No Degeneracy of the Ground State for the Impact Parameter Model

Héctor C. Merino, Juan Héctor Arredondo

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DOI: 10.4236/am.2011.210165                

Abstract

A time dependent Hamiltonian associated to the impact parameter model for the scattering of a light particle and two heavy ones is considered. Existence and non degeneracy of the ground state is shown.

Keywords

Impact Parameter Model, Non Degeneracy of the Ground State

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	J. H. Arredondo, “Asymptotic Transition Probabilities,” Journal of Mathematical Physics, Vol. 30, No. 10, 1989, pp. 2291-2296. doi:10.1063/1.528558
[2]	J. H. Arredondo, “Adiabatic Approximation on the Impact-Parameter Model,” Few-Body Systems, Vol. 10, No. 2, 1991, pp. 59-72. doi:10.1007/BF01352402
[3]	H. S. W. Massey, “Collision between Atoms and Molecules at Ordinary Temperatures,” Reports on Progress in Physics, Vol. 12, No. 1, 1949, p. 248. doi:10.1088/0034-4885/12/1/311
[4]	D. Aharonov, W. van Dam, J. Kempe, Z. Landau, S. Lloyd and O. Regev, “Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation,” 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS'04), Rome, 17-19 October 2004.
[5]	F. Hiroshima, “Topics in the Theory of Schr?dinger Operators,” World Scientific Publishing Co. Pte. Ltd., Singapore, 2004.
[6]	K. Chadan and P. C. Sabatier. “Inverse Problems in Quantum Scattering Theory,” Springer-Verlag, New York, 1989.
[7]	T. Cubel, B. K. Teo, V. S. Malinoysky, J. R. Guest, A. Reinhard, B. Knuffman, P. R. Berman and G. Raithel, “Coherent Population Transfer of Ground-State Atoms into Rydberg States,” Physical Review A, Vol. 72, 2005, pp. 023405 (1-4).
[8]	H. L. Cycon, R. G. Froese, W. Kirsh and B. Simon. “Schr?dinger Operators with Application to Quantum Mechanics and Global Geometry,” Springer-Verlag, New York, 1987.
[9]	E. Farhi, J. Goldstone, S. Gutmann and M. Sipser, “Quantum Computation by Adiabatic Evolution,” 2000. arXiv:quant-ph/0001106v1
[10]	P. D. Hislop and I. M. Sigal, “Introduction to Spectral Theory with Applications to Schr?dinger Operators,” Springer-Verlag, New York, 1996.
[11]	T. Kato, “Perturbation Theory for Linear Operators,” Springer-Verlag, Berlin and New York, 1984.
[12]	J. O. Lee and J. Yin, “A Lower Bound on the Ground State Energy of Dilute Bose Gas,” Journal of Mathematical Physics, Vol. 51, No. 5, 2010, pp. 053302 (1-31).
[13]	E. L. Lieb, R. Seiringer, J. P. Solovej and J. Yngvason, “The Ground State of the Bose Gas,” 2003. arXiv:math-ph/0204027v2
[14]	H. E. Puthoff, “Ground State of Hydrogen as a Zero-Point-Fluctuation-Determined State,” Physical Review D, Vol. 35, No. 10, 1987, pp. 3266-3269. doi:10.1103/PhysRevD.35.3266
[15]	M. Reed and B. Simon, “Methods of Modern Mathematical Physics, Vol. IV,” Academic Press, New York, 1975.
[16]	W. Thirring, “A Course in Mathematical Physics. Vol. III,” Springer-Verlag, New York, 1981.
[17]	I. Wilson-Rae, N. Nooshi, W. Zwerger and T. J. Kippenberg, “Theory of Ground State Cooling of a Mechanical Oscillator Using Dynamical Backaction,” Physical Review Letters, Vol. 99, No. 9, 2007, Article ID: 093901 (1-4).
[18]	P. D. Lax, “Functional Analysis,” John Wiley & Sons Inc., New York, 2002.
[19]	M. Reed and B. Simon, “Methods of Modern Mathematical Physics, Vol. II,” Academic Press, New York, 1975.
[20]	J. H. Arredondo and P. Seibert, “Characterization of the Fundamental State Eigenvalue for Some Time Dependent Hamiltonians,” Aportaciones Matemáticas. Serie Comunicaciones, Vol. 29, No. 3, 2001, pp. 3-10.