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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.

The notion of Loschmidt echo (also called "quantum fidelity") has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of "quantum chaos", quantum computation and decoherence. In this paper, we study this quantity semiclassically (as → 0), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their time evolution. For times not larger than the so-called "Ehrenfest time" C| log |, we are able to estimate semiclassically the Loschmidt Echo as a function of t (time), (Planck constant), and δ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the "return probability" (again on reference states being the coherent states) by the same method, as a function of t and .

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After recalling different formulations of the definition of supersymmetric quantum mechanics given in the literature, we discuss the relationships between them in order to provide an answer to the question raised in the title.

In this paper we perform an exact study of "Quantum Fidelity" (also called Loschmidt Echo) for the time-periodic quantum Harmonic Oscillator of Hamiltonian :Ĥ g (t) :=when compared with the quantum evolution induced byĤ 0 (t) (g = 0), in the case where f is a T -periodic function and g a real constant. The reference (initial) state is taken to be an arbitrary "generalized coherent state" in the sense of Perelomov. We show that, starting with a quadratic decrease in time in the neighborhood of t = 0, this quantum fidelity may recur to its initial value 1 at an infinite sequence of times t k . We discuss the result when the classical motion induced by HamiltonianĤ 0 (t) is assumed to be stable versus unstable. A beautiful relationship between the quantum and the classical fidelity is also demonstrated.

A presentation and a generalization are given of the phenomenon of level rearrangement. This occurs when an attractive long-range potential is perturbed by a short-range attractive potential as its strength is increased. This phenomenon was first discovered in condensed matter physics and has also been studied in the physics of exotic atoms. A similar phenomenon occurs in a model that we propose, inspired by quantum dots, where a short-range interaction is added to a harmonic confinement.