Abstract
In this paper we study point perturbations of the Schrödinger operators within the framework of Krein's theory of self-adjoint extensions. A locality criterion for quadratic forms is proved for such perturbations.
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REFERENCES
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, New-York, 1988.
B. S. Pavlov, “The theory of extensions and solvable models,” Uspekhi Mat. Nauk [Russian Math.Surveys], 42 (1987), no. 6, 99–131.
S. P. Novikov, “Schrödinger operators in periodic fields in dimension two” in: Modern Problems of Mathematics. New Advances [in Russian], 23, VINITI, Moscow (1983), 3–32.
S. Albeverio and V. A. Geiler, “The band structure of the general periodic Schrödinger operator with point interactions,” Comm. Math. Phys., 210 (2000), no. 1, 29–48.
V. A. Geiler and K. V. Pankrashkin, “On fractal structure of the spectrum for periodic point perturbations of the Schrödinger operator with a uniform magnetic field,” Operator Theory: Adv. and Appl., 108 (1999), 259–265.
Yu. G. Shondin, “Semibounded local Hamiltonians for perturbations of the Laplacian on curvres with corner points in ℝ4,” Teoret. Mat. Fiz. [Theoret. and Math. Phys.], 106 (1996), no. 2, 179–199.
B. Simon, “Schrödinger semigroups,” Bull. Amer. Math. Soc., 7 (1982), no. 3, 447–526.
M. G. Krein and G. K. Langer, “On deficiency subspaces and generalized resolvents of an Hermitian operator in Пκ-space,” Funktsional. Anal. i Prilozhen. [Functional Anal. Appl.], 5 (1971), no. 2, 59–71, no. 3, 54–69.
V. A. Geiler, V. A. Margulis, and I. I. Chuchaev, “Zero range operators and Carleman operators,” Sibirsk. Mat. Zh. [Siberian Math. J.], 36 (1995), no. 4, 628–641.
L. Bers, F. John, and M. Schehter, Partial Derivatives Equations, John Wiley and Sons Inc., New York-London-Sydney, 1957.
V. B. Korotkov, Integral Operators [in Russian], Nauka, Novosibirsk, 1983.
M. Sh. Birman, “On the theory of self-adjoint extensions of positive definite operators,” Mat. Sb. [Math. USSR-Sb.], 38 (1956), no. 4, 431–450.
V. A. Derkach and M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps,” J. Funct. Anal., 95 (1991), no. 1, 1–95.
A. Teta, “Quadratic forms for singular pertubations of the Laplacian,” Publ. RIMS Kyoto Univ., 26 (1990), 803–817.
V. D. Koshmanenko, Singular Bilinear Forms in Perturbation Theory of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev, 1993.
V. A. Geiler, “The Schrödinger operator in two dimensions with homogeneous magnetic field and its perturbations by zero range potential,” Algebra i Analiz [St. Petersburg Math. J.], 3 (1991), no. 3, 1–48.
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Pankrashkin, K.V. Locality of Quadratic Forms for Point Perturbations of Schrödinger Operators. Mathematical Notes 70, 384–391 (2001). https://doi.org/10.1023/A:1012352029965
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DOI: https://doi.org/10.1023/A:1012352029965