A singular Sturm–Liouville equation under homogeneous boundary conditions

Given α >0 and f ∈ L2(0, 1), we are interested in the equation − x2αu (x) +u(x) = f (x) on (0, 1], u(1) = 0. We prescribe appropriate (weighted) homogeneous boundary conditions at the origin and prove the existence and uniqueness of H2 loc(0, 1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator Lu := −(x2αu ) +u under those appropriate homogeneous boundary conditions. © 2011 Elsevier Inc. All rights reserved