A Proof of the Gutzwiller Semiclassical Trace Formula Using Coherent States Decomposition

Abstract

The Gutzwiller trace formula links the eigenvalues of the Schrödinger operator H as Planck's constant goes to zero (the semiclassical régime) with the closed orbits of the corresponding classical mechanical system. Gutzwiller gave a heuristic proof of this trace formula, using the Feynman integral representation for the propagator of H. Later, using the theory of Fourier integral operators, mathematicians gave rigorous proofs of the formula in various settings. Here we show how the use of coherent states allows us to give a simple and direct proof.

On the semiclassical functional calculus for h-dependent functions

“…For example, consider a Schwartz function χ : R → R whose Fourier transform has compact support. Then a common approach in the literature is to study the operator χ P (h)−E h in the context of semiclassical Fourier integral operators, E ∈ R being a fixed regular value of p. Writing f E h (x) := χ x−E h , this amounts to studying f E h (P (h)), as in the semiclassical Gutzwiller trace formula [1,4] and in the proof of the semiclassical quantum ergodicity theorem in [3]. Using the same techniques, one can also prove a semiclassical Weyl law for the smallest possible spectral window [E, E + h] with the best possible O(h)-remainder, see [2,3,7].…”

“…Specifically, one considers the extension 3 1], and ψ f ∈ C ∞ c (R) is equal to 1 in a neighborhood of supp f . The main feature of the extension map C ∞ c (R) → C ∞ c (C) defined by (3.7) is that the functionsf in its image are almost analytic, meaning that for each N ∈ N there is a constant C N > 0 such that…”

“…Then, (1.1) can be applied for each f ∈ C ∞ c ([E − ε/2, E + c + ε/2]), and by approximating the characteristic function of [E, E +c] with such functions, one gets the semiclassical Weyl law (1.2) (2πh) n # j ∈ J(h) : E j (h) ∈ [E, E + c] = vol T * X p −1 ([E, E + c]) + o (1) as h → 0, see [2,Cor. 9.7] and [13,Thm.…”

“…The largest class of h-dependent functions that we will consider is given by δ∈[0, 1 2 ) S comp δ , where for each δ ∈ [0, 1 2 ) the symbol S comp δ denotes the set of all families {f h } h∈(0,1] ⊂ C ∞ c (R) such that (1) {f h } h∈(0,1] defines an element of the semiclassical symbol class S δ (1 R ), meaning that…”

Periodic orbits, spectral oscillations, scaling, and vacuum energy: beyond HaMiDeW

“…The development of this subject continues actively in both mathematics and physics (e.g., [7,8]), but the period of greatest activity was around 1970 and can be represented by three classic papers, treating distinct problems: (1) Balian and Bloch [9], Laplacian in a bounded region; (2) Duistermaat and Guillemin [10], Laplace-Beltrami operator on a compact manifold (and generalizations); (3) Gutzwiller [11],…”