Applying reproducing kernels to the evaluation and approximation of the simple and time-dependent imaginary time harmonic oscillator path integrals

Abstract

Reproduction of kernel Hilbert spaces offers an attractive setting for imaginary time path integrals, since they allow to naturally define a probability on the space of paths, which is equal to the probability associated with the paths in Feynman's path integral formulation. This study shows that if the propagator is Gaussian, its variance equals the squared norm of a linear functional on the space of paths. This can be used to rederive the harmonic oscillator propagator, as well as to offer a finite-dimensional perturbative approximation scheme for the time-dependent oscillator wave function and its ground state energy, and its bound error. The error is related to the rate of decay of the Fourier coefficients of the time-dependent part of the potential. When the rate of decay increases beyond a certain threshold, the error in the approximation over a subspace of dimension n is of order (1/n ³).

Keywords:

• Path integral
• Imaginary time
• Finite-dimensional approximation
• Reproducing kernels
• Time-dependent harmonic oscillator

Keywords:

• Mathematics Subject Classifications: Numerical analysis 65Z05 (applications to physics)
• General mathematical topics and methods in quantum theory 81Q30 (Feynman integrals)