Left definite theory for second order differential operators with mixed boundary conditions (Q1891309)

The author considers the boundary-value problem of \([a, b]\) defined by \[ w\ell[ y]\equiv (py')'+ qy= \lambda wy+ wf,\;A\Bigl(\begin{smallmatrix} y(a)\\ py'(a)\end{smallmatrix}\Bigr)+ B\Bigl(\begin{smallmatrix} y(b)\\ py'(b)\end{smallmatrix}\Bigr)= 0, \] where \(1/p\), \(q\), \(w> 0\) are integrable on \([a, b]\), \(q/w\) is bounded below, and the coefficient matrices \(A\), \(B\) satisfy rank \((A, B)= 2\) and the self-adjointness criterion \(AJA^*= BJB^*\) with \(J= \left(\begin{smallmatrix} 0\\ 1\end{smallmatrix} \begin{smallmatrix} - 1\\ 0\end{smallmatrix}\right)\). The operator realization of the problem in \(L^2(a, b; w)\) is self-adjoint with a discrete spectrum. A description is given as a left-definite problem in a space of Sobolev-type \(H^1\) whose inner-product is given by the Dirichlet formula for \(\int^b_a \ell[y]\overline z wdx\). The three classes \(\left(\begin{smallmatrix} - \alpha_{12}\\ -\alpha_{22}\end{smallmatrix} \begin{smallmatrix} \beta_{12}\\ \beta_{22}\end{smallmatrix}\right)\) (where \(A= (\alpha_{ij})\), \(B= (\beta_{ij})\)) being non-singular, singular but non-zero, and zero are treated separately.