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This article is cited in 7 scientific papers (total in 7 papers)
Singular multiparameter differential operators. Expansion theorems
G. A. Isaev
Abstract:
A multiparameter spectral problem of the form
$$
l_j(y_j)+\sum_{k=1}^n\lambda_kb_{jk}(x_j)y_j(x_j)=0,\quad-\infty\leqslant a_j<x_j<b_j\leqslant+\infty,\quad j=1,2,\dots,n,
$$
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.
Received: 18.05.1984
Citation:
G. A. Isaev, “Singular multiparameter differential operators. Expansion theorems”, Mat. Sb. (N.S.), 131(173):1(9) (1986), 52–72; Math. USSR-Sb., 59:1 (1988), 53–73
Linking options:
https://www.mathnet.ru/eng/sm1903https://doi.org/10.1070/SM1988v059n01ABEH003124 https://www.mathnet.ru/eng/sm/v173/i1/p52
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Abstract page: | 439 | Russian version PDF: | 114 | English version PDF: | 14 | References: | 93 |
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