Search

We consider the linear relaxation Boltzmann equation in a semiclassical framework. We construct a family of sharp quasimodes for the associated operator which yields sharp spectral asymptotics for its small spectrum in the low temperature regime. We deduce some information on the long time behavior of the solutions with a sharp estimate on the return to equilibrium as well as a quantitative metastability result. The main novelty is that the collision operator is a pseudo-differential operator in the critical class S 1/2 and that its action on the Gaussian quasimodes yields a superposition of exponentials. [ABSTRACT FROM AUTHOR]

We consider differential operators defined as Friedrichs extensions of quadratic forms with non-smooth coefficients. We prove a two-term optimal asymptotic for the Riesz means of these operators and thereby also reprove an optimal Weyl law under certain regularity conditions. The methods used are then extended to consider more general admissible operators perturbed by a rough differential operator and to obtain optimal spectral asymptotics again under certain regularity conditions. For the Weyl law, we assume that the coefficients are differentiable with Hölder continuous derivatives, while for the Riesz means we assume that the coefficients are twice differentiable with Hölder continuous derivatives. [ABSTRACT FROM AUTHOR]

We analyze the propagation of two-dimensional dispersive and relativistic wavepackets localized in the vicinity of the zero level set T of a slowly varying domain wall modeling the interface separating two insulating media. We propose a semiclassical oscillatory representation of the propagating wavepackets and provide an estimate of their accuracy in appropriate energy norms. We describe the propagation of relativistic modes along T and analyze dispersive modes by a stationary phase method. In the absence of turning points, we show that arbitrary smooth localized initial conditions may be represented as a superposition of such wavepackets. In the presence of turning points, the results apply only for sufficiently high-frequency wavepackets. The theory finds applications for both Dirac systems of equations modeling topologically nontrivial systems as well as Klein--Gordon equations, which are topologically trivial. [ABSTRACT FROM AUTHOR]

We study the asymptotic behaviour of the modified two-dimensional Schrödinger equation (D t − F (D)) u = λ | u | u in the critical regime, where λ ∈ C with ℑ λ ⩾ 0 and F (ξ) is a second order constant coefficients elliptic symbol. For any smooth initial datum of size ε ≪ 1 , we prove that the solution is global-in-time, combining the vector fields method and a semiclassical analysis method introduced by Delort. Moreover, we present the pointwise decay estimates and the large time asymptotic formulas of the solution. [ABSTRACT FROM AUTHOR]

In this article, we study the asymptotic fields of the Yukawa particle-field model of quantum physics in the semiclassical regime ℏ → 0 , with an interaction subject to an ultraviolet cutoff. We show that the transition amplitudes between final (respectively initial) states converge towards explicit quantities involving the outgoing (respectively incoming) wave operators of the nonlinear Schrödinger–Klein–Gordon (S-KG) equation. Thus, we rigorously link the scattering theory of the Yukawa model to that of the Schrödinger–Klein–Gordon equation. Moreover, we prove that the asymptotic vacuum states of the Yukawa model have a phase space concentration property around classical radiationless solutions. Under further assumptions, we show that the S-KG energy admits a unique minimizer modulo symmetries and we identify exactly the semiclassical measure of Yukawa ground states. Some additional consequences of asymptotic completeness are also discussed, and some further open questions are raised. [ABSTRACT FROM AUTHOR]

In this article, we investigate spectral properties of the sublaplacian -ΔG on the Engel group, which is the main example of a Carnot group of step 3.We develop a new approach to the Fourier analysis on the Engel group in terms of a frequency set. This enables us to give fine estimates on the convolution kernel satisfying F(-ΔG)u = U * kF, for suitable scalar functions F, and in turn to obtain proofs of classical functional embeddings, via Fourier techniques. This analysis requires a summability property on the spectrum of the quartic oscillator, which we obtain by means of semiclassical techniques and which is of independent interest. [ABSTRACT FROM AUTHOR]

