JournalsjstVol. 3, No. 2pp. 129–145

A spectral alternative for continuous families of self-adjoint operators

  • Alexander Y. Gordon

    University of North Carolina at Charlotte, USA
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Abstract

We consider a continuous family of self-adjoint operators AsA_s in a separable Hilbert space, the parameter ss being a point of a complete metric space SS. It is well known that isolated simple eigenvalues (assuming that the operators are bounded and the mapping sAss\mapsto A_s is continuous in the norm sense) behave "well'': under small changes of the parameter they do not disappear and change continuously. Unlike this, the eigenvalues embedded in the essential spectrum can display a very "bad'' behavior. It turns out, nevertheless, that if the set of eigenvalues is non-empty for a topologically rich (e.g., open) set of values of the parameter, then the (multi-valued) eigenvalue function has continuous branches. One application is as follows. Suppose a one-dimen\-sional quasi-periodic Schr\"{o}\-dinger operator has Cantor spectrum; then a Baire generic operator in its hull does not have eigenvalues.

Cite this article

Alexander Y. Gordon, A spectral alternative for continuous families of self-adjoint operators. J. Spectr. Theory 3 (2013), no. 2, pp. 129–145

DOI 10.4171/JST/40