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On dynamics of quantum states generated by the Cauchy problem for the Schrödinger equation with degeneration on the half-line. (English) Zbl 1151.35417
J. Math. Sci., New York 151, No. 1, 2741-2753 (2008); translation from Fundam. Prikl. Mat. 12, No. 6, 157-174 (2006).
Summary: The paper considers the Cauchy problem for the Schrödinger equation with operator degenerate on the semiaxis and the family of regularized Cauchy problems with uniformly elliptic operators whose solutions approximate the solution of the degenerate problem. The author studies the strong and weak convergences of the regularized problems and the convergence of values of quadratic forms of bounded operators on solutions of the regularized problems when the regularization parameter tends to zero.

MSC:
35Q40 PDEs in connection with quantum mechanics
35B25 Singular perturbations in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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[1] S. N. Bakhvalov, V. A. Galkin, and Yu. A. Dubinskii, eds., Works of S. N. Kruzhkov: Collection of Papers [in Russian], Fizmatlit, Moscow (2000), pp. 14–38, 39–45, 99–153, 287–316.
[2] G. F. Dell’Antonio, ”On the limits of sequences of normal states,” Commun. Pure Appl. Math., 20, 413–429 (1967). · Zbl 0148.37901
[3] G. Fichera, ”On a unified theory of boundary-value problems for elliptic-parabolic equations of second order,” in: Boundary Problems in Differential Equations, The Univ. of Wisconsin Press, Madison (1960), pp. 97–120. · Zbl 0122.33504
[4] P. Gerard, ”Microlocal defect measures,” Commun. Part. Different. Equations, 16, No. 11, 1761–1794 (1991). · Zbl 0770.35001
[5] A. S. Kholevo, Probabilistic and Statistical Aspects of Quantum Mechanics [in Russian], Nauka, Moscow (1982). · Zbl 0516.60052
[6] I. P. Natanson, Function Theory of Real Variables [in Russian], Nauka, Moscow (1974).
[7] S. M. Nikol’skii, Approximation of Functions of Several Variables and Embedding Theorems [in Russian], Nauka, Moscow (1969).
[8] O. A. Oleinik, ”On linear second-order equations with nonnegative characteristic form,” Mat. Sb., 69(111), No. 1, 111–140 (1966).
[9] M. Reed and B. Simon, Modern Methods of Mathematical Physics [Russian translation], Vol. 1, Mir, Moscow (1977).
[10] V. Zh. Sakbaev, ”On the statement of the Cauchy problem for the Schrödinger equation degenerating on the half-space,” Zh. Vychisl. Mat. Mat. Fiz., 42, No. 11, 161–178 (2002).
[11] V. Zh. Sakbaev, ”On functionals on solutions of the Cauchy problem for the Schrödinger equation with degeneration on the half-line,” Zh. Vychisl. Mat. Mat. Fiz., 44, No. 9, 1654–1673 (2004). · Zbl 1136.35443
[12] V. V. Zhikov, ”On the problem of passing to the limit in divergent nonuniformly elliptic equations,” Funkts. Anal. Prilozh., 35, No. 1, 23–39 (2001). · Zbl 1085.35054
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