Volume 9, no. 1Pages 73 - 91

On the One-Dimensional Harmonic Oscillator with a Singular Perturbation

V.A. Strauss, M.A. Winklmeier
In this paper we investigate the one-dimensional harmonic oscillator with a left-right boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point. The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.
Full text
harmonic oscillator; singular perturbation; selfadjoint extensions; negative eigenvalues.
1. Albeverio S., Gesztesy F., Hoegh-Krohn R., Holden H. Solvable Models in Quantum Mechanics. AMS Chelsea Publishing, Providence, RI, 2005.
2. Berezansky Y.M., Sheftel Z.G., Us G.F. Functional Analysis. Vol. II, Volume 86 of Operator Theory: Advances and Applications. Birkhauser Verlag, Basel, 1996.
3. Berezin F.A., Faddeev L.D. A Remark on Schrodinger's Equation with a Singular Potential. Soviet Mathematics. Doklady, 1961, no. 2, pp. 372-375.
4. Chernoff P.R., Hughes R.J. A New Class of Point Interactions in One Dimension. Journal of Functional Analysis, 1993, vol. 111, no. 1, pp. 97-117. DOI: 10.1006/jfan.1993.1006
5. Eastham M.S.P. The Asymptotic Solution of Linear Differential Systems. Applications of the Levinson Theorem. Oxford, Clarendon Press, 1989.
6. Gadella M., Glasser M.L., Nieto L.M. The Infinite Square Well with a Singular Perturbation. International Journal of Theoretical Physics, 2011, vol. 50, no. 7, pp. 2191-2200. DOI: 10.1007/s10773-011-0690-5
7. Gadella M., Glasser M.L., Nieto L.M. One Dimensional Models with a Singular Potential of the Type -alpha delta(x)+beta delta'(x). International Journal of Theoretical Physics, 2011, vol. 50, no. 7, pp. 2144-2152.
8. Gohberg I., Lancaster P., Rodman L. Indefinite Linear Algebra and Applications. Birkh'auser Verlag, Basel, 2005.
9. Kato T. Perturbation Theory for Linear Operators. Die Grundlehren der mathematischen Wissenschaften, Band 132. New York, Springer-Verlag New York, 1966.
10. Kurasov P. Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients. Journal of Mathematical Analysis and Applications, 1996, vol. 201, no. 1, pp. 297-323. DOI: 10.1006/jmaa.1996.0256
11. Seba P. The Generalized Point Interaction in One Dimension. Czechoslovak Journal of Physics B, 1986, vol. 36, no. 6, pp. 667-673. DOI: 10.1007/BF01597402
12. Triebel H. Higher Analysis. Hochschulbucher fur Mathematik. [University Books for Mathematics]. Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992.
13. Viana-Gomes J., Peres N.M.R. Solution of the Quantum Harmonic Oscillator Plus a Delta-Function Potential at the Origin: the Oddness of Its Even-Parity Solutions. European Journal of Physics, 2011, vol. 32, no. 5, pp. 1377-1384. DOI: 10.1088/0143-0807/32/5/025
14. Weidmann J. Linear Operators in Hilbert Spaces, Volume 68 of Graduate Texts in Mathematics. New York, Berlin, Springer-Verlag, 1980.
15. Weidmann J. Spectral Theory of Ordinary Differential Operators, Volume 1258 of Lecture Notes in Mathematics. Berlin, Springer-Verlag, 1987.
16. Zeldovic Ya.B. Scattering by a Singular Potential in Perturbation Theory and in the Momentum Representation. Soviet Physics. Journal of Experimental and Theoretical Physics, 1960, no. 11, pp. 594-597.