Exact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential
https://doi.org/10.17586/2220-8054-2019-10-6-608-615
Abstract
By taking advantage of Wang’s results on the scalar product of four eigenfunctions of the 1D harmonic oscillator, we explicitly calculate the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by a Gaussian potential, showing that it can be written as a ratio of Gamma functions.
Supporting agencies: The final part of S. Fassari’s contribution to this work has been financially supported by the Government of the Russian Federation through the ITMO University Fellowship and Professorship Programme. S. Fassari would like to thank Prof. Igor Yu. Popov and the entire staff at the Departament of Higher Mathematics, ITMO University, St. Petersburg for their warm hospitality throughout his stay. F. Rinaldi wishes to thank Prof. Igor Yu. Popov for inviting him to the Pierre Duclos Workshop over the years as well as for his interest in our research work
About the Authors
S. Fassari
ITMO University; CERFIM; Univ. degli Studi Guglielmo Marconi
Italy
Italy
Department of Higher Mathematics, ITMO University; Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.
St. Petersburg; PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome
F. Rinaldi
CERFIM; Univ. degli Studi Guglielmo Marconi; Istituto Nazionale di Fisica Nucleare, Sezione di Napoli
Italy
Italy
Dipartimento di Fisica Nucleare, Subnucleare e delle Radiazioni, Univ. degli Studi Guglielmo Marconi.
PO Box 1132, CH-6601 Locarno; Via Plinio 44, I-00193 Rome; Naples
References
1. Landau L.D. and Lifshits E.M., Quantum Mechanics: Non-Relativistic Theory, Pergamon Press.
2. Reed M., Simon B. Fourier Analysis, Methods in Modern Mathematical Physics, Academic Press, New York, 1975.
3. Reed M., Simon B. Analysis of Operators, Methods in Modern Mathematical Physics, Academic Press, New York, 1978.
4. Loeffel J.J., Martin A., Simon B., Wightman A. Pade´ approximants and the anharmonic oscillator, Phys. Lett. B, 1969, 30, P. 656–658.
5. Davydov A.S. Quantum Mechanics, Oxford: Pergamon Press, 1965.
6. Yaris R., Bendler J., Lovett R.A., Bender C.M. and Fedders P.A. Resonance calculations for arbitrary potentials, Phys. Rev. A, 1978, 18, 1816.
7. Caliceti E., Gra S., Maioli M. Perturbation theory of odd anharmonic oscillators, Commun. Math. Phys., 1980, 75, P. 51.
8. Caliceti E., Maioli M. Odd anharmonic oscillators and shape resonances Ann. Inst. Henri Poincare´, 1983, XXXVIII(2), P. 175–186.
9. Fernandez F.M., Guardiola R., Ros J. and Znojil M. Strong-coupling expansions for the PT -symmetric oscillators V (x) = a(ix) + b(ix)2 + c(ix)3. J. Phys. A: Math. Gen., 1998, 31, P. 10105.
10. Grecchi V., Martinez A. The spectrum of the cubic oscillator. Commun. Math. Phys., 2013, 319, P. 479.
11. Kaushal S.K. Small and large solution of the Schro¨dinger equation for the interaction λx2/1 + gx. J. Phys. A Math. Gen., 1979, 12, P. L253.
12. Mitra A.K. On the interaction of the type x2 +λx2/1 + gx. J. Math. Phys., 1979, 19(10), P. 2018–2022.
13. Bessis N., Bessis G. A note on the Schro¨dinger equation for the x2 +λx2/1 + gx2 potential. J. Math. Phys., 1980, 21, P. 2780–2785.
14. Bessis N., Bessis G., Hadinger G. Perturbed harmonic oscillator ladder operators: eigenenergies and eigenfunctions for the x2 +λx2/1 + gx2 interaction. J. Phys. A Math. Gen., 1980, 13, P. 1651.
15. Bhagwat K.V. A harmonic oscillator perturbed by the potential λx2/1 + gx2. J. Phys. A: Math. Gen., 1981, 14, P. 377.
16. Bessis N., Bessis G. Perturbed factorization of the symmetric-anharmonic-oscillator eigenequation. Phys. Rev A, 1992, 46, P. 6824.
17. Hodgson R.J.W. High-precision calculation of the eigenvalues for the x2 +λx2/1 + gx2 potential. J. Phys. A: Math. Gen., 1988, 21, P. 1563.
