Left-definite spaces of singular Sturm-Liouville problems (Q931012)

Consider the left definite (LD) singular Sturm-Liouville problems (SLPs) \[ ly:= -(py')' + qy = \lambda \omega y \quad\text{on } I = [a, b), \qquad -\infty < a < b \leq \infty \] with certain boundary conditions (BCs) that are self-adjoint in the right-definite (RD) equation associated with the SLPs: \[ -(py')' + qy = \lambda | \omega| y \quad\text{on } I, \] where \(1/p, q, \omega \in L_{\text{loc}}^1 (I, \mathbb R), | \omega | > 0, p > 0\) a.e., and \(\omega\) changes sign on \(I\). The associated RD equations are assumed to be limit-circle and non-oscillatory at \(b\). The space \(H_R := L^2 (I, | \omega| )\) is always used to study the RD equation. Let \(D \subset H_R\) be a subspace realizing the SLP, then \((f, g) = ( | \omega | ^{-1} l f, g) = \int_I [-(pf')' + qf] \bar g \,dt\) defines an inner product and a norm \(\| \cdot \| _L\) on \(D\). Denote by \(H_L\) the completion of \(D\) under this norm, it is called the left-definite Hilbert space. The authors give an explicit description of \(H_L\). The construction uses an explicit form of the LD boundary conditions, together with a principle and a non-principle solutions of the SLP involved.