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Mathematics of the USSR-Sbornik, 1988, Volume 59, Issue 1, Pages 53–73
DOI: https://doi.org/10.1070/SM1988v059n01ABEH003124
(Mi sm1903)
 

This article is cited in 7 scientific papers (total in 7 papers)

Singular multiparameter differential operators. Expansion theorems

G. A. Isaev
References:
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
Russian version:
Matematicheskii Sbornik. Novaya Seriya, 1986, Volume 131(173), Number 1(9), Pages 52–72
Bibliographic databases:
UDC: 517.98
MSC: Primary 47A70, 34B25, 35P10; Secondary 42C15, 47E05, 47F05
Language: English
Original paper language: Russian
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
Citation in format AMSBIB
\Bibitem{Isa86}
\by G.~A.~Isaev
\paper Singular multiparameter differential operators. Expansion theorems
\jour Mat. Sb. (N.S.)
\yr 1986
\vol 131(173)
\issue 1(9)
\pages 52--72
\mathnet{http://mi.mathnet.ru/sm1903}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=868601}
\zmath{https://zbmath.org/?q=an:0631.34029|0617.34011}
\transl
\jour Math. USSR-Sb.
\yr 1988
\vol 59
\issue 1
\pages 53--73
\crossref{https://doi.org/10.1070/SM1988v059n01ABEH003124}
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  • https://www.mathnet.ru/eng/sm/v173/i1/p52
  • This publication is cited in the following 7 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Математический сборник (новая серия) - 1964–1988 Sbornik: Mathematics
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    References:93
     
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