Abstract
We present a complete description of the point spectrum of the Laplace operator perturbed by point potentials concentrated at the vertices of regular polygons. We prove a criterion for the absence of points of the point spectrum of a singular perturbed positive self-adjoint operator with the property of cyclicity of defect vectors.
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REFERENCES
S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, Springer, New York (1988).
S. Albeverio and L. Nizhnik, Schrödinger Operators with Number of Negative Eigenvalues Equal to the Number of Point Interactions, Preprint No. 49, Bonn (2002).
S. Albeverio and L. Nizhnik, “On the number of negative eigenvalues of a one-dimensional Schrödinger operator with point interactions,” Lett. Math. Phys., 65, 27–35 (2003).
V. D. Koshmanenko, Singular Bilinear Forms in the Theory of Perturbations of Self-Adjoint Operators [in Russian], Naukova Dumka, Kiev (1993).
W. Karwowski and V. Koshmanenko, “On the definition of singular bilinear forms and singular linear operators,” Ukr. Mat. Zh., 45, No.8, 1084–1089 (1993).
J. F. Brasche, V. Koshmanenko, and H. Neidhardt, “New aspects of Krein’s extension theory,” Ukr. Mat. Zh., 46, No.1, 37–54 (1994).
S. Albeverio and V. Koshmanenko, “On the form-sum approximations of singularly perturbed positive self-adjoint operators,” J. Funct. Anal., 169, 32–51 (1999).
W. Karwowski and V. Koshmanenko, “Singular quadratic forms: regularization by restriction,” J. Funct. Anal., 143, 205–220 (1997).
L. P. Nizhnik, “On rank one singular perturbations of self-adjoint operators,” Meth. Funct. Anal. Top., 7, No.3, 54–66 (2001).
S. Albeverio and P. Kurasov, Singular Perturbations of Differential Operators and Solvable Schrödinger Type Operators, Cambridge University Press, Cambridge (2000).
M. G. Krein, “Theory of self-adjoint extensions of semibounded Hermitian operators and its applications. I,” Mat. Sb., 20, No.3, 431–495 (1947).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).
V. Koshmanenko, “A variant of inverse negative eigenvalues problem in singular perturbation theory, ” Meth. Funct. Anal. Top., 8, No.1, 49–69 (2002).
V. D. Koshmanenko and O. V. Samoilenko, “Singular perturbations of finite rank. Point spectrum,” Ukr. Mat. Zh., 49, No.9, 1186–1212 (1997).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 56, No. 8, pp. 1128–1134, August, 2004.
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Dudkin, M.E. Point spectrum of the schrödinger operator with point interactions at the vertices of regular N-gons. Ukr Math J 56, 1343–1352 (2004). https://doi.org/10.1007/s11253-005-0061-6
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DOI: https://doi.org/10.1007/s11253-005-0061-6