An impulsive problem for quadratic pencil of Schrödinger operators on semi-axis

• Published in: Journal of inequalities and applications Vol. 2025; no. 1; pp. 29 - 14
• Main Author: Cebesoy, Serifenur
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 04.03.2025; Springer Nature B.V; SpringerOpen
• Subjects: Analysis; Applications of Mathematics; Applied mathematics; Asymptotics; Eigenvalues; Impulsive boundary value problems; Impulsive conditions; Jost function; Mathematics; Mathematics and Statistics; Operators (mathematics); Quadratic pencil of Schrödinger equations; Resolvent operator; Schrodinger equation; Singularity (mathematics); 34L40; Jost function; 34K08; Impulsive boundary value problems; 34L05; Eigenvalues; 34L25; Resolvent operator; Asymptotics; Spectral singularities; Quadratic pencil of Schrödinger equations; Impulsive conditions
• ISSN: 1029-242X; 1025-5834; 1029-242X
• DOI: 10.1186/s13660-024-03247-6

The present article examines a quadratic pencil of Schrödinger operator given with an impulsive condition. The novelty of this study lies in its focus on the half-line and inclusion of an arbitrary single discontinuity point, distinguishing it from previous works in the literature. First, some solutions to the impulsive equation associated with so-called operator L are presented. Next, the Jost solutions and resolvent operator of L are defined. The primary work includes deriving an asymptotic equation for the function related to the Wronskian of the Jost solutions, providing sufficient conditions to ensure the finiteness of eigenvalues and spectral singularities, and demonstrating that their multiplicities are finite.