Feynman path integrals and the trace formula for the Schrödinger operators

Abstract

We study Schrödinger operators of the form $$H = - \frac{{h^2 }}{2}\Delta + \frac{1}{2}x \cdot A^2 x + V(x)$$ on ℝ d , whereA 2 is a strictly positive symmetricd×d matrix andV(x) is a continuous real function which is the Fourier transform of a bounded measure. If λ n are the eigenvalues ofH we show that the theta function $$\theta (t) = \sum\limits_n {\exp \left( { - \frac{i}{h}t\lambda _n } \right)}$$ is explicitly expressible in terms of infinite dimensional oscillatory integrals (Feynman path integrals) over the Hilbert space of closed trajectories. We use these explicit expressions to give the asymptotic behaviour of θ(t) for smallh in terms of classical periodic orbits, thus obtaining a trace formula for the Schrödinger operators. This then yields an asymptotic expansion of the spectrum ofH in terms of the periodic orbits of the corresponding classical mechanical system. These results extend to the physical case the recent work on Poisson and trace formulae for compact manifolds.