Spectrum and Spectral Singularities of a Quadratic Pencil of a Schrödinger Operator with Boundary Conditions Dependent on the Eigenparameter

In this paper, we consider the following quadratic pencil of Schrödinger operators L (λ) generated in L 2 ( ℝ + ) by the equation with the boundary condition y ′ ( 0 ) y ( 0 ) = β 1 λ + β 0 α 1 λ + α 0 , where p ( x )and q ( x ) are complex valued functions and α 0 , α 1 , β 0 , β 1 are complex numb...

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Bibliographic Details
Published in:Acta mathematica Sinica. English series 2023-11, Vol.39 (11), p.2164-2180
Main Authors: Zhu, Xiang, Zheng, Zhao Wen, Li, Kun
Format: Article
Language:English
Subjects:
Boundary conditions
Complex numbers
Eigenvalues
Formulas (mathematics)
Mathematics
Mathematics and Statistics
Operators (mathematics)
Singularities
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Summary:In this paper, we consider the following quadratic pencil of Schrödinger operators L (λ) generated in L 2 ( ℝ + ) by the equation with the boundary condition y ′ ( 0 ) y ( 0 ) = β 1 λ + β 0 α 1 λ + α 0 , where p ( x )and q ( x ) are complex valued functions and α 0 , α 1 , β 0 , β 1 are complex numbers with α 0 β 1 − α 1 β 0 ≠ 0 . It is proved that L (λ) has a finite number of eigenvalues and spectral singularities, and each of them is of a finite multiplicity, if the conditions and sup 0 ≤ x < + ∞ { e ε x [ | p ′ ( x ) | + | q ′ ′ ( x ) | ] } < + ∞ hold, where ε > 0.
ISSN:1439-8516
1439-7617
DOI:10.1007/s10114-023-1413-6