Left definite Sturm-Liouville problems with eigenparameter dependent boundary conditions
The authors study the Sturm-Liouville (SL) problem \[ -(py')'+qy = \lambda ry, \quad (a_j \lambda + b_j)y(j) = (c_j \lambda + d_j)(py')(j), \] with \((a_j,b_j,c_j,d_j) \neq 0, j=0,1.\) In their previous works the authors assumed a ``right definiteness'' condition: \(r>0\). Here, so called ``left definiteness'' conditions are supposed: (i) \(p>0, q \geq 0,\) on \([0,1]\) and \(1/p, q, r \in L^1[0,1], r \not \equiv 0\); (ii) either (in sections 3 and 4) \(a_0=c_0=0\) and the matrix with elements computed by means of \(a_1, b_1, c_1, d_1\) is positive definite or (in section 5) matrices whose elements are created by \(a_j, b_j, c_j, d_j\) are positive definite for \(j=0,1\). Then they introduce a modified Prüfer angle which takes advantage of the sign conditions on \(p\) and \(q\). Eigenvalues for the above SL problem when one boundary condition (at \(x=0\)) is \(\lambda \)-independent are studied in section 4. The authors prove oscillation properties of associated eigenfunctions, the interlacing and asymptotics of the eigenvalues and they give a variational principle governing them. In particular, they show that the eigenvalues are closely related to those of a ``right hand asymptotic'' SL problem where the boundary condition at \(x=1\) is \(\lambda \)-independent. Section 5 is devoted to problems in which both boundary conditions are \(\lambda \)-dependent while various exceptional cases are presented in section 6.
scientific article; zbMATH DE number 1858743