Spectral properties, including spestral singularities, of a quadratic pencil of Schrödinger operators on the whole real axis

Abstract

We consider the Quadratic Pencil of Schrodinger Operator L
generated in L²-(R) by the differential expression

l (y) = –y"; +
[q(x) + 2λp(x)–λ²=]
y, x ∈ R=(–∞,∞)

where p and q are complex valued functions.
Using the uniqueness theorems
of an­alytic functions. we investigate the dependence of the structure of
eigenvalues and spectral sirtgularities of L on the behavior of p and q at
infinity. We also obtain the conditions on p and q under which the operator L has a finite number of eigenvalues and spectral singularities finite
multiplicities The results about the discrete spectrum of L are applied to
non-selfadjoint Sturn-Liouville and Klein-Gordon op­erators on the whole real
axis.

Mathematics Subject Classification (2000): 34B20. 34B05. 34B25, 47A70,
47E05.

Key words: Singular operators. Sturn-Liouville problem. boundary value
problem. Weyl theory


Quaestiones Mathematicae 26(2003), 15-30