We study the Agmon estimate and the exponential decay of eigenfunctions for multi-dimensional discrete Schrödinger operators with emphasis on the microlocal analysis on the torus. We first consider the semiclassical setting where semiclassical continuous Schrödinger operators are discretized with the mesh width proportional to the semiclassical parameter. For a general class of potentials, we prove the Agmon estimate for eigenfunctions using the Agmon metric which is a Finsler metric rather than a Riemannian metric. Klein and Rosenberger (2008) proved this by a different argument in the case of a potential minimum. Our argument seems to be simpler. We also prove the Agmon estimate and the optimal anisotropic exponential decay of eigenfunctions for discrete Schrödinger operators in the non-semiclassical standard setting. [ABSTRACT FROM AUTHOR]

The main result of this article gives scaling asymptotics of the Wigner distributions W φ N γ , φ N γ of isotropic harmonic oscillator orbital coherent states φ N γ concentrating along Hamiltonian orbits γ in shrinking tubes around γ in phase space. In particular, these Wigner distributions exhibit a hybrid semi-classical scaling. That is, simultaneously, we have an Airy scaling when the tube has radius N − 2 / 3 normal to the energy surface ΣE, and a Gaussian scaling when the tube has radius N − 1 / 2 tangent to ΣE. [ABSTRACT FROM AUTHOR]

On an asymptotically conical manifold, we prove time decay estimates for the flow of the Schrödinger wave and Klein–Gordon equations via some differentiability properties of the spectral measure. To keep the paper at a reasonable length, we limit ourselves to the low-energy part of the spectrum, which is the one that dictates the decay rates. With this paper, we extend sharp estimates that are known in the asymptotically flat case (see Bouclet and Burq in Duke Math J 170(11):2575–2629, 2021, ) to this more general geometric framework and therefore recover the same decay properties as in the Euclidean case. The first step is to prove some resolvent estimates via a limiting absorption principle. It is at this stage that the proof of the previously mentioned authors fails, in particular when we try to recover a low-frequency positive commutator estimate. Once the resolvent estimates are established, we derive regularity for the spectral measure that in turn is applied to obtain the decay of the flows. [ABSTRACT FROM AUTHOR]

We study quantum particles in interaction with a force-carrying field, in the quasi-classical limit. This limit is characterized by the field having a very large number of excitations (it is therefore macroscopic), while the particles retain their quantum nature. We prove that the interacting microscopic dynamics converges, in the quasi-classical limit, to an effective dynamics where the field acts as a classical environment that drives the quantum particles. [ABSTRACT FROM AUTHOR]

We define a Schrödinger operator on the half-space with a discontinuous magnetic field having a piecewise-constant strength and a uniform direction. Motivated by applications in the theory of superconductivity, we study the infimum of the spectrum of the operator. We give sufficient conditions on the strength and the direction of the magnetic field such that the aforementioned infimum is an eigenvalue of a reduced model operator on the half-plane. We use the Schrödinger operator on the half-space to study a new semiclassical problem in bounded domains of the space, considering a magnetic Neumann Laplacian with a piecewise-constant magnetic field. We then make precise the localization of the semiclassical ground state near specific points at the discontinuity jump of the magnetic field. [ABSTRACT FROM AUTHOR]

Quantum-classical molecular dynamics, as a partial classical limit of the full quantum Schrödinger equation, is a widely used framework for quantum molecular dynamics. The underlying equations are nonlinear in nature, containing a quantum part (representing the electrons) and a classical part (standing for the nuclei). An accurate simulation of the wave function typically requires a time step comparable to the rescaled Planck constant h, resulting in a formidable cost when h ≪ 1. We prove an additive observable error bound of Schwartz observables for the proposed time-splitting schemes based on semiclassical analysis, which decreases as h becomes smaller. Furthermore, we establish a uniform-in-h observable error bound, which allows for an O (1) time step to accurately capture the physical observable regardless of the size of h. Numerical results verify our estimates. [ABSTRACT FROM AUTHOR]

