Semiclassical Estimates Research Articles

Polymer quantization of the Einstein-Rosen wormhole throat

We present a polymer quantization of spherically symmetric Einstein gravity in which the polymerized variable is the area of the Einstein-Rosen wormhole throat. In the classical polymer theory, the singularity is replaced by a bounce at a radius that depends on the polymerization scale. In the polymer quantum theory, we show numerically that the area spectrum is evenly-spaced and in agreement with a Bohr-Sommerfeld semiclassical estimate, and this spectrum is not qualitatively sensitive to issues of factor ordering or boundary conditions except in the lowest few eigenvalues. In the limit of small polymerization scale we recover, within the numerical accuracy, the area spectrum obtained from a Schrodinger quantization of the wormhole throat dynamics. The prospects of recovering from the polymer throat theory a full quantum-corrected spacetime are discussed.

On the semiclassical 3D Neumann Laplacian with variable magnetic field

In this paper we are interested in the semiclassical estimates of the spectrum of the Neumann Laplacian in dimension 3. This work aims to present a complementary case to the one presented in the paper of Helffer and Morame in the case of constant magnetic field. More precisely, in the case when the magnetic field is variable and under the most generic condition for which boundary localizations can be observed, we prove a three terms upper bound for the lowest eigenvalue and establish some semiclassical behaviour of the spectrum.

Pólya's conjecture in the presence of a constant magnetic field

We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Pólya's conjecture is not true in the presence of a magnetic field.

Semiclassical Resolvent Estimates in Chaotic Scattering

We prove resolvent estimates for semiclassical operators such as − h 2 Δ + V(x) in scattering situations. Provided the set of trapped classical trajectories supports a chaotic flow and is sufficiently filamentary, the analytic continuation of the resolvent is bounded by h − M in a strip whose width is determined by a certain topological pressure associated with the classical flow. This polynomial estimate has applications to local smoothing in Schrodinger propagation and to energy decay of solutions to wave equations.

Resonance eigenfunctions in chaotic scattering systems

We study the semiclassical structure of resonance eigenstates of open chaotic systems. We obtain semiclassical estimates for the weight of these states on different regions in phase space. These results imply that the long-lived right (left) eigenstates of the non-unitary propagator are concentrated in the semiclassical limit ħ → 0 on the backward (forward) trapped set of the classical dynamics. On this support the eigenstates display a self-similar behaviour which depends on the limiting decay rate.

Relativistic Scott correction for atoms and molecules

AbstractWe prove the first correction to the leading Thomas‐Fermi energy for the ground state energy of atoms and molecules in a model where the kinetic energy of the electrons is treated relativistically. The leading Thomas‐Fermi energy, established in [25], as well as the correction given here, are of semiclassical nature. Our result on atoms and molecules is proved from a general semiclassical estimate for relativistic operators with potentials with Coulomb‐like singularities. This semiclassical estimate is obtained using the coherent state calculus introduced in [36]. The paper contains a unified treatment of the relativistic as well as the nonrelativistic case. © 2009 Wiley Periodicals, Inc.

Phase-Diffusion Dynamics in Weakly Coupled Bose-Einstein Condensates

We study the phase sensitivity of collisional phase diffusion between weakly coupled Bose-Einstein condensates, using a semiclassical picture of the two-mode Bose-Hubbard model. When weak coupling is allowed, zero relative phase locking is attained in the Josephson-Fock transition regime, whereas a pi relative phase is only locked in Rabi-Josephson point. Our analytic semiclassical estimates agree well with the numerical results.

Numerical and semiclassical analysis of some generalized Casimir pistons

The Casimir force due to a scalar field on a piston in a cylinder of radius $r$ with a spherical cap of radius $R>r$ is computed numerically in the world-line approach. A geometrical subtraction scheme gives the finite interaction energy that determines the Casimir force. The spectral function of convex domains is obtained from a probability measure on convex surfaces that is induced by the Wiener measure on Brownian bridges the convex surfaces are the hulls of. The vacuum force on the piston by a scalar field satisfying Dirichlet boundary conditions is attractive in these geometries, but the strength and short-distance behavior of the force depends crucially on the shape of the piston casing. For a cylindrical casing with a hemispherical head, the force for $a/R\sim 0$ does not depend on the dimension of the casing and numerically approaches $\sim - 0.00326(4)\hbar c/a^2$. Semiclassically this asymptotic force is due to short, closed and non-periodic trajectories that reflect once off the piston near its periphery. The semiclassical estimate $-\hbar c/(96\pi a^2)(1+2\sqrt{R^2-r^2}/a)$ for the force when $a/r\ll r/R\leq 1$ reproduces the numerical results within statistical errors.

