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Discrete spectrum and principal functions of non-selfadjoint differential operator

Gülen Başcanbaz Tunca, Elgiz Bairamov (1999)

Czechoslovak Mathematical Journal

In this article, we consider the operator L defined by the differential expression ( y ) = - y ' ' + q ( x ) y , - < x < in L 2 ( - , ) , where q is a complex valued function. Discussing the spectrum, we prove that L has a finite number of eigenvalues and spectral singularities, if the condition sup - < x < exp ϵ | x | | q ( x ) | < , ϵ > 0 holds. Later we investigate the properties of the principal functions corresponding to the eigenvalues and the spectral singularities.

Eigensystem of an L 2-perturbed harmonic oscillator is an unconditional basis

James Adduci, Boris Mityagin (2012)

Open Mathematics

For any complex valued L p-function b(x), 2 ≤ p < ∞, or L ∞-function with the norm ‖b↾L ∞‖ < 1, the spectrum of a perturbed harmonic oscillator operator L = −d 2/dx 2 + x 2 + b(x) in L 2(ℝ1) is discrete and eventually simple. Its SEAF (system of eigen- and associated functions) is an unconditional basis in L 2(ℝ).

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