We study a commonly-used second-kind boundary-integral equation for solving the Helmholtz exterior Neumann problem at high frequency, where, writing Γ for the boundary of the obstacle, the relevant integral operators map L 2 (Γ) to itself. We prove new frequency-explicit bounds on the norms of both the integral operator and its inverse. The bounds on the norm are valid for piecewise-smooth Γ and are sharp up to factors of log k (where k is the wavenumber), and the bounds on the norm of the inverse are valid for smooth Γ and are observed to be sharp at least when Γ is smooth with strictly-positive curvature. Together, these results give bounds on the condition number of the operator on L 2 (Γ) ; this is the first time L 2 (Γ) condition-number bounds have been proved for this operator for obstacles other than balls. [ABSTRACT FROM AUTHOR]

This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural. [ABSTRACT FROM AUTHOR]

In this paper we give an overview over some aspects of the modern mathematical theory of Ruelle resonances for chaotic, i.e. uniformly hyperbolic, dynamical systems and their implications in physics. First we recall recent developments in the mathematical theory of resonances, in particular how invariant Ruelle distributions arise as residues of weighted zeta functions. Then we derive a correspondence between weighted and semiclassical zeta functions in the setting of negatively curved surfaces. Combining this with results of Hilgert, Guillarmou and Weich yields a high frequency interpretation of invariant Ruelle distributions as quantum mechanical matrix coefficients in constant negative curvature. We finish by presenting numerical calculations of phase space distributions in the more physical setting of three-disk scattering systems. [ABSTRACT FROM AUTHOR]

We show that for almost all perturbations in a one-parameter family of KAM Hamiltonians on a smooth compact surface and for almost all KAM Lagrangian tori Λω, we can find a semiclassical measure with positive mass on Λω. [ABSTRACT FROM AUTHOR]

In a previous paper, we introduced a class of "semiclassical functions of isotropic type," starting with a model case and applying Fourier integral operators associated with canonical transformations. These functions are a substantial generalization of the "oscillatory functions of Lagrangian type" that have played major role in semiclassical and microlocal analysis. In this paper we exhibit more clearly the nature of these isotropic functions by obtaining oscillatory integral expressions for them. Then, we use these to prove that the classes of isotropic functions are equivariant with respect to the action of general FIOs (under the usual cleanintersection hypothesis). The simplest examples of isotropic states are the "coherent states," a class of oscillatory functions that has played a pivotal role in mathematics and theoretical physics beginning with their introduction by of Schrödinger in the 1920's. We prove that every oscillatory function of isotropic type can be expressed as a superposition of coherent states, and examine some implications of that fact. We also show that certain functions of elliptic operators have isotropic functions for Schwartz kernels. This lead us to a result on an eigenvalue counting function that appears to be new (Corollary 4.5). [ABSTRACT FROM AUTHOR]

For the Helmholtz equation posed in the exterior of a Dirichlet obstacle, we prove that if there exists a family of quasimodes (as is the case when the exterior of the obstacle has stable trapped rays), then there exist near-zero eigenvalues of the standard variational formulation of the exterior Dirichlet problem (recall that this formulation involves truncating the exterior domain and applying the exterior Dirichlet-to-Neumann map on the truncation boundary). Our motivation for proving this result is that (a) the finite-element method for computing approximations to solutions of the Helmholtz equation is based on the standard variational formulation, and (b) the location of eigenvalues, and especially near-zero ones, plays a key role in understanding how iterative solvers such as the generalized minimum residual method (GMRES) behave when used to solve linear systems, in particular those arising from the finite-element method. The result proved in this paper is thus the first step towards rigorously understanding how GMRES behaves when applied to discretizations of high-frequency Helmholtz problems under strong trapping (the subject of the companion paper [P. Marchand et al., Adv. Comput. Math., to appear]). [ABSTRACT FROM AUTHOR]

Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties. [ABSTRACT FROM AUTHOR]

This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set with Dirichlet conditions on the boundary. Assuming that the magnetic field is positive and a few generic conditions, we establish the simplicity of the eigenvalues and provide accurate asymptotic estimates involving Segal--Bargmann and Hardy spaces associated with the magnetic field. [ABSTRACT FROM AUTHOR]

We prove quantum ergodicity for the eigenfunctions of the pseudo-Laplacian on surfaces with hyperbolic cusps and ergodic geodesic flows. [ABSTRACT FROM AUTHOR]

