Eigenoscillations in an angular domain and spectral properties of functional equations

• DOI: 10.1017/s0956792521000115
• Publication Date: 2021-05-06T06:30:32Z
• Authors: M. A. LYALINOV
• Keywords: Eigenfunction; Operator (biology); Spectral Theory

This work studies functional difference equations of the second order with a potential belonging to special class meromorphic functions. The depend on spectral parameter. Consideration this type is motivated by applications in diffraction theory and construction eigenfunctions for Laplace operator angular domains. In particular, such describe eigenoscillations acoustic waves domains ‘semitransparent’ boundary conditions. For negative values parameter, we study essential discrete spectrum properties corresponding solutions. based reduction integral symmetric kernel. A sufficient condition formulated that ensures existence spectrum. obtained results are applied studying behaviour adjacent Robin-type conditions their common boundary. At infinity, vanish exponentially as was expected. However, rate decay depends observation direction. vicinity some directions, regime switched from one another asymptotic described Fresnel-type integral.