Semiclassical Estimates Research Articles

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator P = - h^{2} \partial^{2}_{x} + V(x;h) on L^{2}(\R) , with V(x;h) a finite signed measure, and where h >0 is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for P , and a limiting absorption resolvent estimate with sharp h -dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

Screwon spectral statistics and dispersion relation in the quantum Rajeev–Ranken model

The Rajeev–Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves (screwons) of wavenumber k in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model based on a quadratic Hamiltonian on a nilpotent/Euclidean Poisson algebra is Liouville integrable. Upon adopting canonical variables in a slightly extended phase space, the model was interpreted as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. Here, we examine the spectral statistics and dispersion relation of quantized screwons via numerical diagonalization validated by variational and perturbative approximations. We also derive a semiclassical estimate for the cumulative level distribution which compares favorably with the one from numerical diagonalization. The spectrum shows level crossings typical of an integrable system. The ith unfolded nearest neighbor spacings are found to follow Poisson statistics for small i. Nonoverlapping spacing ratios also indicate that successive spectral gaps are independently distributed. After displaying universal linear behavior over energy windows of short lengths, the spectral rigidity saturates at a length and value that scales with the square-root of energy. For strong coupling λ and intermediate k, we argue that reduced screwon energies can depend only on the product λk. Numerically, we find power law dependences on λ and k with an approximately common exponent 2/3 provided the angular momentum quantum number l is small compared to the number of nodes n in the radial wavefunction. On the other hand, for the ground state n=l=0, the common exponent becomes 1.

Fermionic semiclassical Lp estimates

We generalize the semiclassical Lp estimates of Koch, Tataru and Zworski in the setting of Schrödinger operators with confining potentials to density matrices. This is motivated by the problem of the concentration of free fermionic particles in a trapping potential. Our proof relies on semiclassical and many-body tools. As an application, we provide bounds on spectral clusters. We also discuss the optimality of the one-body and many-body bounds through explicit examples of quasimodes.

Geometric bounds for the magnetic Neumann eigenvalues in the plane

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of R2 with uniform magnetic field β>0 and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy λ1 and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields β=β(x) on a simply connected domain in a Riemannian surface.In particular: we prove the upper bound λ1<β for a general plane domain for a constant magnetic field, and the upper bound λ1<maxx∈Ω‾⁡|β(x)| for a variable magnetic field when Ω is simply connected.For smooth domains, we prove a lower bound of λ1 depending only on the intensity of the magnetic field β and the rolling radius of the domain.The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when k→∞ and consists of the semiclassical limit 2πk|Ω| plus an oscillating term.We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which λ1 is always small.

Sharp Resolvent Estimate for the Damped-Wave Baouendi–Grushin Operator and Applications

In this article we study the semiclassical resolvent estimate for the non-selfadjoint Baouendi-Grushin operator on the two-dimensional torus $\mathbb{T}^2=\mathbb{R}^2/(2\pi\mathbb{Z})^2$ with H\"older dampings. The operator is subelliptic degenerating along the vertical direction at $x=0$. We exhibit three different situations: (i) the damping region verifies the geometric control condition with respect to both the non-degenerate Hamiltonian flow and the vertical subelliptic flow; (ii) the undamped region contains a horizontal strip; (iii) the undamped part is a line. In all of these situations, we obtain sharp resolvent estimates. Consequently, we prove the optimal energy decay rate for the associated damped waved equations. For (i) and (iii), our results are in sharp contrast to the Laplace resolvent since the optimal bound is governed by the quasimodes in the subelliptic regime. While for (ii), the optimality is governed by the quasimodes in the elliptic regime, and the optimal energy decay rate is the same as for the classical damped wave equation on $\mathbb{T}^2$. Our analysis contains the study of adapted two-microlocal semiclassical measures, construction of quasimodes and refined Birkhoff normal-form reductions in different regions of the phase-space. Of independent interest, we also obtain the propagation theorem for semiclassical measures of quasimodes microlocalized in the subelliptic regime.

Disorder suppression in topological semiconductor-superconductor junctions

Disorder in a proximitizing bulk superconductor can scatter quasiparticles in a putative topological superconductor and eventually destroy the topological superconducting state. We use a scattering approach and a random-matrix calculation to estimate the disorder scattering time in a topological Josephson junction. We find that the disorder scattering rate from the bulk of the superconductor, even in the strong coupling limit, is suppressed in the ratio of Fermi momenta between the semiconductor and superconductor. This suppression of disorder scattering is accompanied by near perfect Andreev reflection at such semiconductor/superconductor interfaces, which can be used as a signature of such clean proximity effect. We also find that these results can be understood by a semiclassical estimate of scattering. We discuss limits in other systems such as the semiconductor nanowire where disorder scattering is suppressed according to similar classical estimates.

