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A note on eigenvalues of ordinary differential operators.

Alan Ho (1997)

Revista Matemática Iberoamericana

In this follow-up on the work of Fefferman-Seco [FS] an improved condition for the discrete eigenvalues of the operator -d2 / dx2 + V(x) is established for V(x) satisfying certain hypotheses. The eigenvalue condition in [FS] establishes eigenvalues of this operator to within a small error. Through an obervation due to C. Fefferman, the order of accuracy can be improved if a certain condition is true. This paper improves on the result obtained in [FS] by showing that this condition does indeed hold....

Asymptotic analysis of non-self-adjoint Hill operators

Oktay Veliev (2013)

Open Mathematics

We obtain uniform asymptotic formulas for the eigenvalues and eigenfunctions of the Sturm-Liouville operators L t (q) with a potential q ∈ L 1[0,1] and t-periodic boundary conditions, t ∈ (−π, π]. Using these formulas, we find sufficient conditions on the potential q such that the number of spectral singularities in the spectrum of the Hill operator L(q) in L 2(−∞,∞) is finite. Then we prove that the operator L(q) has no spectral singularities at infinity and it is an asymptotically spectral operator...

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.

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