The Kubo–Martin–Schwinger (KMS) condition is a widely studied fundamental property in quantum statistical mechanics which characterizes the thermal equilibrium states of quantum systems. In the seventies, Gallavotti and Verboven, proposed an analogue to the KMS condition for infinite classical mechanical systems and highlighted its relationship with the Kirkwood–Salzburg equations and with the Gibbs equilibrium measures. In this paper, we prove that in a certain limiting regime of high temperature the classical KMS condition can be derived from the quantum condition in the simple case of the Bose–Hubbard dynamical system on a finite graph. The main ingredients of the proof are Golden–Thompson inequality, Bogoliubov inequality and semiclassical analysis. [ABSTRACT FROM AUTHOR]

We study the dynamics of the three-dimensional polaron---a quantum particle coupled to bosonic fields--in the quasi-classical regime. In this case, the fields are very intense and the corresponding degrees of freedom can be treated semiclassically. We prove that in such a regime the effective dynamics for the quantum particles is approximated by the one generated by a time-dependent point interaction, i.e., a singular time-dependent perturbation of the Laplacian supported in a point. As a by-product, we also show that the unitary dynamics of a time-ependent point interaction can be approximated in strong operator topology by the one generated by time-dependent Schrödinger operators with suitably rescaled regular potentials. [ABSTRACT FROM AUTHOR]

We study the asymptotic behavior of eigenvalues and eigenmodes of the Witten Laplacian on a smooth compact Riemannian manifold without boundary. We show that they converge to the Pollicott-Ruelle spectral data of the corresponding gradient flow acting on appropriate anisotropic Sobolev spaces. As an application of our methods, we also construct a natural family of quasimodes satisfying the Witten-Helffer-Sjostrand tunneling formulas and the Fukaya conjecture on Witten deformation of the wedge product. [ABSTRACT FROM AUTHOR]

We discuss the problem of recovering geometric objects from the spectrum of a quantum integrable system. In the case of one degree of freedom, precise results exist. In the general case, we report on the recent notion of good labelings of asymptotic lattices. [ABSTRACT FROM AUTHOR]

We obtain new semiclassical estimates for pseudodifferential operators with low regular symbols. Such symbols appear naturally in a Cauchy Problem related to recent weak solutions to the unstable Muskat problem constructed via convex integration. In particular, our new estimates reveal the tight relation between the speed of opening of the mixing zone and the regularity of the interphase. [ABSTRACT FROM AUTHOR]

We explore the effect of sampling rates when measuring data given by Mf for special operators M arising in Thermoacoustic Tomography. We start with sampling requirements on Mf given f satisfying certain conditions. After this we discuss the resolution limit on f posed by the sampling rate of Mf without assuming any conditions on these sampling rates. Next we discuss aliasing artifacts when Mf is known to be under sampled in one or more of its variables. Finally, we discuss averaging of measurement data and resulting aliasing and artifacts, along with a scheme for anti-aliasing. [ABSTRACT FROM AUTHOR]

We consider the nonlinear Hartree and Vlasov equations around a translation-invariant (homogeneous) stationary state in infinite volume, for a short range interaction potential. For both models, we consider time-dependent solutions which have a finite relative energy with respect to the reference translation-invariant state. We prove the convergence of the Hartree solutions to the Vlasov ones in a semi-classical limit and obtain as a by-product global well-posedness of the Vlasov equation in the (relative) energy space. [ABSTRACT FROM AUTHOR]

We consider Toeplitz operators associated with the renormalized Bochner Laplacian on high tensor powers of a positive line bundle on a compact symplectic manifold. We study the asymptotic behavior, in the semiclassical limit, of low-lying eigenvalues and the corresponding eigensections of a self-adjoint Toeplitz operator under assumption that its principal symbol has a non-degenerate minimum with discrete wells. As an application, we prove upper bounds for low-lying eigenvalues of the Bochner Laplacian in the semiclassical limit. [ABSTRACT FROM AUTHOR]