18. Lai C.S., Lin H.E. On the Schro¨dinger equation for the x2 +λx2/1 + gx2 interaction. J. Phys. A: Math. Gen., 1982, 15, P. 1495.
19. Blecher M.H., Leach P.G.L. The Schro¨dinger equation for the x2 +λx2/1 + gx2 interaction. J. Phys. A: Math. Gen., 1987, 20, P. 5923.
20. Angeletti A., Castagnari C., Zirilli F. Asymptotic eigenvalue degeneracy for a class of one-dimensional Fokker-Planck operators. J. Math. Phys., 1985, 26, P. 678.
21. Fassari S. A note on the eigenvalues of the Hamiltonian of the harmonic oscillator perturbed by the potential λx2/1 + gx2. Rep. Math. Phys., 1996, 37(2), P. 283–293.
22. Fassari S., Inglese G. On the eigenvalues of the Hamiltonian of the harmonic oscillator with the interaction λx2/1 + gx2. II Rep. Math. Phys., 1997, 39(1), P. 77–86.
23. Carin˜ena J.F., Ran˜ada M.F., Santander M. A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour. Annals of Physics, 2007, 322(2), P. 434–459.
24. Gadreau P., Safouhi H. Double exponential sinc-collocation method for solving the energy eigenvalues of harmonic oscillators perturbed by a rational function. J. Math. Phys., 2017, 26(10), P. 101509.
25. Avakian M.P., Pogosyan G.S., Sissakian A.N., Ter-Antonyan V.M. Spectroscopy of a singular linear oscillator. Physics Letters A, 1987, 124(4-5), P. 233–236.
26. Goldstein J., Lebiedzik C., Robinett R.W. Supersymmetric quantum mechanics: Examples with Dirac δ-functions. American Journal of Physics, 62(7), P. 612–618.
27. Fassari S., Inglese G. On the spectrum of the harmonic oscillator with a δ−type perturbation. Helv. Phys. Acta, 1994, 67, P. 650–659.
28. Demiralp E. Bound states of n-dimensional harmonic oscillator decorated with Dirac delta functions. J. Phys. A: Math. Theor., 2005, 22, P. 478393.
29. Uncu H., Tarhan D., Demiralp E., Mu¨stecaplog¨lu O¨ .E. Phys. Rev. A., 2007, 76, P. 013618.
30. Patil S.H. Harmonic oscillator with a δ-function potential. Eur. J. Phys., 2006, 27, P. 899.
31. Goold J., ODonoghue D. and Busch Th. Low-density, one-dimensional quantum gases in the presence of a localized attractive potential. J. Phys. B: At. Mol. Opt. Phys., 2008, 41, P. 215301.
32. Shea P., Van Zyl B.P. and Bhaduri R.K. The two-body problem of ultra-cold atoms in a harmonic trap. Am. J. Phys., 2009, 77, P. 5115.
33. Filatova T.A. and Shafarevich A.I. Semiclassical spectral series of the Schro¨dinger operator with a delta-potential on a straight line and on a sphere. Theor. Math. Phys., 2010, 164, P. 106480.
34. Mityagin B. The spectrum of a harmonic oscillator operator perturbed by point interactions. Int. J. Theor. Phys., 2014, 53, P. 118.
35. Mityagin B.S., Siegl P. Root system of singular perturbations of the harmonic oscillator type operators. Lett. Math. Phys., 2016, 106, P. 147– 167.
36. Ferkous N., Boudjedaa T. Bound States Energies of a Harmonic Oscillator Perturbed by Point Interactions. Commun. Theor. Phys., 2017, 67, P 241.
37. Fassari S., Inglese G. On the spectrum of the harmonic oscillator with a δ−type perturbation. II Helv. Phys. Acta, 1997, 70, P. 858–865.
38. Fassari S., Rinaldi F. On the spectrum of the Schro¨dinger Hamiltonian of the one-dimensional harmonic oscillator perturbed by two identicalт attractive point interactions. Rep. Math. Phys., 2012, 69(3), P. 353–370.
39. Demiralp E. Properties of a pseudo-Hermitian Hamiltonian for the harmonic oscillator decorated with Dirac delta interactions. Czech. J. Phys., 2005, 55, P. 10814.
