Abstract
The one-dimensional Schrödinger operator with potential growing at infinity and with a semiclassical small parameter is considered. We obtain estimates via powers of the small parameter for the remainder in the expansion of smooth sufficiently rapidly decaying functions in the exact and asymptotic eigenfunctions. For the asymptotic eigenfunctions, we use a global representation in the form of an Airy function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
L. D. Landau and E. M. Lifshits, Theoretical Physics, Vol. 3: Quantum Mechanics. Nonrelativistic Theory (Nauka, Moscow, 1974) [in Russian].
V. P. Maslov, Théorie des perturbations et méthodes asymptotiques (Dunod, Paris, 1972).
V. P. Maslov and M. V. Fedoryuk, Semi-Classical Approximation in Quantum Mechanics (Reidel, Dordrecht, 1981).
S. Yu. Slavyanov, Asymptotic Solutions of the One-Dimensional Schrödinger Equation (Amer. Math. Soc., Providence, RI, 1996).
A. Yu. Anikin, S. Yu. Dobrokhotov, V. E. Nazaikinskii, and A. V. Tsvetkova, “Uniform asymptotic solution in the form of an Airy function for semiclassical bound states in one-dimensional and radially symmetric problems,” Theoret. and Math. Phys. 201 (3), 1742–1770 (2019).
A. Erdélyi, Asymptotic Expansions (Dover Publ., New York, 1956).
F. W. J. Olver, Asymptotics and Special Functions (Academic Press, New York–London, 1974).
M. Born and V. Fock, “Beweis des adiabaten Satzes,” Z. Phys. 51, 165–180 (1928).
T. Kato, “On the adiabatic theorem of quantum mechanics,” J. Phys. Soc. Jpn. 5, 435–439 (1950).
S. B. Kuksin and A. I. Neishtadt, “On quantum averaging, quantum KAM, and quantum diffusion,” Russian Math. Surveys 68 (2), 335–348 (2013).
A. Yu. Anikin, S. Yu. Dobrokhotov, and A. V. Tsvetkova, “Airy function and transition between the semiclassical and harmonic oscillator approximations for one-dimensional bound states,” Theoret. and Math. Phys. 204 (2), 984–992 (2020).
W. N. Everitt and M. Giertz, “Some properties of the domains of certain differential operators,” Proc. London Math. Soc. 23 (3), 301–324 (1971).
W. N. Everitt and M. Giertz, “Some inequalities associated with certain differential operators,” Math. Z. 126, 308–326 (1972).
W. N. Everitt and M. Giertz, “On some properties of the powers of a formally self-adjoint differential expression,” Proc. London Math. Soc. 24 (3), 149–170 (1972).
W. N. Everitt and M. Giertz, “On some properties of the domains of powers of certain differential operators,” Proc. London Math. Soc. 24 (3), 756–768 (1972).
M. A. Naimark, Linear Differential Operators (Nauka, Moscow, 1969) [in Russian].
E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations (Oxford Univ. Press, Oxford, 1958).
N. Dunford and J. T. Schwartz, Linear Operators, Vol. 2: Spectral Theory (Interscience Publ., New York, 1963).
J. Heading, An Introduction to Phase-Integral Methods (Methuen & Co., Ltd., John Wiley & Sons, Inc., London, New York, 1962).
M. V. Fedoryuk, Asymptotic Methods for Linear Ordinary Differential Equations (URSS, Moscow, 2009) [in Russian].
B. Helffer and B. Robert, “Puits de potentiel généralisés et asymptotique semi-classique,” Ann. Inst. H. Poincaré Phys. Théor. 41 (3), 291–331 (1984).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Matematicheskie Zametki, 2022, Vol. 112, pp. 644–664 https://doi.org/10.4213/mzm13670.
Rights and permissions
About this article
Cite this article
Anikin, A.Y., Dobrokhotov, S.Y. & Shkalikov, A.A. On Expansions in the Exact and Asymptotic Eigenfunctions of the One-Dimensional Schrödinger Operator. Math Notes 112, 623–641 (2022). https://doi.org/10.1134/S0001434622110013
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0001434622110013