Symmetries of Schrödinger Operator with Point Interactions
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The transformations of all the Schrödinger operators with point interactions in dimension one under space reflection P, time reversal T and (Weyl) scaling Wλ are presented. In particular, those operators which are invariant (possibly up to a scale) are selected. Some recent papers on related topics are commented upon.
- Albeverio, S., Brzeźniak, Z., Dąbrowski, L. (1994) Time dependent propagator for point interactions. J. Phys. A 27: pp. 4933-4943
- Albeverio, S., Brzeźniak, Z., Dąbrowski, L. (1995) Fundamental solution of the heat and Schrödinger equation with point interactions. J. Funct. Anal. 128: pp. 220-254
- Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H. (1988) Solvable Models in Quantum Mechanics. Springer-Verlag, Berlin
- Albeverio, S., Gesztesy, F., Holden, H. (1993) Comment on a recent note on the Schrödinger equation with a δ′-interaction. J. Phys. A: Math. Gen. 26: pp. 3903-3904
- Albeverio, S., Kurasov, P. (1997) Rank one perturbations, approximations and selfadjoint extensions. J. Funct. Anal. 148: pp. 152-169
- Albeverio, S., Kurasov, P. (1997) Rank one perturbations of not semibounded operators. Integral Equations Operator Theory 27: pp. 349-400
- Berezin, F. A., Faddeev, L. D. (1961) A remark on Schrödinger equation with a singular potential. Soviet Math. Dokl. 2: pp. 372-375
- Cheon, T. and Shigehara, T.: Realizing discontinuous wave function with renormalized shortrange potentials, quant-ph/9709035.
- Chernoff, P., Hughes, R. (1993) A new class of point interactions in one dimension. J. Funct. Anal. 111: pp. 97-117
- Coutinho, F. A. B., Nogami, Y., Fernando Perez, J. (1997) Generalized point interactions in onedimensional quantum mechanics. J. Phys. A: Math. Gen. 30: pp. 3937-3945
- Demkov, Y. N., Kurasov, P. B., Ostrovsky, V. N. (1995) Doubly periodical in time and energy exactly soluble system with two interacting systems of states. J. Phys. A: Math. Gen. 28: pp. 4361-4380
- Demkov, Y. N., Ostrovskii, V. N. (1975) Zero-range Potentials and their Applications in Atomic Physics. Leningrad Univ. Press, Leningrad
- Gesztesy, F., Holden, H. (1987) A new class of solvable models in quantum mechanics describing point interactions on the line. J. Phys. A: Math. Gen. 20: pp. 5157
- Gesztesy, F., Kirsch, W. (1985) One-dimensional Schrödinger operators with interactions singular on a discrete set. J. Reine Angew. Math. 362: pp. 28-50
- Griffiths, D. J. (1993) Boundary conditions at the derivative of a delta function. J. Phys. A: Math. Gen. 26: pp. 2265-2267
- Hassi, S., Langer, H. and de Snoo, H.: Selfadjoint extensions for a class of symmetric operators with defect numbers (1,1), 15th OT Conference Proc., 1995, pp. 115–145.
- Hassi, S., de Snoo, H. (1997) On rank one perturbations of selfadjoint operators. Integral Equations Operator Theory 29: pp. 288-300
- Kiselev, A., Simon, B. (1995) Rank one perturbations with infinitesimal coupling. J. Funct. Anal. 130: pp. 345-356
- Kurasov, P. (1996) Distribution theory for the discontinuous test functions and differential operators with the generalized coefficients. J. Math. Anal. Appl. 201: pp. 297-323
- Kurasov, P. and Boman, J.: Finite rank singular perturbations and distributions with discontinuous test functions, accepted for publication in Proc. Amer. Math. Soc.
- Kurasov, P. and Elander, N.: On the δ′-potential in one dimension, Preprint MSI 93-7, ISSN-1100-214X (1993).
- Kurasov, P., Scrinzi, A., Elander, N. (1994) On the δ′-potential arising in exterior complex scaling. Phys. Rev. A 49: pp. 5095-5097
- Krein, M. G. (1946) On the resolvents of a Hermitian operator with deficiency indices (m, m). Dokl. Akad. Nauk SSSR 52: pp. 657-660
- Pavlov, B. S. (1987) The theory of extensions and explicitly-soluble models. Russian Math. Surveys 42: pp. 127-168
- Reed, M., Simon, B. (1975) Methods of Modern Mathematical Physics. Academic Press, New York
- Seba, P. (1986) The generalized point interaction in one dimension. Czech. J. Phys. B 36: pp. 667-673
- Seba, P. (1986) Some remarks on the δ′-interaction in one dimension. Rep. Math. Phys. 24: pp. 111-120
- Shubin Christ, C., Stolz, G. (1994) Spectral theory of one-dimensional Schrödinger operators with point interactions. J. Math. Anal. Appl. 184: pp. 491-516
- Simon, B. (1995) Spectral analysis of rank one perturbations and applications. CRM Proc. Lectures Notes 8. Amer. Math. Soc., Providence, pp. 109-149
- Symmetries of Schrödinger Operator with Point Interactions
Letters in Mathematical Physics
Volume 45, Issue 1 , pp 33-47
- Cover Date
- Print ISSN
- Online ISSN
- Kluwer Academic Publishers
- Additional Links
- Schrödinger operators
- extension theory
- point interactions
- exactly solvable models.
- Author Affiliations
- 1. Institute of Mathematics, Ruhr-Universität, D-44780, Bochum; SFB 237; BiBoS; Cerfim (Locarno); Acc.Arch, USI (Mendrisio)
- 2. SISSA, 34014, Trieste, Italy
- 3. Alexander von Humboldt fellow, Institute of Mathematics, Ruhr-Universität, Bochum Department of Mathematics, Stockholm University; Department of Mathematical and Computational Physics, St. Petersburg University, Italy