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Estimates for the periodic and semi-periodic eigenvalues of Hill’s equation. (English) Zbl 0977.34077

The Schrödinger equation on a finite interval (with the standard periodic or anti-periodic boundary conditions) is of current use in condensed matter physics etc. Its spectral analysis is well developed. The paper (based on the first author’s PhD dissertation) contributes by the derivation of the asymptotic form of the eigenvalues without smoothness conditions imposed upon the (integrable) potential. The method is based on Hochstadt’s trick (re-arrangement of boundary conditions to the Dirichlet ones, somewhere within the interval) and theorem (the new eigenvalues are bracketed by the old ones). In this manner, the authors vary Hochstadt’s point and minimize and maximize his eigenvalues (obtained by the co-author’s Riccati-equation technique). After fairly complicated calculations (with full details available in the corresponding PhD thesis) they arrive at the known estimates, in this way having got rid of the redundant smoothness assumptions.

MSC:

34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34L20 Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions for ordinary differential operators
34C25 Periodic solutions to ordinary differential equations
34B24 Sturm-Liouville theory
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