40. Haag D., Cartarius H. and Wunner G. A Bose-Einstein condensate with PT -symmetric double delta function loss and gain in a harmonic trap: a test of rigorous estimates. Acta Polytech., 2014, 54, P. 11621.
41. Single F., Cartarius H., Wunner G. and Main J. Coupling approach for the realization of a PT -symmetric potential for a Bose-Einstein condensate in a double well. Phys. Rev. A, 2014, 90, P. 042123.
42. Albeverio S., Dabrowski L. and Kurasov P. Symmetries of Schro¨dinger operators with point interactions. Lett. Math. Phys., 1998, 45, P. 3347.
43. Albeverio S. and Kurasov P. Singular Perturbations of Differential Operators: Solvable Type Operators. Cambridge, Cambridge University Press, 2000.
44. Gadella M., Glasser M.L., Nieto L.M. One-dimensional models with a singular potential of the type −αδ + βδ0. Int. J. Theor. Phys., 2011, 50, P. 2144–2152.
45. Maldonado-Villamizar F.H. Semitransparent one-dimensional potential: a Green’s function approach. Phys. Scr., 2015, 90, P. 065202.
46. Albeverio S., Fassari S., Rinaldi F. A remarkable spectral feature of the Schro¨dinger Hamiltonian of the harmonic oscillator perturbed by an attractive δ0-interaction centred at the origin: double degeneracy and level crossing. J. Phys. A: Math. Theor., 2013, 46, P. 385305.
47. Albeverio S., Fassari S., Rinaldi F. The Hamiltonian of the harmonic oscillator with an attractive δ0-interaction centred at the origin as approximated by the one with a triple of attractive interactions. J. Phys. A: Math. Theor., 2016, 49, P. 025302.
48. Cheon T. and Shigehara T. Fermion-Boson duality of one-dimensional quantum particles with generalized contact interactions. Phys. Rev. Lett., 1999, 82, P. 2536.
49. Nouicer K., Chetouani L. Variational treatment of Gaussian potentials. Acta Physica Slovaca, 1999, 49(3), P. 309–318.
50. Fassari S., Gadella M., Glasser M.L., Nieto L.M. and Rinaldi F. Level crossings of eigenvalues of the Schro¨dinger Hamiltonian of the isotropic harmonic oscillator perturbed by a central point interaction in different dimensions. Nanosystems, Physics, Chemistry, Mathematics, 2018, 9(2), P. 179–186.
51. Fassari S., Gadella M., Glasser M.L., Nieto L.M. Spectroscopy of a one-dimensional V-shaped quantum well with a point impurity. Annals of Physics, 2018, 389, P. 48–62.
52. Reed M., Simon B. Functional Analysis, Methods in Modern Mathematical Physics. Academic Press, New York, 1972.
53. Muchatibaya G., Fassari S., Rinaldi F., Mushanyu J. A note on the discrete spectrum of Gaussian wells (I): the ground state energy in one dimension. Adv. Math. Phys., 2016, Article ID 2125769.
54. Fassari S., Gadella M., Nieto L.M., Rinaldi F. On the spectrum of the 1D Schro¨dinger Hamiltonian perturbed by an attractive Gaussian potential. Acta Polytech., 2017, 57, P. 385–390.
55. Klaus M. A remark about weakly coupled one-dimensional Schro¨dinger operators. Helv. Phys. Acta, 1979, 52, P. 223.
56. Fassari S. An estimate regarding one-dimensional point interactions. Helv. Phys. Acta, 1995, 68, P. 121–125.
57. Wang W.-M. Pure Point Spectrum of the Floquet Hamiltonian for the Quantum Harmonic Oscillator Under Time Quasi-Periodic Perturbations. Commun. Math. Phys., 2008, 277, P. 459–496.
58. Albeverio S., Fassari S., Gadella M., Nieto L.M. and Rinaldi F. The Birman–Schwinger Operator for a Parabolic Quantum Well in a ZeroThickness Layer in the Presence of a Two-Dimensional Attractive Gaussian Impurity. Front. Phys., 2019, 7(102).
Review
For citations:
Fassari S., Rinaldi F. Exact calculation of the trace of the Birman–Schwinger operator of the one-dimensional harmonic oscillator perturbed by an attractive Gaussian potential. Nanosystems: Physics, Chemistry, Mathematics. 2019;10(6):608-615. https://doi.org/10.17586/2220-8054-2019-10-6-608-615
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