Right-definite multiparameter Sturm-Liouville problems with eigenparameter-dependent boundary conditions. (English) Zbl 1022.34009

Bhattacharyya, Tirthankar; Košir, Tomaž; Plestenjak, Bor

Proc. Edinb. Math. Soc., II. Ser. 45, No. 3, 565-578 (2002).

The authors study the distribution of eigenvalues of the multiparameter Sturm-Liouville system defined by \[ -y_{j}^{\prime \prime }+q_{j}y_{j}=\left( \sum_{j=1}^{n}\lambda _{k}r_{jk}\right) y_{j},\quad j=1,\dots n, \] with eigenparameter-dependent boundary conditions defined by \[ ( a_{j0}\lambda _{j}+b_{j0})y_j(0)= (c_{j0}\lambda _{j}+d_{j0}) y_j'(0), \] where \(q_{i}\), \(r_{jk}\in C^{1}[0,1]\) and \(r_{jk}\) satisfy the Minkowsky condition \(\int_{0}^{1}r_{jk}(x)|y(x)|^{2} dx\leq 0\) for \(j\neq k\) while \(\sum_{k=1}^{n} \int_{0}^{1}r_{jk}(x)|y(x)|^{2} dx>0\) for \(j=1, 2,\dots , n\). This gives rise to a right-definite regular Sturm-Liouville system whose spectrum, made of the vectors \((\lambda_{1}, \lambda _{2}, \dots , \lambda_{n})\in \mathbb{R}^{n}\), is discrete. Using the oscillation properties of the eigenfunctions, the intersection of hypersurfaces of eigenvalues are examined in the cases when either one or both boundary conditions depend on the eigenvalue parameter.

Reviewer: Amin Boumenir (Carrollton)

MSC:

34B08	Parameter dependent boundary value problems for ordinary differential equations
34B24	Sturm-Liouville theory

Keywords:

Sturm-Liouville; right-definite; multiparameter eigenvalue problem