Geodesic
On the Similarity of Some Differential Operators to Self-Adjoint Ones
Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 292-302.

Voir la notice de l'article provenant de la source Math-Net.Ru

The paper is devoted to the study of the similarity to self-adjoint operators of operators of the form $L=-\frac {\operatorname {sign}x}{|x|^\alpha p(x)}\,\frac {d^2}{dx^2}$, $\alpha >-1$, in the space $L_2(\mathbb R)$ with weight $|x|^\alpha p(x)$. As is well known, the answer to this problem in the case $p(x)\equiv 1$ is positive; it was obtained by using delicate methods of the theory of Hilbert spaces with indefinite metric. The use of a general similarity criterion in combination with methods of perturbation theory for differential operators allows us to generalize this result to a much wider class of weight functions $p(x)$. 
@article{MZM_2002_72_2_a12,
     author = {M. M. Faddeev and R. G. Shterenberg},
     title = {On the {Similarity} of {Some} {Differential} {Operators} to {Self-Adjoint} {Ones}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {292--302},
     publisher = {mathdoc},
     volume = {72},
     number = {2},
     year = {2002},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a12/}
}
TY  - JOUR
AU  - M. M. Faddeev
AU  - R. G. Shterenberg
TI  - On the Similarity of Some Differential Operators to Self-Adjoint Ones
JO  - Matematičeskie zametki
PY  - 2002
SP  - 292
EP  - 302
VL  - 72
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a12/
LA  - ru
ID  - MZM_2002_72_2_a12
ER  - 
%0 Journal Article
%A M. M. Faddeev
%A R. G. Shterenberg
%T On the Similarity of Some Differential Operators to Self-Adjoint Ones
%J Matematičeskie zametki
%D 2002
%P 292-302
%V 72
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a12/
%G ru
%F MZM_2002_72_2_a12
M. M. Faddeev; R. G. Shterenberg. On the Similarity of Some Differential Operators to Self-Adjoint Ones. Matematičeskie zametki, Tome 72 (2002) no. 2, pp. 292-302. http://geodesic.mathdoc.fr/item/MZM_2002_72_2_a12/

[1] Fleige A., Najman B., “Nonsingularity of critical points of some differential and difference operators”, Operators Theories: Advances and Appl., 106, 1998, 147–155 | MR | Zbl

[2] Langer H., “Spectral function of definitizable operators in Krein spaces”, Functional Analysis, Proceedings (Dubrovnik, 1981), Lecture Notes in Math., 948, Springer-Verlag, Berlin, 1982, 1–46

[3] Faddeev M. M., Shterenberg R. G., “O podobii nekotorykh singulyarnykh differentsialnykh operatorov samosopryazhennym”, Issledovaniya po lineinym operatoram i teorii funktsii. 28, Zapiski nauchnykh seminarov POMI, 270, ed. S. V. Kislyakov, POMI RAN, SPb., 2000, 336–349 | MR | Zbl

[4] Karabash I. M., “O $J$-samosopryazhennykh differentsialnykh operatorakh, podobnykh samosopryazhennym”, Matem. zametki, 68:6 (2000), 943–944 | MR | Zbl

[5] Malamud M. M., “O podobii treugolnogo operatora diagonalnomu”, Issledovaniya po lineinym operatoram i teorii funktsii. 28, Zapiski nauchnykh seminarov POMI, 270, ed. S. V. Kislyakov, POMI RAN, SPb., 2000, 201–241 | MR | Zbl

[6] Sakhnovich L. A., “Neunitarnye operatory s absolyutno nepreryvnym spektrom”, Izv. AN SSSR. Ser. matem., 33:1 (1969), 52–64 | MR | Zbl

[7] Sekefalvi-Nad B., Foyash Ch., Garmonicheskii analiz operatorov v gilbertovom prostranstve, Mir, M., 1970

[8] Van Casteren J., “Operators similar to unitary or self-adjoint ones”, Pacific J. Math., 104 (1983), 241–255 | MR | Zbl

[9] Naboko S. N., “Ob usloviyakh podobiya unitarnym i samosopryazhennym operatoram”, Funktsion. analiz i ego prilozh., 18:1 (1984), 16–27 | MR | Zbl

[10] Agranovich Z. S., Marchenko V. A., Obratnaya zadacha teorii rasseyaniya, Kharkov, 1960

[11] Blaschak V. V., “O razlozhenii po glavnym funktsiyam nesamosopryazhennogo differentsialnogo operatora vtorogo poryadka na vsei osi”, Dokl. AN URSR, 1965, no. 4, 416–419

[12] Blaschak V. V., “O differentsialnom operatore vtorogo poryadka na vsei osi so spektralnymi osobennostyami”, Dokl. AN URSR, 1966, no. 1, 38–41

[13] Lyantse V. E., “O nesamosopryazhennom differentsialnom operatore vtorogo poryadka na poluosi”, Dokl. AN SSSR, 154:5 (1964), 1030–1033 | MR | Zbl