  We consider the eigenvalue equation for the Laplace-Beltrami operator acting
on scalar functions on the non-compact Eguchi-Hanson space. The corresponding
differential equation is reducible to a confluent Heun equation with Ince
symbol [0,2,1_2]. We construct approximations for the eigenfunctions and their
asymptotic scattering phases with the help of the Liouville-Green approximation
(WKB). Furthermore, for specific discrete eigenvalues obtained by a continued
T-fraction we construct the solution by the Frobenius method and determine its
scattering phase by a monodromy computation.
