Abstract
The Cauchy problem for the Schrödinger equation with an operator degenerating on a half-line and a family of regularized Cauchy problems with uniformly elliptic operators, whose solutions approximate the solution to the degenerate problem, are considered. A set-valued mapping is investigated that takes a bounded operator to a set of partial limits of values of its quadratic form on solutions of the regularized problems when the regularization parameter tends to zero. The dynamics of quantum states are determined by applying an averaging procedure to the set-valued mapping.
Similar content being viewed by others
References
G. Fichera, “On a Unified Theory of Boundary Value Problems for Elliptic-Parabolic Equations of Second Order,” in Boundary Problems in Differential Equations (Univ. Wisconsin Press, Madison, 1960), pp. 97–120.
O. A. Oleinik, “Second-Order Linear Equations with a Nonnegative Characteristic Form,” Mat. Sb. 69(1), 111–140 (1966).
M. I. Vishik and L. A. Lyusternik, “Regular Degeneration and Boundary Layer for Linear Differential Equations with a Small Parameter,” Usp. Mat. Nauk 12(5), 3–122 (1957).
M. I. Freidlin, “Ito Stochastic Equations and Degenerate Elliptic Equations,” Izv. Akad. Nauk SSSR, Ser. Mat. 26, 653–676 (1962).
Works by S.N. Kruzhkov: Collected Papers, Ed. by S.N. Bakhvalov (Fizmatlit, Moscow, 2000) [in Russian].
V. V. Zhikov, “To the Problem of Passage to the Limit in Divergent Nonuniformly Elliptic Equations,” Funkts. Anal. Ego Prilozh. 35(1), 23–39 (2001).
V. V. Zhikov, “Remarks on the Uniqueness of a Solution of the Dirichlet Problem for Second-Order Elliptic Equations with Lower-Order Terms,” Funkts. Anal. Ego Prilozh. 38(3), 15–28 (2004).
J.-L. Lions and E. Magenes, Problemes aux limites non homogènes et applications (Dunod, Paris, 1968; Mir, Moscow, 1971).
V. Zh. Sakbaev, “On a Formulation of the Cauchy Problem for the Schrödinger Equation Degenerate on a Half-Space,” Zh. Vychisl. Mat. Mat. Fiz. 42, 1700–1711 (2002) [Comput. Math. Math. Phys. 42, 1636–1646 (2002)].
V. Zh. Sakbaev, “Functionals on Solutions of the Cauchy Problem for the Schrödinger Equation Degenerate on a Half-Line,” Zh. Vychisl. Mat. Mat. Fiz. 44, 1654–1673 (2004) [Comput. Math. Math. Phys. 44, 1573–1591 (2004)].
P. Gerard, “Microlocal Defect Measures,” Commun. Part. Differential Equations 16, 1761–1794 (1991)
R. G. Cooke, Infinite Matrices and Sequence Spaces (Macmillan, London, 1950; Fizmatlit, Moscow, 1960).
A. S. Kholevo, Probabilistic and Statistical Aspects of Quantum Mechanics (Inst. Komp’yuternykh Issledovanii, Moscow, 2003) [in Russian].
Yu. N. Orlov, Fundamentals of Quantization of Degenerate Dynamical Systems (MFTI, Moscow, 2004) [in Russian].
V. V. Kozlov, “Dynamics of Systems with Nonintegrable Couplings, Part III,” Vestn. Mosk. Gos. Univ., Ser. 1: Mat. Mekh., No. 3, 102–111 (1983).
V. V. Zhikov, S. M. Kozlov, and O. A. Olleinik, Homogenization of Differential Operators and Integral Functionals (Nauka, Moscow, 1993; Springer-Verlag, Berlin, 1994).
G. F. Dell’Antonio, “On the Limits of Sequences of Normal States,” Commun. Pure Appl. Math. 20, 413–429 (1967).
V. Zh. Sakbaev, “On the Dynamics of Probability Measures Generated by the Cauchy Problem for the Schrödinger Equation with Degeneration on a Half-Line,” in Some Problems in Fundamental and Applied Mathematics (MFTI, Moscow, 2004), pp. 119–143 [in Russian].
K. Kuratowski, Topology (Academic, New York, 1966; Moscow: Mir, Moscow, 1966).
M. C. Reed and B. Simon, Methods of Modern Mathematical Physics, Vol. 1: Functional Analysis (Academic, New York, 1972; Mir, Moscow, 1977).
K. Yoshida, Functional Analysis (Mir, Moscow, 1967; Springer-Verlag, Berlin, 1980).
G. P. Tolstov, Measure and Integral (Nauka, Moscow, 1976) [in Russian].
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.Zh. Sakbaev, 2006, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2006, Vol. 46, No. 4, pp. 683–699.
Rights and permissions
About this article
Cite this article
Sakbaev, V.Z. Set-valued mappings specified by regularization of the Schrödinger equation with degeneration. Comput. Math. and Math. Phys. 46, 651–665 (2006). https://doi.org/10.1134/S0965542506040117
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0965542506040117