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 $$ l_j(y_j)+\sum_{k=1}^n\lambda_kb_{jk}(x_j)y_j(x_j)=0,\quad-\infty\leqslant a_j<x_j<b_j\leqslant+\infty,\quad j=1,2,\dots,n, $$ 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.

Bibliographie

[1] Isaev G. A., “Voprosy teorii samosopryazhennykh mnogoparametricheskikh zadach”, Spektralnaya teoriya operatorov, Tr. 2-i Vsesoyuznoi letnei matem. shkoly po spektr. teorii oper. (Zagulba, 1975), Elm, Baku, 1979, 87–102 | MR

[2] Vladimirov V. S., Uravneniya matematicheskoi fiziki, Nauka, M., 1971 | MR | Zbl

[3] Mors F. M., Feshbakh G., Metody teoreticheskoi fiziki, t. 1, IL, M., 1958

[4] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii, t. 3. Ellipticheskie i avtomorfnye funktsii, funktsii Lame i Mate, Nauka, M., 1967 | MR

[5] Arscott F. M., Periodic differential equations, Pergamon Press, Oxford, 1964 | Zbl

[6] Berezanskii Yu. M., Razlozhenie po sobstvennym funktsiyam samosopryazhennykh operatorov, Naukova Dumka, Kiev, 1965 | MR

[7] Rid M., Saimon B., Metody sovremennoi matematicheskoi fiziki, t. 1, Mir, M., 1977 | MR

[8] Atkinson F. V., Multiparameter eigenvalue problems, v. 1, Acad. Press, New York, 1972 | MR | Zbl

[9] Atkinson F. V., “Multiparameter spectral theory”, Bull. Amer. Math. Soc., 74 (1968), 1–27 | DOI | MR | Zbl

[10] Binding P., Browne P. J., “A variational approach to multiparameter eigenvalue problems for matrices”, SIAM J. Math. Anal., 8:5 (1977), 763–777 | DOI | MR | Zbl

[11] Hilbert D., Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen, Berlin, 1924

[12] Kurant R., Gilbert D., Metody matematicheskoi fiziki, t. 1, Gostekhizdat, M.-L., 1951

[13] Dixon A. C., “Harmonics expansions of functions of two variable”, Proc. London Math. Soc. Ser. II, 5 (1907), 411–478 | DOI | MR | Zbl

[14] Ains E. L., Obyknovennye differentsialnye uravneniya, ONTI, Kharkov, 1939

[15] Faierman M., The expansion theorem in multi-parameter Sturm–Lionville theory, Lect. Notes Math., 415, 1974, p. 137–142 | MR | Zbl

[16] Isaev G. A., Allakhverdiev B. P., “Ostsillyatsionnye teoremy dlya mnogoparametricheskikh spektralnykh zadach, svyazannykh s differentsialnymi uravneniyami vtorogo poryadka”, Spektralnaya teoriya operatorov, 3, Elm, Baku, 1980, 202–221 | MR

[17] Faierman M., “The completeness and expansion theorems associated with the multiparameter eigen-value problem in ordinary differential equations”, J. Diff. Equat., 5 (1969), 197–213 | DOI | MR | Zbl

[18] Browne P. J., “Abstract multi-parameter theory. I”, J. Math. Anal. AppL., 60 (1977), 259–273 | DOI | MR | Zbl

[19] Sleeman B. D., Multi-parameter spectral theory in Hilbert space, Res. Notes Math., Pitman, 1978, No 22 | Zbl

[20] Isaev G. A., “Geneticheskie operatory i mnogoparametricheskie spektralnye zadachi”, DAN SSSR, 268:4 (1983), 785–788 | MR | Zbl

[21] Isaev G. A., “Vvedenie v obschuyu mnogoparametricheskuyu spektralnuyu teoriyu”, Spektralnaya teoriya operatorov, 3, Elm, Baku, 1980, 142–201 | MR

[22] Browne P. J., “A singular multi-parameter eigenvalue problem in second order ordinary differential equations”, J. Diff. Equat., 12 (1972), 81–94 | DOI | MR | Zbl

[23] Browne P. J., Multi-parameter problems, Lect. Notes Math., 415, 1974, p. 78–84 | MR | Zbl

[24] Isaev G. A., “Razlozhenie po sobstvennym funktsiyam samosopryazhennykh singulyarnykh mnogoparametricheskikh differentsialnykh operatorov”, DAN SSSR, 260:4 (1981), 786–790 | MR | Zbl

[25] Levitan B. M., Sargsyan I. S., Vvedenie v spektralnuyu teoriyu, Nauka, M., 1970 | MR | Zbl

[26] Koddington E. A., Levinson N., Teoriya obyknovennykh differentsialnykh uravnenii, IL, M., 1958

[27] Ganning R., Rossi Kh., Analiticheskie funktsii mnogikh kompleksnykh peremennykh, Mir, M., 1969 | MR

[28] Danford N., Shvarts Dzh. T., Lineinye operatory, t. 2. Spektralnaya teoriya, Mir, M., 1966

[29] McShane E. L., Integration, Princeton, 1944 | MR | Zbl

[30] Shilov G. E., Gurevich B. L., Integral, mera, proizvodnaya. Obschaya teoriya, Nauka, M., 1967

[31] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[32] Isaev G. A., “K mnogoparametricheskoi spektralnoi teorii”, DAN SSSR, 229:2 (1976), 284–287 | MR | Zbl

[33] Naimark M. A., Lineinye differentsialnye operatory, Nauka, M., 1969 | MR