Eigenspaces of the quantum isotropic Harmonic Oscillator H ̂ ℏ : = − ℏ 2 2 Δ + 1 2 | | x | | 2 on R d have extremally high multiplicites and the eigenspace projections Π ℏ , E N (ℏ) have special asymptotic properties. This article gives a detailed study of their Wigner distributions W ℏ , E N (ℏ) (x , ξ). Heuristically, if E N (ℏ) = E , W ℏ , E N (ℏ) (x , ξ) is the "quantization" of the energy surface ΣE, and should be like the delta-function δ Σ E on ΣE; rigorously, W ℏ , E N (ℏ) (x , ξ) tends in a weak* sense to δ Σ E . But its pointwise asymptotics and scaling asymptotics have more structure. The main results give Bessel asymptotics of W ℏ , E N (ℏ) (x , ξ) in the interior H (x , ξ) < E of ΣE; interface Airy scaling asymptotics in tubes of radius ℏ 2 / 3 around ΣE, with (x , ξ) either in the interior or exterior of the energy ball; and exponential decay rates in the exterior of the energy surface. [ABSTRACT FROM AUTHOR]

Inspired by special and general relativistic systems that can have Hamiltonians involving square roots or more general fractional powers, in this article, we address the question of how a suitable set of coherent states for such systems can be obtained. This becomes a relevant topic if the semiclassical sector of a given quantum theory is to be analysed. As a simple setup, we consider the toy model of a deparametrised system with one constraint that involves a fractional power of the harmonic oscillator Hamiltonian operator, and we discuss two approaches to finding suitable coherent states for this system. In the first approach, we consider Dirac quantisation and group averaging, as have been used by Ashtekar et al., but only for integer powers of operators. Our generalisation to fractional powers yields in the case of the toy model a suitable set of coherent states. The second approach is inspired by coherent states based on a fractional Poisson distribution introduced by Laskin, which however turn out not to satisfy all properties to yield good semiclassical results for the operators considered here and in particular do not satisfy a resolution of identity as claimed. Therefore, we present a generalisation of the standard harmonic oscillator coherent states to states involving fractional labels, which approximate the fractional operators in our toy model semiclassically more accurately and satisfy a resolution of identity. In addition, motivated by the way the proof of the resolution of identity is performed, we consider these kind of coherent states also for the polymerised harmonic oscillator and discuss their semiclassical properties.

The internal control problem for the Kadomtsev-Petviashvili II equation, better known as KP-II, is the object of study in this paper. The controllability in L²(T²) from a vertical strip is proved using the Hilbert uniqueness method through the techniques of semiclassical and microlocal analysis. Additionally, a negative result for the controllability in L²(T²) from a horizontal strip is also shown. [ABSTRACT FROM AUTHOR]

This article is concerned with compositions in the context of three standard quantizations in the framework of Fock spaces, namely, anti-Wick, Wick and Weyl quantizations. The first one is a composition of states also known as a Wick product and is closely related to the standard scattering identification operator encountered in Quantum Electrodynamics for issues on time dynamics (see [29-13]). Anti-Wick quantization and Segal–Bargmann transforms are implied here for that purpose. The other compositions are for observables (operators in some specific classes) for the Wick and Weyl symbols. For the Wick and Weyl symbols of the composition of two operators, we obtain an absolutely converging series and for the Weyl symbol, the remainder terms up to any orders of the expansion are controlled, still in the Fock space framework. [ABSTRACT FROM AUTHOR]

