Geodesic
Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space
Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 3, pp. 466-472 
Yu. G. Shondin. Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space. Teoretičeskaâ i matematičeskaâ fizika, Tome 92 (1992) no. 3, pp. 466-472. http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a8/
@article{TMF_1992_92_3_a8,
     author = {Yu. G. Shondin},
     title = {Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {466--472},
     year = {1992},
     volume = {92},
     number = {3},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a8/}
}
TY  - JOUR
AU  - Yu. G. Shondin
TI  - Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 1992
SP  - 466
EP  - 472
VL  - 92
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a8/
LA  - en
ID  - TMF_1992_92_3_a8
ER  - 
%0 Journal Article
%A Yu. G. Shondin
%T Perturbation of differential operators on high-codimension manifold and the extension theory for symmetric linear relations in an indefinite metric space
%J Teoretičeskaâ i matematičeskaâ fizika
%D 1992
%P 466-472
%V 92
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_1992_92_3_a8/
%G en
%F TMF_1992_92_3_a8

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

The problem of realization of nontrivial perturbations supported on thin sets of “codimension” $\nu$ in $R^n$ for elliptic operators of order $m$, when $\nu\geqslant 2m$, is formulated as one of construction of the self-adjoint extensions of some symmetric linear relation in an indefinite metric space. The self-adjoint extensions and their resolvents are described. It is found that the same extensions can be obtained as a result of extensions of some symmetric operator in $L_2(R^n)$ with outgoing to a larger indefinite metric space. But such operator is picked out already by the “nonlocal” boundary conditions. Applications to quantum models of point interactions are discussed.

[1] Pavlov B. S., “The extension theory and explicitly soluble models”, Uspechi Mat. Nauk, 42:6 (1987), 99–131 | MR | Zbl

[2] Shirokov Y. M., “Algebra of three-dimensional generalized functions”, Teor. Mat. Fiz., 40:3 (1979), 348–354 | DOI | MR | Zbl

[3] Albeverio S., Gestezy F., Hoegh-Krohn R., Holden H., Solvable models in quantum mechanics, Springer, New York, 1988 | MR | Zbl

[4] Hepp K., Theorie de la renormalization, Springer, New York, 1969 | MR | Zbl

[5] Berezin F. A., “To the Lee model”, Mat. Sborn., 60:4 (1963), 425–446 | MR | Zbl

[6] Shondin Yu. G., “Quantum mechanical models in $R^n$ connected to the extensions of the energy operator in Pontrjagin space”, Teor. Mat. Fiz., 74:3 (1988), 331–344 | DOI | MR | Zbl

[7] Krein M. G., Langer H., “About defect subspaces and generalized resolvents of Hermitean operator in the space $\Pi_\kappa$”, Funkt. Anal. i Priloz., 5:2 (1971), 59–71 ; 3, 54–69 | MR | Zbl | MR | Zbl

[8] Krein M. G., Langer H., “Über die $Q$-Funktion eines $\pi$-hermiteschen Operators in Raume $\Pi_\kappa$”, Acta Sci. Math., 34 (1973), 191–230 | MR | Zbl

[9] Arens R., “Operational calculus of linear relations”, Pacific J. Math., 11:1 (1961), 9–23 | DOI | MR | Zbl