Abstract
In this paper, the sufficient conditions on summability degree p are found under which the Schrödinger operator with a potential, being singular on manifolds, is a positive one in Banach spaces Lp. It is also demonstrated that the domains of various degrees on this operator form an interpolation pair. In addition to this, the sufficient conditions on p are established which ensure that fractional powers σ, 0 < σ < 1, of this operator are bounded from \(W_{p}^{{2\sigma }}\) to Lp.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
V. A. Il’in, “Kernels of fractional order,” Mat. Sb. Novaya Ser. 41, 459–480 (1957).
Sh. A. Alimov, “Fractional powers of elliptic operators and isomorphism of classes of differentiable functions,” Differ. Uravn. 8, 1609–1626 (1972).
A. R. Khalmukhamedov, “Complex powers of the Schrödinger operator with singular potential,” in Contemporary Mathematics, Ed. by B. Russo, A. G. Aksoy, R. Ashurov, and Sh. Ayupov, Topics in Functional Analysis and Algebra, Vol. 672 (2016), pp. 205–215. https://doi.org/10.1090/conm/672/13473
V. A. Kostin and M. N. Nebol’sina, “Well-posedness of boundary value problems for a second-order equation,” Dokl. Math. 80, 650–652 (2009). https://doi.org/10.1134/S1064562409050044
A. V. Balakrishnan, “An operational calculus for infinitesimal generators of semigroups,” Trans. Am. Math. Soc 91, 330–353 (1959). https://doi.org/10.1090/S0002-9947-1959-0107179-0
A. V. Balakrishnan, “Fractional powers of closed operators and the semi-groups generated by them,” Pacif. J. Math. 10, 419–437 (1960).
M. A. Krasnosel’skii and P. E. Sobolevskii, “Fractional powers of operators acting in Banach spaces,” Dokl. Akad. Nauk SSSR 129, 499–502 (1959).
T. Kato, “Note on fractional powers of linear operators,” Proc. Jpn. Acad. 36, 94–96 (1960). https://doi.org/10.3792/pja/1195524082
Sh. Alimov, “Complex powers of the Schrödinger operator with singular potential,” Eurasian J. Math., No. 2, 4–12 (2007).
Sh. A. Alimov and A. R. Khalmukhamedov, “Estimates of spectral function of Schrödinger operator with potential satisfying the Kato condition,” Vestn. Nats. Univ. Uzbekistana, No. 3, 46–50 (2005).
M. A. Krasnosel’skii, P. P. Zabreiko, P. E. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions (Nauka, Moscow, 1966).
H. Triebel, Interpolation Theory, Function Spaces, Differential Operators (North-Holland, Amsterdam, 1978).
A. R. Khalmukhamedov and T. N. Alikulov, “On imaginary powers of Schrödinger operator in Banach space with potential singular on manifolds,” Vestn. Nats. Univ. Uzbekistana, No. 3, 110–119 (2007).
S. G. Mikhlin, Multidimensional Singular Integrals and Integral Equations (Fizmatgiz, Moscow, 1962).
S. M. Nikol’skii, Approximation of Multivariate Functions and Embedding Theorems, 2nd ed. (Nauka, Moscow, 1977).
S. G. Mikhlin, Linear Partial Differential Equations (Vysshaya Shkola, Moscow, 1977).
A. Zygmund, Trigonometric Series (Cambridge Univ. Press, New York, 1959).
Author information
Authors and Affiliations
Corresponding authors
Ethics declarations
The author declares that he has no conflicts of interest.
Additional information
Translated by A.V. Shishulin
About this article
Cite this article
Alikulov, T.N., Khalmukhamedov, A.R. On Fractional Powers of the Schrödinger Operator with a Potential Singular on Manifolds. Russ Math. 67, 8–15 (2023). https://doi.org/10.3103/S1066369X2305002X
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.3103/S1066369X2305002X