We develop a modified semi-classical approach to the approximate solution of Schrödinger'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 2 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 Schrödinger'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/Schrödinger perturbation theory.To the orders considered (and, conjecturally, to all orders) our eigenvalue expansions agree with those of Rayleigh/Schrödinger 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.

Elegant 'microlocal' methods have long since been extensively developed for the analysis of conventional Schrödinger eigenvalue problems. For technical reasons though these methods have not heretofore been applicable to quantum field theories.In this article however we initiate a 'Euclidean signature semi-classical' program to extend the scope of these analytical techniques to encompass the study of selfinteracting scalar fields in 1 + 1, 2 + 1 and 3 + 1 dimensions. The basic microlocal approach entails, first of all, the solution of a single, nonlinear equation of Hamilton-2 Jacobi type followed by the integration (for both ground and excited states) of a sequence of linear 'transport' equations along the 'flow' generated by the 'fundamental solution' to the aforementioned Hamilton-Jacobi equation. Using a combination of the direct method of the calculus of variations, elliptic regularity theory and the Banach space version of the implicit function theorem we establish, in a suitable function space setting, the existence, uniqueness and global regularity of this needed 'fundamental solution' to the relevant, Euclidean signature Hamilton-Jacobi equation for the systems under study. Our methods are applicable to (massive) scalar fields with polynomial self-interactions of renormalizable type. They can, as we shall show elsewhere, also be applied to Yang-Mills fields in 2 + 1 and 3 + 1 dimensions.3

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It has long been realized that the natural 'orbit space' for non-abelian Yang-Mills dynamics (i.e., the reduced configuration space of gauge equivalence classes of spatial connections) is a positively curved (infinite dimensional) Riemannian manifold. Expanding upon this result I.M. Singer was led to propose that strict positivity of the corresponding Ricci tensor (computable from the rigorously defined curvature tensor through a suitable zeta function regularization procedure) could play a fundamental role in establishing that the associated Schrödinger operator admits a spectral gap.

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