A singular Sturm–Liouville equation under homogeneous boundary conditions

Given α > 0 and f ∈ L 2 ( 0 , 1 ) , we are interested in the equation { − ( x 2 α u ′ ( 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 H loc 2 ( 0 , 1 ] solutions. We study the regularity at the origin of such solutions. We perform a spectral analysis of the differential operator L u : = − ( x 2 α u ′ ) ′ + u under those appropriate homogeneous boundary conditions.