Spectral Asymptotics for Large Skew-Symmetric Perturbations of the Harmonic Oscillator

Originally motivated by a stability problem in Fluid Mechanics, we study the spectral and pseudospectral properties of the differential operator H ǫ = −∂ 2 x + x 2 + iǫ −1 f (x) on L 2 (R), where f is a real-valued function and ǫ > 0 a small parameter. We define Σ(ǫ) as the infimum of the real part of the spectrum of H ǫ , and Ψ(ǫ) −1 as the supremum of the norm of the resolvent of H ǫ along the imaginary axis. Under appropriate conditions on f , we show that both quantities Σ(ǫ), Ψ(ǫ) go to infinity as ǫ → 0, and we give precise estimates of the growth rate of Ψ(ǫ). We also provide an example where Σ(ǫ) ≫ Ψ(ǫ) if ǫ is small. Our main results are established using variational hypocoercive methods, localization techniques and semiclassical subelliptic estimates.

Agmon-Type Estimates for a Class of Difference Operators

We analyze a general class of self-adjoint difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct a Finslerian distance $d$ induced by $H_\varepsilon$ and show that short integral curves are geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with a rate controlled by the Finsler distance to the well. This is analog to semiclassical Agmon estimates for Schr\"odinger operators.

Semiclassical analytic estimation of charged boson nebulae mass and the gravitationally radiated flux

In the present paper, we use the first-order approximate solutions to the non-linear system of Klein–Gordon–Maxwell–Einstein equations describing the minimally coupled charged spinless field to a spherically symmetric spacetime to analyze a becoming boson star. In the far future and long-range approximation, the analytical time-dependent mass formula has allowed us to point out several significant moments in the evolution of the boson nebula and to predict the necessary period to turn the collapsing star into a Schwarzschild black hole. Finally, we have semiclassically derived the transition amplitude for the graviton emission as well as the gravitational analog of the Umov–Poynting vector.

Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrödinger and Gross–Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, arxiv: math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].

Semiclassical spectral estimates for Schrödinger operators at a critical energy level. Case of a degenerate minimum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on R n . We assume here that the potential has a totally degenerate critical point associated to a local minimum. The main result, which computes the contribution of this equilibrium, is valid for all time in a compact and establishes the existence of a total asymptotic expansion whose top order coefficient depends only on the germ of the potential at the critical point.

Stability of quantum motion in regular systems: A uniform semiclassical approach

We study the stability of quantum motion of classically regular systems in the presence of small perturbations. On the basis of a uniform semiclassical theory we derive the fidelity decay which displays a quite complex behavior, from Gaussian to power law decay t(-alpha), with 1 <or= alpha <or= 2. Semiclassical estimates are given for the time scales separating the different decaying regions, and numerical results are presented which confirm our theoretical predictions.

A Phase-Space Study of the Quantum Loschmidt Echo in the Semiclassical Limit

The notion of Loschmidt echo (also called fidelity) has been introduced in order to study the (in)-stability of the quantum dynamics under perturbations of the Hamiltonian. It has been extensively studied in the past few years in the physics literature, in connection with the problems of chaos, quantum computation and decoherence. In this paper, we study this quantity semiclassically (as $\hbar \to 0$), taking as reference quantum states the usual coherent states. The latter are known to be well adapted to a semiclassical analysis, in particular with respect to semiclassical estimates of their evolution. For times not larger than the so-called Ehrenfest time $C | \log \hbar |$, we are able to estimate semiclassically the Loschmidt Echo as a function of $t$ (time), $\hbar$ (Planck constant), and $\delta$ (the size of the perturbation). The way two classical trajectories merging from the same point in classical phase-space, fly apart or come close together along the evolutions governed by the perturbed and unperturbed Hamiltonians play a major role in this estimate. We also give estimates of the return probability (again on reference states being the coherent states) by the same method, as a function of $t$ and $\hbar$.

