## 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} ).

Original language | English |
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Pages (from-to) | 793-810 |

Number of pages | 18 |

Journal | Applicable Analysis |

Volume | 85 |

Issue number | 6-7 |

DOIs | |

State | Published - Jun 2006 |

## Keywords

- Finite-dimensional approximation
- General mathematical topics and methods in quantum theory 81Q30 (Feynman integrals)
- Imaginary time
- Mathematics Subject Classifications: Numerical analysis 65Z05 (applications to physics)
- Path integral
- Reproducing kernels
- Time-dependent harmonic oscillator

## ASJC Scopus subject areas

- Analysis
- Applied Mathematics