Singular multiparameter differential operators. Expansion theorems

G. A. Isaev

Sbornik. Mathematics, Tome 59 (1988) no. 1, pp. 53-73

Résumé

A multiparameter spectral problem of the form $$ l_j(y_j)+\sum_{k=1}^n\lambda_kb_{jk}(x_j)y_j(x_j)=0,\quad-\infty\leqslant a_j\leqslant+\infty,\quad j=1,2,\dots,n, $$ is considered, where \begin{gather*} l_j(y_j)=(-1)^{k_j}(p_{j0}(x_j)y_j^{(k_j)}(x_j))^{(k_j)}+(-1)^{k_j-1}(p_{j1}(x_j)y_j^{(k_j-1)}(x_j))^{(k_j-1)}+\dots+ \\ +p_{j,2k_j}(x_j)y_j(x_j), \\ p_{js_j}\in C^{(2k_j-s_j)}((a_j,b_j)),\qquad b_{jk}\in C((a_j,b_j)),\qquad p_{j0}(x_j)\ne0, \end{gather*} and at least for one of these equations the endpoints $a_j$ and $b_j$ are singular, $$ s_j=0,1,\dots,2k_j,\qquad j=1,2,\dots,n,\qquad k=1,2,\dots,n, $$ all the functions $p_{js_j}$ and $b_{jk}$ are real-valued, and the following natural independence condition holds: $$ \det\{b_{jk}(x_j)\}_{j,k=1}^n>0,\qquad x_j\in(a_j,b_j). $$ The Parseval equality and the corresponding theorem on expansion in the eigenfunctions of this multiparameter problem are proved. The main results give, in a particular case, the solution of the problem on singular multiparameter operators of the Sturm–Liouville type on $(-\infty,\infty)$ posed by P. J. Browne in 1974. Bibliography: 33 titles.