Functional difference equations and eigenfunctions of a Schrödinger operator with δ' -interaction on a circular...

Area of Science:

• Mathematical Physics
• Partial Differential Equations
• Spectral Theory

Background:

• The Laplace operator with singular potentials on surfaces presents complex analytical challenges.
• Understanding eigenfunction behavior is crucial for wave phenomena and quantum mechanics.

Purpose of the Study:

• To investigate the asymptotic behavior of eigenfunctions for the Laplace operator with a singular potential on a circular conical surface in 3D space.
• To develop and apply analytical techniques for characterizing these eigenfunctions at large distances.

Main Methods:

• Utilized incomplete separation of variables to derive an integral representation of the Kontorovich-Lebedev (KL) type.
• Reduced an auxiliary functional difference equation to an integral equation with a bounded self-adjoint operator.
• Employed the saddle point technique on a Sommerfeld-type integral for asymptotic analysis.

Main Results:

• Obtained an integral representation for eigenfunctions in terms of solutions to a functional difference equation.
• Derived an asymptotic expression that simplifies to an exponentially decreasing form outside singular directions.
• Identified a dependence on the parabolic cylinder (Weber) function near singular directions.

Conclusions:

• The study provides a detailed characterization of eigenfunction asymptotics for Laplace operators with conical singularities.
• The findings highlight the significant influence of singular directions on the large-distance behavior of eigenfunctions.
• The developed methods offer a framework for analyzing similar problems in mathematical physics.