The Hill operators Ly=-y''+v(x)y, considered with singular complex valued
\pi-periodic potentials v of the form v=Q' with Q in L^2([0,\pi]), and subject
to periodic, antiperiodic or Neumann boundary conditions have discrete spectra.
For sufficiently large n, the disc {z: |z-n^2|<n} contains two periodic (if n
is even) or antiperiodic (if n is odd) eigenvalues \lambda_n^-, \lambda_n^+ and
one Neumann eigenvalue \nu_n. We show that rate of decay of the sequence
|\lambda_n^+-\lambda_n^-|+|\lambda_n^+ - \nu_n| determines the potential
smoothness, and there is a basis consisting of periodic (or antiperiodic) root
functions if and only if for even (respectively, odd) n, \sup_{\lambda_n^+\neq
\lambda_n^-}{|\lambda_n^+-\nu_n|/|\lambda_n^+-\lambda_n^-|} < \infty.Comment: arXiv admin note: substantial text overlap with arXiv:1207.094

Batal, Ahmet

English

arXiv

The Hill-Schrödinger operators, considered with singular complex valued periodic potentials, and subject to the periodic, anti-periodic or Neumann boundary conditions, have discrete spectra. For su ciently large integer n, the disk with radius n and with center square of n, contains two periodic (if n is even) or anti-periodic (if n is odd) eigenvalues and one Neumann eigenvalue. We construct two spectral deviations by taking the di erence of two periodic (or anti-periodic) eigenvalues and the difference of a periodic (or anti-periodic) eigenvalue and the Neumann eigenvalue. We show that asymptotic decay rates of these spectral deviations determine the smoothness of the potential of the operator, and there is a basis consisting of periodic (or anti-periodic) root functions if and only if the supremum of the absolute value of the ratio of these deviations over even (respectively, odd) n is  nite. We also show that, if the potential is locally square integrable, then in the above results one can replace the Neumann eigenvalues with the eigenvalues coming from a special class of boundary conditions more general than the Neumann boundary conditions

Sabanci University Research Database

Characterization of potential smoothness and riesz basis property of hill-schrödinger operators with singular periodic potentials in terms of periodic, antiperiodic and neumann spectra

http://research.sabanciuniv.edu/31361/1/AhmetBatal_10026049.pdf

Characterization of potential smoothness and Riesz basis property of
  Hill-Scr\"odinger operators with singular periodic potentials in terms of
  periodic, antiperiodic and Neumann spectra

Characterization of potential smoothness and Riesz basis property of Hill-Scr\"odinger operators with singular periodic potentials in terms of periodic, antiperiodic and Neumann spectra

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