Abstract
Two physical applications of the Laplace operator perturbed on a set of zero measure are suggested. The approach is based on the theory of self-adjoint extensions of symmetrical operators. The first applicatio is a solvable model of scattering of a plane wave by a perturbed thin cylinder. “Nonlocal” extensions are described. The model parameters can be chosen such that the model solution is an approximation of the corresponding “realistic” solution. The second application is the description of the time evolution of a one-dimensional quasi-Chaplygin medium, which can be reduced using a hodograph transform to the ill-posed problem of the Laplace operator perturbed on a set of codimension two inR 3. Stability and instability conditions are obtained.
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References
M. V. Fedoryuk,Math. USSR Izv.,26, 153–184 (1986).
M. V. Fedoryuk,Math. USSR Izv.,18, 145–161 (1982).
V. M. Babich, B. A. Samokish and D. B. Dement’ev,Zap. Nauchn. Sem. POMI,221, 67–74 (1995).
E. Fermi,Ric. Sci.,7, 13–52 (1936).
F. A. Berezin and L. D. Faddeev,Sov. Math. Dokl.,2, 372–375 (1961).
S. Albeverio, F. Gesztesy, R. Hoegh-Krohn and H. Holden,Solvable Models in Quantum Mechanics, Springer, Berlin (1988).
B. S. Pavlov,Usp. Mat. Nauk,42, No. 6, 99–131 (1987).
D. E. Potter,Nucl. Fusion,18, 813–820 (1978).
A. Shyam and M. Srinivasan,Appl. Phys.,17, 245–252 (1978).
Ya. B. Zel’dovich and I. D. Novikov,The Structure and Evolution of the Universe [in Russian], Nauka, Moscow (1975); English transl.,Relativistic Astrophysics, Vol. 2,The Structure and Evolution of the Universe, Univ. Chicago Press, Chicago (1983).
S. K. Zhdanov and B. A. Trubnikov,Quasi-Gaseous Unstable Media [in Russian], Nauka, Moscow (1991).
A. S. Blagoveshchenskii and K. K. Lavrent’ev,Vestn. Leningr. Gos. Univ., No. 1, 9–16 (1977).
L. Hörmander,The Analysis of Linear Partial Differential Operators, Vol. 3, Pseudo-Differential Operators, Springer, Berlin (1985).
Yu. G. Shondin,Theor. Math. Phys.,105, 1189–1200 (1995).
S. P. Novikov and B. Yu. Sternin,Sov. Math. Dokl. 7, 1373–1376 (1966).
V. A. Geyler, B. S. Pavlov and I. Yu. Popov,J. Math. Phys. 37, 5171–5194 (1996).
I. Yu. Popov,J. Math. Phys. 33, 3794–3801 (1992).
Yu. G. Shondin,Theor. Math. Phys.,74, 220–230 (1988).
J. Bognar,Indefinite Inner Product Spaces, Springer, Berlin (1975).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 119, No. 2, pp. 295–307, May, 1999.
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Zubok, D.A., Popov, I.Y. Two physical applications of the Laplace operator perturbed on a null set. Theor Math Phys 119, 629–639 (1999). https://doi.org/10.1007/BF02557355
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DOI: https://doi.org/10.1007/BF02557355