Collisional losses of ultracold molecules due to intermediate complex formation

Understanding the sources of losses and chemical reactions of ultracold alkali-metal molecules is among the critical elements needed for their application in precision measurements and quantum technologies. Recent experiments with nonreactive systems have reported unexpectedly large loss rates, posing a challenge for theoretical explanation. Here, we examine the dynamics of intermediate four-atom complexes formed in bimolecular collisions. We calculate the nuclear spin--rotation, spin--spin, and quadrupole coupling constants for bialkali tetramers using ab intio quantum-chemical methods. We show that the nuclear spin--spin and quadrupole couplings are strong enough to couple different rotational manifolds to increase the density of states and lifetimes of the collision complexes, which is consistent with experimental results. We propose further experiments to confirm our predictions.

The tunneling splitting and the Kramers theory of activated processes

The study of tunneling splitting is fundamental to get insight into the dynamics of a multitude of molecular systems. In this paper, a novel approach to the analysis of the ground-state tunneling splitting is presented and explicitly applied to one-dimensional systems, with the aim of benchmarking its applicability in the simplest case. The isomorphism between the Fokker-Planck-Smoluchowski operator and the Born-Oppenheimer quantum Hamiltonian is the key element of this method. The localization function approach, used in the field of stochastic processes to study the Kramers problem, leads to a simple, yet asymptotically justified, integral approximation for the tunneling splitting. The comparison with exact values of the tunneling splittings shows a better accuracy than WKB semiclassical estimates for the family of considered model systems; demonstrating that the method may improve numerical estimates beside offering an alternative point of view on the features controlling the tunneling splitting in simple potential profiles.

Balanced metrics for Kähler-Ricci solitons and quantized Futaki invariants

We show that a Kähler-Ricci soliton on a Fano manifold can always be smoothly approximated by a sequence of relative anticanonically balanced metrics, also called quantized Kähler-Ricci solitons. The proof uses a semiclassical estimate on the spectral gap of an equivariant Berezin transform to extend a strategy due to Donaldson, and can be seen as the quantization of a method due to Tian and Zhu, using quantized Futaki invariants as obstructions for quantized Kähler-Ricci solitons. As corollaries, we recover the uniqueness of Kähler-Ricci solitons up to automorphisms, and show how our result also applies to Kähler-Einstein Fano manifolds with general automorphism group.

On the semiclassical spectrum of the Dirichlet–Pauli operator

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.

Sharp tunneling estimates for a double-well model in infinite dimension

We consider the stochastic quantization of a quartic double-well energy functional in the semiclassical regime and derive optimal asymptotics for the exponentially small splitting of the ground state energy. Our result provides an infinite-dimensional version of some sharp tunneling estimates known in finite dimensions for semiclassical Witten Laplacians in degree zero. From a stochastic point of view it proves that the L2 spectral gap of the stochastic one-dimensional Allen-Cahn equation in finite volume satisfies a Kramers-type formula in the limit of vanishing noise. We work with finite-dimensional lattice approximations and establish semiclassical estimates which are uniform in the dimension. Our key estimate shows that the constant separating the two exponentially small eigenvalues from the rest of the spectrum can be taken independently of the dimension.

Reduction and coherent states

We apply a quantum version of dimensional reduction to Gaussian coherent states in Bargmann space to obtain squeezed states on complex projective spaces. This leads to a definition of a family of squeezed spin states with excellent semi-classical properties, governed by a symbol calculus. We prove semiclassical norm estimates and a propagation result.

Semiclassical evolution with low regularity

We prove semiclassical estimates for the Schrödinger-von Neumann evolution with C1,1 potentials and density matrices whose square root have either Wigner functions with low regularity independent of the dimension, or matrix elements between Hermite functions having long range decay. The estimates are settled in different weak topologies and apply to initial density operators whose square root have Wigner functions 7 times differentiable, independently of the dimension. They also apply to the N-body quantum dynamics uniformly in N and to concentrating pure and mixed states without any regularity assumption. In a appendix, we finally estimate the dependence in the dimension of the constant appearing on the Calderón-Vaillancourt Theorem.

