Eigenfunctions and their asymptotic behaviour
at large distances for the Laplace operator with
singular potential, the support of which is on a
circular conical surface in three-dimensional space,
are studied. Within the framework of incomplete
separation of variables an integral representation
of the Kontorovich–Lebedev (KL) type for the
eigenfunctions is obtained in terms of solution of
an auxiliary functional difference equation with a
meromorphic potential. Solutions of the functional
difference equation are studied by reducing it to
an integral equation with a bounded self-adjoint
integral operator. To calculate the leading term of
the asymptotics of eigenfunctions, the KL integral
representation is transformed to a Sommerfeld-type
integral which is well adapted to application of the
saddle point technique. Outside a small angular
vicinity of the so-called singular directions the
asymptotic expression takes on an elementary form
of exponent decreasing in distance. However, in
an asymptotically small neighbourhood of singular
directions, the leading term of the asymptotics also
depends on a special function closely related to the
function of parabolic cylinder (Weber function).