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Name:
Znojil, Miloslav
Reviewer number:
9689
Email:
znojil@ujf.cas.cz
Item's zbl-Number:
DE 015 568 717
Author(s):
Coskun, Haskiz; Harris, B. J.
Shorttitle:
Estimates for the eigenvalues for Hill's equation
Source:
Proc. R. Soc. Edinb., Sect. A, Math. 130, No. 5, 991-998 (2000)
Classification:
34L20Asymptotic distribution of eigenvalues, asymptotic theory of eigenfunctions
34L40Particular operators Dirac, one-dimensional Schroedinger, etc.
34B30Special equations Mathieu, Hill, Bessel, etc.
34C25Periodic solutions
Primary Classification:
Secondary Classification:
Keywords:
Hill's equation, spectral asymptotics, periodic and semi-periodic boundary conditions, non-smooth potential, Sturm-Liouville problems with continuous spectra
Review:

Schroedinger 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 the
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 the Hochstad'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.
Remarks to the editors: