Modified semi-classical methods for nonlinear quantum oscillations problems

Journal Article · Mon Oct 15 04:00:00 EDT 2012 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.4755836· OSTI ID:22093764

Moncrief, Vincent ^[1]; Marini, Antonella ^[2]; Maitra, Rachel ^[3]

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1. Department of Physics and Department of Mathematics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520 (United States)
2. Department of Mathematics, Yeshiva University, 500 West 185th Street, New York, New York 10033, USA and Department of Mathematics, University of L'Aquila, Via Vetoio, 67010 L'Aquila, AQ (Italy)
3. Department of Physics, Albion College, 611 E. Porter Street, Albion, Michigan 49224 (United States)

We develop a modified semi-classical approach to the approximate solution of Schroedinger's equation for certain nonlinear quantum oscillations problems. In our approach, at lowest order, the Hamilton-Jacobi equation of the conventional semi-classical formalism is replaced by an inverted-potential-vanishing-energy variant thereof. With suitable smoothness, convexity and coercivity properties imposed on its potential energy function, we prove, using methods drawn from the calculus of variations together with the (Banach space) implicit function theorem, the existence of a global, smooth 'fundamental solution' to this equation. Higher order quantum corrections thereto, for both ground and excited states, can then be computed through the integration of associated systems of linear transport equations, derived from Schroedinger's equation, and formal expansions for the corresponding energy eigenvalues obtained therefrom by imposing the natural demand for smoothness on the (successively computed) quantum corrections to the eigenfunctions. For the special case of linear oscillators our expansions naturally truncate, reproducing the well-known exact solutions for the energy eigenfunctions and eigenvalues. As an explicit application of our methods to computable nonlinear problems, we calculate a number of terms in the corresponding expansions for the one-dimensional anharmonic oscillators of quartic, sectic, octic, and dectic types and compare the results obtained with those of conventional Rayleigh/Schroedinger perturbation theory. To the orders considered (and, conjecturally, to all orders) our eigenvalue expansions agree with those of Rayleigh/Schroedinger theory whereas our wave functions more accurately capture the more-rapid-than-gaussian decay known to hold for the exact solutions to these problems. For the quartic oscillator in particular our results strongly suggest that both the ground state energy eigenvalue expansion and its associated wave function expansion are Borel summable to yield natural candidates for the actual exact ground state solution and its energy. Our techniques for proving the existence of the crucial 'fundamental solution' to the relevant (inverted-potential-vanishing-energy) Hamilton-Jacobi equation have the important property of admitting interesting infinite dimensional generalizations. In a project paralleling the present one we shall show how this basic construction can be carried out for the Yang-Mills equations in Minkowski spacetime.

• OSTI ID: 22093764
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 10 Vol. 53; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ANHARMONIC OSCILLATORS
APPROXIMATIONS
BANACH SPACE
COERCIVE FORCE
EIGENFUNCTIONS
EIGENVALUES
EXACT SOLUTIONS
EXCITED STATES
GROUND STATES
HAMILTON-JACOBI EQUATIONS
HARMONIC OSCILLATORS
MINKOWSKI SPACE
NONLINEAR PROBLEMS
OSCILLATIONS
PERTURBATION THEORY
SCHROEDINGER EQUATION
SPACE-TIME
TRANSPORT THEORY
WAVE FUNCTIONS
YANG-MILLS THEORY