Quadratic forms for singular perturbations of the Laplacian. (English) Zbl 0735.35048

The negative Laplacian is defined in \(L_ 2(\mathbb{R}^ 3)\) and singular perturbations of it, ported by points, regular curves and regular surfaces, are considered. The aim is to develop an approach, based on the theory of quadratic forms, for a description of Schrödinger operators in \(L_ 2(\mathbb{R}^ 3)\) with interactions supported by particular sets of Lebesgue measure zero in \(\mathbb{R}^ 3\). The appropriate quadratic forms are constructed and a complete characterisation of the domain, and the action of associated operators is given together with an explicit construction of the resolvent.


35J10 Schrödinger operator, Schrödinger equation
35B25 Singular perturbations in context of PDEs
35P05 General topics in linear spectral theory for PDEs
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[1] Albeverio, S., Fenstad, I.E., H0egh-Krohn, R. and Lindstrom, T., Non standard Methods in Stochastic Analysis and Mathematical Physics, Academic Press, New York- San Francisco-London, 1986.
[2] Albeverio, S., Fenstad, J.E., H^egh-Krohn, R., Karwowski, W. and Lindstrom, T., Schrodinger operators with potentials supported by null sets, preprint in preparation. · Zbl 0795.35088
[3] Brasche, J.F., Perturbations of self-adjoint operators suppoted by null sets, Dis- sertation zur Erlangung des Doktorgrades der Fakultat fur Mathematik der Universitat Bielefeld, 1988. · Zbl 0691.47016
[4] Albeverio, S., Gesztesy, F., H0egh-Krohn, R. and Holden, H., Solvable Models in Quantum Mechanics, Springer- Verlag, Berlin-Heidelberg-New York, 1988. · Zbl 0679.46057
[5] Antoine, J.P., Gesztesy, F. and Shabani, J., Exactly solvable models of sphere inter- actions in quantum mechanics, /. Phys., A20 (1987), 3687-3712.
[6] Albeverio, S., Fenstad, I.E., H^egh-Krohn, R., Karwowski, W. and Lindstrom, T., Perturbations of the Laplacian supported by null sets, with applications to polymer measure and quantum fields, Phys. Lett., 104A (1984), 396-400.
[7] Figari, R., Holden, H. and Teta, A., A law of large numbers and a central limit theorem for the Schrodinger operator with zero-range potentials, /. Stat. Phys., 51 (1988), 205-214. · Zbl 1086.81510
[8] Kato, T., Perturbation Theory for Linear Operators, Springer- Verlag, Berlin-Heidel- berg-New York, 1980. · Zbl 0435.47001
[9] Simon, B.s Quantum Mechanics for Hamiltonian Defined as Quadratic Forms, Princeton U.P., Princeton, NJ, 1971. · Zbl 0232.47053
[10] Reed, M. and Simon, B., Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York-San Francisco-London, 1975. · Zbl 0308.47002
[11] Thomas, L.E., Birman-Schwinger bounds for the Laplacian with point interactions, /. Math. Phys., 20 (1979), 1848-1850. · Zbl 0423.35068
[12] Kellogg, O.D., Foundations of Potential Theory, Springer-Verlag, Berlin, 1929. · Zbl 0053.07301
[13] Grossmann, A. and Wu, T.T., A class of potentials with extremely narrow reso- nances, Chin. J. Phys., 25 (1987), 129-139.
[14] Koshmanenko, V.D., Singular perturbations defined by forms, BiBoS preprint Nr. 314/88.
[15] Svendsen, E.G., The effect of submanifolds upon essential self-adjointness and defi- ciency indices”, /. Math. Anal. Appl., 80 (1981), 551-565. · Zbl 0473.47039
[16] Rozanov, Yu. A., On the Schrodinger type equation with generalized potential, Math. USSR Sbornik, 55 (1986), 475-484. · Zbl 0649.34005
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