Control of nonadiabatic dissociation dynamics with the use of laser-induced wave packet interferences

Based on wave packet interferences induced by a stationary laser field, a simple way of controlling nonadiabatic dissociation dynamics is proposed. We treat a simple two-state model of diatomic molecules. In this model, there exist two dissociative potential energy curves which cross and are strongly coupled at an internuclear distance, and thus dissociations into one channel are predominant. We propose a control scheme to selectively dissociate a molecule into any favorite channel by choosing the laser frequency and intensity appropriately. The semiclassical estimation of desirable laser parameters can be performed easily by regarding the dissociation processes as nonadiabatic transitions between the Floquet states. The agreement between the semiclassical estimation and the quantum wave packet calculation is found to be satisfactory in the high frequency region (> or =1000 cm(-1)) where the Floquet state picture is valid. In the low frequency region (<1000 cm(-1)), on the other hand, there are discrepancies between them due to the invalidity of the Floquet picture and the dissociation probability is sensitive to the laser phase. This control scheme is applied to the predissociation dynamics of NaI, NaI-->Na+I.

Fractal conductance fluctuations in electron billiards: a random matrix theory approach

Motivated by recent experiments and theoretical works that contradict the original explanation for fractal conductance fluctuations (FCF) in electron billiards, based on the mixed structure of the classical phase space, we propose an alternative approach to investigate FCF using Random Matrix Theory (RMT). By means of a semiclassical estimate for value of the magnetic correlation field BC we conclude that most of the experiments on FCF were performed for magnetic fields around or greater than BC. This strongly suggests that the appropriate explanation for the observed FCF should rely on the absence of long-range correlations and not on the structure of the classical phase space. This idea is supported by a numerical study of parametric variations within the framework of RMT, which validates our surmise that the observed FCF actually reflect a diffusive scenario for electronic transport.

Calculation of absolute scattering phase shifts

Use of a simple semiclassical estimate of Bessel function phase for arbitrary argument avoids the modulo π errors which the conventional approach tends to introduce into the calculation of absolute quantal phase shifts for high partial waves. This has significant implications for the direct quantum mechanical calculation of second virial coefficients at high temperatures.

Asymptotic rate of quantum ergodicity in chaotic Euclidean billiards

The quantum unique ergodicity (QUE) conjecture of Rudnick and Sarnak is that every eigenfunction ϕn of the Laplacian on a manifold with uniformly hyperbolic geodesic flow becomes equidistributed in the semiclassical limit (eigenvalue En ∞); that is, “strong scars” are absent. We study numerically the rate of equidistribution for a uniformly hyperbolic, Sinai-type, planar Euclidean billiard with Dirichlet boundary condition (the “drum problem”) at unprecedented high E and statistical accuracy, via the matrix elements 〈ϕn, Âϕm〉 of a piecewise-constant test function A. By collecting 30,000 diagonal elements (up to level n ≈ 7 × 105) we find that their variance decays with eigenvalue as a power 0.48 ± 0.01, close to the semiclassical estimate ½ of Feingold and Peres. This contrasts with the results of existing studies, which have been limited to En a factor 102 smaller. We find strong evidence for QUE in this system. We also compare off-diagonal variance as a function of distance from the diagonal, against Feingold-Peres (or spectral measure) at the highest accuracy (0.7%) thus far in any chaotic system. We outline the efficient scaling method and boundary integral formulae used to calculate eigenfunctions. © 2006 Wiley Periodicals, Inc.

Semi-classical spectral estimates for Schrödinger operators at a critical level. Case of a degenerate maximum of the potential

We study the semi-classical trace formula at a critical energy level for a Schrödinger operator on R n . We assume here that the potential has a totally degenerate critical point associated to a local maximum. The main result, which establishes the contribution of the associated equilibrium in the trace formula, is valid for all time in a compact subset of R and includes the singularity in t = 0 . For these new contributions the asymptotic expansion involves the logarithm of the parameter h. Depending on an explicit arithmetic condition on the dimension and the order of the critical point, this logarithmic contribution can appear in the leading term.