A singular Sturm–Liouville equation under homogeneous boundary conditions

Given α>0α>0 and f∈L2(0,1)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 Hloc2(0,1] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator Lu:=−(x2αu′)′+uLu:=−(x2αu′)′+u under those appropriate homogeneous boundary conditions.