TY - JOUR

T1 - Functional difference equations and eigenfunctions of a Schrödinger operator with δ'−interaction on a circular conical surface

AU - Lyalinov, Mikhail A.

PY - 2020/9/30

Y1 - 2020/9/30

N2 - Eigenfunctions and their asymptotic behaviourat large distances for the Laplace operator withsingular potential, the support of which is on acircular conical surface in three-dimensional space,are studied. Within the framework of incompleteseparation of variables an integral representationof the Kontorovich–Lebedev (KL) type for theeigenfunctions is obtained in terms of solution ofan auxiliary functional difference equation with ameromorphic potential. Solutions of the functionaldifference equation are studied by reducing it toan integral equation with a bounded self-adjointintegral operator. To calculate the leading term ofthe asymptotics of eigenfunctions, the KL integralrepresentation is transformed to a Sommerfeld-typeintegral which is well adapted to application of thesaddle point technique. Outside a small angularvicinity of the so-called singular directions theasymptotic expression takes on an elementary formof exponent decreasing in distance. However, inan asymptotically small neighbourhood of singulardirections, the leading term of the asymptotics alsodepends on a special function closely related to thefunction of parabolic cylinder (Weber function).

AB - Eigenfunctions and their asymptotic behaviourat large distances for the Laplace operator withsingular potential, the support of which is on acircular conical surface in three-dimensional space,are studied. Within the framework of incompleteseparation of variables an integral representationof the Kontorovich–Lebedev (KL) type for theeigenfunctions is obtained in terms of solution ofan auxiliary functional difference equation with ameromorphic potential. Solutions of the functionaldifference equation are studied by reducing it toan integral equation with a bounded self-adjointintegral operator. To calculate the leading term ofthe asymptotics of eigenfunctions, the KL integralrepresentation is transformed to a Sommerfeld-typeintegral which is well adapted to application of thesaddle point technique. Outside a small angularvicinity of the so-called singular directions theasymptotic expression takes on an elementary formof exponent decreasing in distance. However, inan asymptotically small neighbourhood of singulardirections, the leading term of the asymptotics alsodepends on a special function closely related to thefunction of parabolic cylinder (Weber function).

KW - functional difference equations

KW - asymptotics

KW - Robin Laplacians

KW - δ′ −interaction

KW - eigenfunctions of integral operator

UR - http://dx.doi.org/10.1098/rspa.2020.0179

M3 - Article

VL - 476

JO - PROC. R. SOC. - A.

JF - PROC. R. SOC. - A.

SN - 0950-1207

IS - 2241

ER -