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Register a new userA semi-theoretical method to detect classical monodromy
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Added by
Juan Jos\\'e Omiste
Publication date
2021
fields
Physics
and research's language is
English
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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.
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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 develop physically admissible lattice models in the harmonic approximation which define by Hamiltons variational principle fractional Laplacian matrices of the forms of power law matrix functions on the n -dimensional periodic and infinite lattice in n=1,2,3,..n=1,2,3,.. dimensions. The present model which is based on Hamiltons variational principle is confined to conservative non-dissipative isolated systems. The present approach yields the discrete analogue of the continuous space fractional Laplacian kernel. As continuous fractional calculus generalizes differential operators such as the Laplacian to non-integer powers of Laplacian operators, the fractional lattice approach developed in this paper generalized difference operators such as second difference operators to their fractional (non-integer) powers. Whereas differential operators and difference operators constitute local operations, their fractional generalizations introduce nonlocal long-range features. This is true for discrete and continuous fractional operators. The nonlocality property of the lattice fractional Laplacian matrix allows to describe numerous anomalous transport phenomena such as anomalous fractional diffusion and random walks on lattices. We deduce explicit results for the fractional Laplacian matrix in 1D for finite periodic and infinite linear chains and their Riesz fractional derivative continuum limit kernels.
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.
Expository paper on the relations between perturbation theory of pseudo-differential operators, finiteness theorems and deformations of Lagrangian varieties.
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.
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