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Register a new userAnalytic geometry and semi-classical analysis
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Mauricio D. Garay
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Expository paper on the relations between perturbation theory of pseudo-differential operators, finiteness theorems and deformations of Lagrangian varieties.
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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 methods developped by the geometers in the seventies for the heat expansions.
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 of several physically significant, interacting quantum field theories. Our main focus is on applying these `Euclidean-signature semi-classical methods to self-interacting (real) scalar fields of renormalizable type in 2, 3 and 4 spacetime dimensions and to Yang-Mills fields in 3 and 4 spacetime dimensions. A central argument in favor of our program is that the asymptotic methods for Schr{o}dinger operators developed in the microlocal literature are far superior, for the quantum mechanical systems to which they naturally apply, to the conventional WKB methods of the physics literature and that these methods can be modified, by techniques drawn from the calculus of variations and the analysis of elliptic boundary value problems, to apply to certain (bosonic) quantum field theories. Unlike conventional (Rayleigh/ Schr{o}dinger) perturbation theory these methods allow one to avoid the artificial decomposition of an interacting system into an approximating `unperturbed system and its perturbation and instead to keep the nonlinearities (and, if present gauge invariances) of an interacting system intact at every level of the analysis.
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.
An integrable Hamiltonian system presents monodromy if the action-angle variables cannot be defined globally. We consider a classical system with azimuthal symmetry and explore the topology structure of its phase space. Based on the behavior of closed orbits around singular points or regions of the energy-momentum plane, a semi-theoretical method is derived to detect classical monodromy. The validity of the monodromy test is numerically illustrated for several systems with azimuthal symmetry.
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 self-adjoint realization of the formal Hamiltonian $-frac{hbar^{2}}{2m},frac{d^{2},}{dx^{2}} + beta,delta_{0}$, with $delta_{0}$ the derivative of Diracs delta distribution at $x = 0$ and $beta$ a real parameter. We show that in the semi-classical limit such a quantum evolution can be approximated (w.r.t. the $L^{2}(mathbb{R})$-norm, uniformly for any $t in mathbb{R}$ away from the collision time) by $e^{frac{i}{hbar} A_{t}} e^{it L_{B}} phi^{hbar}_{x}$, where $A_{t} = frac{p^{2}t}{2m}$, $phi_{x}^{hbar}(xi) := psi^{hbar}_{xi}(x)$ and $L_{B}$ is a suitable self-adjoint extension of the restriction to $mathcal{C}^{infty}_{c}({mathscr M}_{0})$, ${mathscr M}_{0} := {(q,p) in mathbb{R}^{2},|,q eq 0}$, of ($-i$ times) the generator of the free classical dynamics. While the operator $L_{B}$ here utilized is similar to the one appearing in our previous work [C. Cacciapuoti, D. Fermi, A. Posilicano, The semi-classical limit with a delta potential, Annali di Matematica Pura e Applicata (2020)] regarding the semi-classical limit with a delta potential, in the present case the approximation gives a smaller error: it is of order $hbar^{7/2-lambda}$, $0 < lambda < 1/2$, whereas it turns out to be of order $hbar^{3/2-lambda}$, $0 < lambda < 3/2$, for the delta potential. We also provide similar approximation results for both the wave and scattering operators.
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