Vincent Moncrief, Antonella Marini, Rachel Maitra
We develop a modified semi-classical approach to the approximate solution of
Schrodinger's equation for certain nonlinear quantum oscillations problems. At
lowest order, the Hamilton-Jacobi equation of the conventional semi-classical
formalism is replaced by an inverted-potential-vanishing-energy variant
thereof. Under smoothness, convexity and coercivity hypotheses on its potential
energy function, we prove, using the calculus of variations together with the
Banach space implicit function theorem, the existence of a global, smooth
`fundamental solution'. Higher order quantum corrections, for ground and
excited states, are computed through the integration of associated systems of
linear transport equations, and formal expansions for the corresponding energy
eigenvalues obtained by imposing smoothness on the quantum corrections to the
eigenfunctions. For linear oscillators our expansions naturally truncate,
reproducing the well-known solutions for the energy eigenfunctions and
eigenvalues. As an application, we calculate a number of terms in the
corresponding expansions for the one-dimensional anharmonic oscillators of
quartic, sectic, octic, and dectic types and find that our eigenvalue
expansions agree with those of Rayleigh/Schrodinger theory, whereas our wave
functions more accurately capture the more-rapid-than-gaussian decay. For the
quartic oscillator our results strongly suggest that 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 Hamilton Jacobi equation admit infinite
dimensional generalizations. In a parallel project we shall show how this
construction can be carried out for the Yang-Mills equations in Minkowski
spacetime.
View original:
http://arxiv.org/abs/1201.5311
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