This paper is concerned with resolvent estimates on the real axis for the Helmholtz equation posed in the exterior of a bounded obstacle with Dirichlet boundary conditions when the obstacle is trapping. There are two resolvent estimates for this situation currently in the literature: (i) in the case of elliptic trapping the general "worst case" bound of exponential growth applies, and examples show that this growth can be realized through some sequence of wavenumbers; (ii) in the prototypical case of hyperbolic trapping where the Helmholtz equation is posed in the exterior of two strictly convex obstacles (or several obstacles with additional constraints) the nontrapping resolvent estimate holds with a logarithmic loss. This paper proves the first resolvent estimate for parabolic trapping by obstacles, studying a class of obstacles the prototypical example of which is the exterior of two squares (in two dimensions) or two cubes (in three dimensions), whose sides are parallel. We show, via developments of the vector-field/multiplier argument of Morawetz and the first application of this methodology to trapping configurations, that a resolvent estimate holds with a polynomial loss over the nontrapping estimate. We use this bound, along with the other trapping resolvent estimates, to prove results about integral equation formulations of the boundary value problem in the case of trapping. Feeding these bounds into existing frameworks for analyzing finite and boundary element methods, we obtain the first wavenumber-explicit proofs of convergence for numerical methods for solving the Helmholtz equation in the exterior of a trapping obstacle. [ABSTRACT FROM AUTHOR]

Let f: Rd → R be a smooth function, and let (Xt)t≥ 0 be the stochastic process solution to the overdamped Langevin dynamics dXt = f(Xt)dt +√h dBt. Let Ω ⊂ Rd be a smooth bounded domain, and assume that f| Ω is a double-well potential with degenerate barriers. In this work, we study in the small temperature regime, i.e., when h → 0+, the asymptotic repartition of the quasi-stationary distribution of (Xt)t\geq 0 in Ω within the two wells of f| Ω. We show that this distribution generically concentrates in precisely one well of f| Ω when h → 0+ but can nevertheless concentrate in both wells when f| Ω admits sufficient symmetries. This phenomenon corresponds to the so-called tunneling effect in semiclassical analysis. We also investigate in this setting the asymptotic behavior when h → 0+ of the first exit point distribution from Ω of (Xt)t\geq 0 when X0 is distributed according to the quasi-stationary distribution. [ABSTRACT FROM AUTHOR]

The aim of this paper is to discuss the Besov spaces bounds for semiclassical Fourier integral operator. We give the conditions that the symbol and the phase function must satisfy for this operator to be bounded. [ABSTRACT FROM AUTHOR]

Let C be a smooth planar curve. We assume that C is simple, closed, smooth, symmetric with respect to an axis and its curvature attains its minimum at exactly two points away from the axis of symmetry. In a tubular neighborhood about C, we study the Laplace operator with a magnetic flux and mixed boundary conditions. As the thickness of the domain tends to 0, we establish an explicit asymptotic formula for the splitting of the first two eigenvalues (tunneling effect). [ABSTRACT FROM AUTHOR]

This paper focuses on the derivation and analysis of the frozen Gaussian approximation (FGA) for the Dirac equation in the semiclassical regime. Unlike the strictly hyperbolic system studied in [J. Lu and X. Yang, Comm. Pure Appl. Math., 65 (2012), pp. 759-789], the Dirac equation possesses eigenfunction spaces of multiplicity two, which demands more delicate expansions for deriving the amplitude equations in FGA. Moreover, we prove that the nonrelativistic limit of the FGA for the Dirac equation is the FGA of the Schrödinger equation, which shows that the nonrelativistic limit is asymptotically preserved after one applies FGA as the semiclassical approximation. Numerical experiments including the Klein paradox are presented to illustrate the method and confirm part of the analytical results. [ABSTRACT FROM AUTHOR]

We provide an exact version of the Egorov Theorem for a class of Schrödinger operators in L 2 (T) , where T = R / 2 π Z is the one-dimensional torus. We show that the classical Hamiltonian, after the symplectic transformation to action coordinates, can be composed with a toroidal semiclassical ψ do in order to recover the Schrödinger operator. This result turns out to be strictly related to the Bohr-Sommerfeld quantization rules as well as to the inverse spectral problem and the periodic homogenization of Hamilton–Jacobi equations. [ABSTRACT FROM AUTHOR]

Abstract This paper establishes an aspect of Bohr's correspondence principle, i.e. that quantum mechanics converges in the high frequency limit to classical mechanics, for commuting semiclassical unitary operators. We prove, under minimal assumptions, that the semiclassical limit of the convex hulls of the quantum spectrum of a collection of commuting semiclassical unitary operators converges to the convex hull of the classical spectrum of the principal symbols of the operators. [ABSTRACT FROM AUTHOR]