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We consider the semi-classical limit of the quantum evolution of Gaussian coherent states whenever the Hamiltonian $mathsf H$ is given, as sum of quadratic forms, by $mathsf H= -frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}},dot{+},alphadelta_{0}$, with $alphainmathbb R$ and $delta_{0}$ the Dirac delta-distribution at $x=0$. We show that the quantum evolution can be approximated, uniformly for any time away from the collision time and with an error of order $hbar^{3/2-lambda}$, $0!<!lambda!<!3/2$, by the quasi-classical evolution generated by a self-adjoint extension of the restriction to $mathcal C^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0}:={(q,p)!in!mathbb R^{2},|,q! ot=!0}$, of ($-i$ times) the generator of the free classical dynamics; such a self-adjoint extension does not correspond to the classical dynamics describing the complete reflection due to the infinite barrier. Similar approximation results are also provided for the wave and scattering operators.
We consider the quantum evolution $e^{-ifrac{t}{hbar}H_{beta}} psi_{xi}^{hbar}$ of a Gaussian coherent state $psi_{xi}^{hbar}in L^{2}(mathbb{R})$ localized close to the classical state $xi equiv (q,p) in mathbb{R}^{2}$, where $H_{beta}$ denotes a sel
The spherically symmetric potential $a ,delta (r-r_0)+b,delta (r-r_0)$ is generalised for the $d$-dimensional space as a characterisation of a unique selfadjoint extension of the free Hamiltonian. For this extension of the Dirac delta, the spectrum
Expository paper on the relations between perturbation theory of pseudo-differential operators, finiteness theorems and deformations of Lagrangian varieties.
In this article we discuss our ongoing program to extend the scope of certain, well-developed microlocal methods for the asymptotic solution of Schr{o}dingers equation (for suitable `nonlinear oscillatory quantum mechanical systems) to the treatment
in the recent paper [Journal of Physics A, 43474-0288 (2011)], B. Helffer and R. Purice compute the second term of a semi-classical trace formula for a Schrodinger operator with magnetic field. We show how to recover their formula by using the method