection of the asymptotic coefficients with the potential is established. Based upon these, the inverse problem is solved, this problem is to determine the potential from the asymptotics of the reflection coefficients in a high energy region (a crystallography problem).
The book consists of the following chapters: 1) Introduction. 2) Perturbation theory for a polyharmonic operator in the case of \(2\ell>n\). 3) Perturbation theory for the polyharmonic operator in the case \(4\ell>n+1\). 4) Perturbation theory for Schrödinger operator with a periodic potential. 5) The interaction of a free wave with a semibounded crystal. 6) References: 199 titles.

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35J10 Schrödinger operator, Schrödinger equation
35B20 Perturbations in context of PDEs
35P15 Estimates of eigenvalues in context of PDEs
35P20 Asymptotic distributions of eigenvalues in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
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