Improved resolvent bounds for radial potentials

We prove semiclassical resolvent estimates for the Schrödinger operator in \({\mathbb {R}}^d\), \(d\ge 3\), with real-valued radial potentials \(V\in L^\infty ({\mathbb {R}}^d)\). In particular, we show that if \(V(x)={{\mathcal {O}}}\left( \langle x\rangle ^{-\delta }\right) \) with \(\delta >2\), then the resolvent bound is of the form \(\exp \left( Ch^{-4/3}\right) \) with some constant \(C>0\). We also get resolvent bounds when \(1<\delta \le 2\). For slowly decaying \(\alpha \)—Hölder potentials, we get better resolvent bounds of the form \(\exp \left( Ch^{-4/(\alpha +3)}\right) \).

Semiclassical resolvent estimates for Hölder potentials

We first prove semiclassical resolvent estimates for the Schrodinger operator in R d , d ≥ 3, with real-valued potentials which are Holder with respect to the radial variable. Then we extend these resolvent estimates to exterior domains in R d , d ≥ 2, and real-valued potentials which are Holder with respect to the space variable. As an application, we obtain the rate of the decay of the local energy of the solutions to the wave equation with a refraction index which may be Holder, Lipschitz or just L ∞ .

A semi-classical estimate for the q-parameter and decay time with Tsallis entropy of black holes in quantum geometry

In this letter, using the non-extensive entropy of Tsallis, we study some properties of the Schwarzschild black holes (BHs), based on the loop quantum gravity (LQG), some novel characteristics and results of the Schwarzschild BH can be obtained in Mejrhit and Ennadifi (Phys Lett B 794:45–49, 2019). Here we find that these findings are strikingly identical to ones obtained by Hawking and Page in anti-de Sitter space within the original of the Boltzmann entropy formula. By using the semi-classical estimate analysis on the energy at this minimum M_{min}, an approximate relationship between the q and gamma parameters of BHs can be found, (qapprox frac{sqrt{3}gamma }{pi ln 2}+1), which is remarkable approaching to q-parameters of cosmic ray spectra and quarks coalescing to hadrons in high energy.

Semiclassical estimates for pseudodifferential operators and the Muskat problem in the unstable regime

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.

Geometric quantization of Hamiltonian flows and the Gutzwiller trace formula

We use the theory of Berezin-Toeplitz operators of Ma and Marinescu to study the quantum Hamiltonian dynamics associated with classical Hamiltonian flows over closed prequantized symplectic manifolds in the context of geometric quantization of Kostant and Souriau. We express the associated evolution operators via parallel transport in the quantum spaces over the induced path of almost complex structures, and we establish various semi-classical estimates. In particular, we establish a Gutzwiller trace formula for the Kostant-Souriau operator and compute explicitly the leading term. We then describe a potential application to contact topology.

Sensitivity of semiclassical vibrational spectroscopy to potential energy surface accuracy: A test on formaldehyde

A set of permutationally invariant potential energy surfaces for the electronic ground state of formaldehyde is built at several levels of electronic theory and atomic orbital basis sets starting from a database of more than 34,000 ab initio energies. Preliminarily, the reliability of the fitted surfaces is determined by comparing the calculated harmonic frequencies with the corresponding ab initio values. Then, semiclassical estimates of the quantum frequencies of vibration are presented, and their dependence on the employed level of theory, type of atomic orbital basis set, and complexity of the fit is investigated. Comparisons of semiclassical results to experimental data provide a further assessment of the quality of the analytical surfaces and show that anharmonic frequencies are influenced by the precision of the fit, while accurate frequency values are obtained also with density functional theory. Results and conclusions support the use of ab initio “on-the-fly” semiclassical dynamics as a means of spectroscopic investigation when high-level analytical potential energy surfaces are not available.

Semiclassical Resolvent Estimates for $$L^\infty $$ Potentials on Riemannian Manifolds

We prove semi-classical resolvent estimates for the Schr{\"o}dinger operator with a real-valued L $\infty$ potential on non-compact, connected Riemannian manifolds which may have a compact smooth boundary. We show that the resolvent bound depends on the structure of the man-ifold at infinity. In particular, we show that for compactly supported real-valued L $\infty$ potentials and asymptoticaly Euclidean manifolds the resolvent bound is of the form exp(Ch --4/3 log(h --1)), while for asymptoticaly hyperbolic manifolds it is of the form exp(Ch --4/3), where C > 0 is some constant.