On the Jost Solutions of A Class of the Quadratic Pencil of the Sturm-Liouville Equation

Abstract

In this study we construct new integral representations of Jost-type solutions of the quadratic pencil of the Sturm-Liouville equation with the piece-wise constant coefficient on the entire real line. Our aim is to express the special solutions of the Sturm-Liouville quadratic pencil in the form of some integral operators which kernels is related with the potential function of the Sturm-Liouville equation. This problem is technically diffucult due to the discontinous coefficient which causes the kernel function to also have a jump discontinuity.

Keywords

• Sturm-Liouville equation
• operator pencil
• transformation operator
• diferantial equation with discontinuous cofficients
• Jost solution

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