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We deduce Levinson\'{}s theorem in non-relativistic quantum mechanics in one dimension as a sum rule for the spectral density constructed from asymptotic data. We assume a self-adjoint hamiltonian which guarantees completeness; the potential needs not to be isotropic and a zero-energy resonance is automatically taken into account. Peculiarities of this one-dimension case are explained because of the ``critical'' character of the free case $u(x) = 0$, in the sense that any atractive potential forms at least a bound state. We believe this method is more general and direct than the usual one in which one proves the theorem first for single wave modes and performs analytical continuation., Comment: Presented in the XXV ICGTMP. Cocoyoc (Mexico), August 2004

We study the quantum-mechanical tunneling phenomenon in models which include the existence of non-equivalent vacua. For such a purpose we evaluate the euclidean propagator between two minima of the potential at issue in terms of the quadratic fluctuations over the corresponding instantons. The effect of the multi-instanton configurations are included by means of the alternate dilute-gas approximation., Comment: 12 pages. Presented at the Seventh International Wigner Symposium. August 2001. College Park, MD, USA

We consider in detail how the quantum-mechanical tunneling phenomenon occurs in a well-behaved octic potential. Our main tool will be the euclidean propagator just evaluated between two minima of the potential at issue. For such a purpose we resort to the standard semiclassical approximation which takes into account the fluctuations over the instantons, i.e. the finite-action solutions of the euclidean equation of motion. As regards the one-instanton approach, the functional determinant associated with the so-called stability equation is analyzed in terms of the asymptotic behaviour of the zero-mode. The conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the two different frequencies derived from the minima of the potential involved in the computation. The second instanton of the model is studied in a similar way. The physical effects of the multi-instanton configurations are included in this context by means of the alternate dilute-gas approximation where the two instantons participate to provide us with the final expression of the propagator., Comment: RevTex, 13 pages

We consider the quantum tunneling phenomenon in a well-behaved triple-well potential. As required by the semiclassical approximation we take into account the quadratic fluctuations over the instanton which represents as usual the localised finite-action solution of the euclidean equation of motion. The determinants of the quadratic differential operators at issue are evaluated by means of the Gelfang-Yaglom method. In doing so the explicit computation of the conventional ratio of determinants takes as reference the harmonic oscillator whose frequency is the average of the individual frequencies derived from the non-equivalent minima of the potential. Eventually the physical effects of the multi-instanton configurations are included in this approach. As a matter of fact we obtain information about the energies of the ground-state and the two first excited levels of the discrete spectrum at issue., Comment: 12 pages, RevTex

We review the euclidean path-integral formalism in connection with the one-dimensional non-relativistic particle. The configurations which allow to construct a semiclassical approximation classify themselves into either topological (instantons) and non-topological (bounces) solutions. The quantum amplitudes consist on an exponential associated with the classical contribution multiplied by the fluctuation factor which is given by a functional determinant. The eigenfunctions as well as the energy eigenvalues of the quadratic operators at issue can be written in closed form due to the shape-invariance property. Accordingly we resort to the zeta-function method to compute the functional determinants in a systematic way. The effect of the multi-instantons configurations is also carefully considered. To illustrate the instanton calculus in a relevant model we go to the double-well potential. The second popular case is the periodic-potential where the initial levels split into bands. The quantum decay rate of the metastable states in a cubic model is evaluated by means of the bounce-like solution., Comment: To appear in Commun. Math. Scienc

We consider in detail the quantum-mechanical problem associated with the motion of a one-dimensional particle under the action of the double-well potential. Our main tool will be the euclidean (imaginary time) version of the path-integral method. Once we perform the Wick rotation, the euclidean equation of motion is the same as the usual one for the point particle in real time, except that the potential at issue is turned upside down. In doing so, our double-well potential becomes a two-humped potential. As required by the semiclassical approximation we may study the quadratic fluctuations over the instanton which represents in this context the localised finite-action solutions of the euclidean equation of motion. The determinants of the quadratic differential operators are evaluated by means of the zeta-function method. We write in closed form the eigenfunctions as well as the energy eigenvalues corresponding to such operators by using the shape-invariance symmetry. The effect of the multi-instantons configurations is also included in this approach., Comment: 27 pages, RevTex