Volume 14, no. 1Pages 75 - 90

On a Limit Pass From Two-Point to One-Point Interaction in a One Dimensional Quantum Mechanical Problem Giving Rise to a Spontaneous Symmetry Breaking

A. Restuccia, A. Sotomayor, V.A. Strauss
We analyze, by means of singular potentials defined in terms of Dirac functions and their derivatives, a one dimensional symmetry breaking in quantum mechanics. From a mathematical point of view we use a technique of selfadjoint extensions applied to a symmetric differential operator with a domain containing smooth functions which vanish at two inner points of the real line. As is well known, the latter leads to a two-point boundary problem. We compute the resolvent of the corresponding extension and investigate its behavior in the case in which the inner points change their positions. The domain of these extensions can contain some functions with non differentiability or discontinuity at the points mentioned before. This fact can be interpreted as a presence of singular potentials like shifted Dirac delta functions and/or their first derivative centered at the same points. Then, we study the existence of broken-symmetry bound states. For some given entanglement boundary conditions we can show the existence of a ground state, which leads to a spontaneous symmetry breaking. We also prove that within a frame of Pontryagin spaces this type of symmetry breaking is saved if the distance between the mentioned above interior points tends to zero and then we can reformulate this result in terms of a larger Hilbert space.
Full text
operator theory; resolvent; solution of wave equation: bound states; spontaneous symmetry breaking; Pontryagin spaces.
1. Restuccia A., Sotomayor A., Strauss V. Non-Local Interactions in Quantum Mechanics Modelled by Shifted Dirac Delta Functions. Journal of Physics: Conference Series, 2016, no. 1043, 9 p.
2. Restuccia A., Sotomayor A., Strauss V. On a Model of Spontaneous Symmetry Breaking in Quantum Mechanics. Bulletin of South Ural State University. Mathematical Modelling, Programming and Computer Software, 2020, vol. 13, no. 3, pp. 5-16. DOI: 10.14529/mmp200301
3. Burrau O. Berechnung des Energiewertes des Wasserstoffmolekel-Ions (H^2_+) im Normalzustand. Naturwissenschaften, 1927, vol. 15, no. 1, pp. 16-17. DOI: 10.1007/BF01504875
4. Kronig R. de L., Penney W.G. Quantum Mechanics of Electrons in Crystal Lattices. Proceedings of the Royal Society, 1931, vol. 130A, pp. 499-513. DOI 10.1098/rspa.1931.0019
5. Albeverio S., Gesztesy F., Hoegh-Kron R., Holden H. Solvable Models in Quantum Mechanics. White River Junction: Chelsea Publishing, 2004.
6. Kurasov P. Distribution Theory for Discontinuous Test Functions and Differential Operators with Generalized Coefficients. Journal of Mathematical Analysis and Applications, 1996, no. 201, pp. 297-323.
7. Zurek W.H. Decoherence, Einselection, and the Quantum Origins of the Classical. Reviews of Modern Physics, 2003, vol. 75, no. 3, pp. 715-776. DOI: 10.1103/RevModPhys.75.715
8. Weinberg S. Lectures on Quantum Mechanics. Cambridge, Cambridge University Press, 2012.
9. Mal'cev A.I. Foundations of Linear Algebra. Freeman and Company, San Francisco, 1963.
10. Kurasov P., Luger A. Reflectionless Potentials and Point Interactions in Pontryagin Spaces. Letters in Mathematical Physics, 2005, vol. 73, pp. 109-122. DOI: 10.1007/s11005-005-0002-1
11. Yariv, A. An Introduction Theory to Theory and Applications of Quantum Mechanics, New York, Wiley and Sons, 1982.
12. Azizov T.Ya., Iokhvidov I.S. Linear Operators in Spaces with Indefinite Metric, New York, Wiley and Sons, 1989.