This paper is devoted to semiclassical estimates of the eigenvalues of the Pauli operator on a bounded open set whose boundary carries Dirichlet conditions. 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.
This paper is devoted to the semiclassical analysis of the spectrum of the Dirichlet-Pauli operator on an annulus. We assume that the magnetic field is strictly positive and radial. We give an explicit asymptotic expansion at the first order of the lowest eigenvalues of this operator in the semiclassical limit. In particular, we exhibit the Aharonov-Bohm effect that has been revealed, for constant magnetic field, in a recent paper by Helffer and Sundqvist.
We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semi-classical parameter. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and estimate this decay rate. This extends our results, discussing the results of a recent paper by Ekholm--Kovařík--Portmann, to include also non-simply connected domains.
We discuss the results of a recent paper by Ekholm, Kovařík and Portmann in connection with a question of C. Guillarmou about the semiclassical expansion of the lowest eigenvalue of the Pauli operator with Dirichlet conditions. We exhibit connections with the properties of the torsion function in mechanics, the exit time of a Brownian motion and the analysis of the low eigenvalues of some Witten Laplacian.
We consider the semi-classical Dirichlet Pauli operator in bounded connected domains in the plane. Rather optimal results have been obtained in previous papers by Ekholm-Kovařík-Portmann and Helffer-Sundqvist for the asymptotics of the ground state energy in the semi-classical limit when the magnetic field has constant sign. In this paper, we focus on the case when the magnetic field changes sign. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and give lower bounds and upper bounds for this d…
This paper is a contribution to semiclassical analysis for abstract Schrödinger type operators on locally compact spaces: Let $X$ be a metrizable seperable locally compact space, let $μ$ be a Radon measure on $X$ with a full support. Let $(t,x,y)\mapsto p(t,x,y)$ be a strictly positive pointwise consistent $μ$-heat kernel, and assume that the generator $H_p\geq 0$ of the corresponding self-adjoint contraction semigroup in $L^2(X,μ)$ induces a regular Dirichlet form. Then, given a function $Ψ: (0,1)\to (0,\infty)$ such that the limit $\lim_{t\to 0+}p(t,x,x)Ψ(t)$ exists for all $x\in X$, we prov…
Let $M$ be a compact Riemannian manifold with smooth boundary, and let $R(λ)$ be the Dirichlet-to-Neumann operator at frequency $λ$. We obtain a leading asymptotic for the spectral counting function for $λ^{-1}R(λ)$ in an interval $[a_1, a_2)$ as $λ\to \infty$, under the assumption that the measure of periodic billiards on $T^*M$ is zero. The asymptotic takes the form \begin{equation*} N(λ; a_1,a_2) = \bigl(κ(a_2)-κ(a_1)\bigr)\mathsf{vol}'(\partial M) λ^{d-1}+o(λ^{d-1}), \end{equation*} where $κ(a)$ is given explicitly by \begin{equation*} κ(a) = \frac{ω_{d-1}}{(2π)^{d-1}} \biggl( -\frac{1}{2π…
We consider the Dirichlet Laplace operator on open, quasi-bounded domains of infinite volume. For such domains semiclassical spectral estimates based on the phase-space volume - and therefore on the volume of the domain - must fail. Here we present a method how one can nevertheless prove uniform bounds on eigenvalues and eigenvalue means which are sharp in the semiclassical limit. We give examples in horn-shaped regions and so-called spiny urchins. Some results are extended to Schrödinger operators defined on quasi-bounded domains with Dirichlet boundary conditions.
Let -Δdenote the Dirichlet Laplace operator on a bounded open set in \mathbb{R}^d. We study the sum of the negative eigenvalues of the operator -h^2 Δ- 1 in the semiclassical limit h \to 0+. We give a new proof that yields not only the first term of the asymptotic formula but also the second term involving the surface area of the boundary of the set. The proof is valid under weak smoothness assumptions on the boundary.
We consider non-interacting particles subject to a fixed external potential $V$ and a self-generated magnetic field $B$. The total energy includes the field energy $β\int B^2$ and we minimize over all particle states and magnetic fields. In the case of spin-1/2 particles this minimization leads to the coupled Maxwell-Pauli system. The parameter $β$ tunes the coupling strength between the field and the particles and it effectively determines the strength of the field. We investigate the stability and the semiclassical asymptotics, $h\to0$, of the total ground state energy $E(β, h, V)$. The rele…
The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in $\mathbb{R}^3$ confined by a local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians.
We compute three-term semiclassical asymptotic expansions of counting functions and Riesz-means of the eigenvalues of the Laplacian on spheres and hemispheres, for both Dirichlet and Neumann boundary conditions. Specifically for Riesz-means we prove upper and lower bounds involving asymptotically sharp shift terms, and we extend them to domains of $\mathbb S^d$. We also prove a Berezin-Li-Yau inequality for domains contained in the hemisphere $\mathbb S^2_+$. Moreover, we consider polyharmonic operators for which we prove analogous results that highlight the role of dimension for Pólya-type in…
The goal of this paper is to derive estimates of eigenvalue moments for Dirichlet Laplacians and Schrödinger operators in regions having infinite cusps which are geometrically nontrivial being either curved or twisted; we are going to show how those geometric properties enter the eigenvalue bounds. The obtained inequalities reflect the essentially one-dimensional character of the cusps and we give an example showing that in an intermediate energy region they can be much stronger than the usual semiclassical bounds.
In this article, we consider the semiclassical Schrödinger operator $P = - h^{2} Δ+ V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $λ_{k} ( P )$ as $h \to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $λ_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = \{ 0 \}$, we are able to control the eigenvalues $λ_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where …
We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists an almost periodic magnetic potential which generates the magnetic field $b - b_{0}$, $b_{0}$ being the mean value of $b$. Next, we assume that $V = 0$, and investigate the zero modes of $H$. As expected, if $b_{0} \neq 0$, then generically $\operatorname{dim} \operatorname{Ker} H = \infty$. If $b_{0} = 0$, then for each $m \in {\mathbb N} \cup \{ \infty \}$, we construct almost peri…
Semiclassical asymptotics of the Bloch--Torrey operator in two dimensions
Czech Technical University in Prague（布拉格捷克理工大学）；Univ Angers, CNRS, LAREMA, Institut Universitaire de France, SFR MATHSTIC（昂热大学，法国国家科学研究中心，LAREMA，法国高等研究院，MATHSTIC科研联合基金）
Let $\mathcal{P}_{Ω,tA}$ denoted the Pauli operator on a bounded open region $Ω\subset\mathbb{R}^2$ with Dirichlet boundary conditions and magnetic potential $A$ scaled by some $t>0$. Assume that the corresponding magnetic field $B=\mathrm{curl}\,A$ satisfies $B\in L\log L(Ω)\cap C^α(Ω_0)$ where $α>0$ and $Ω_0$ is an open subset of $Ω$ of full measure (note that, the Orlicz space $L\log L(Ω)$ contains $L^p(Ω)$ for any $p>1$). Let $\mathsf{N}_{Ω,tA}(λ)$ denote the corresponding eigenvalue counting function. We establish the strong field asymptotic formula \[ \mathsf{N}_{Ω,tA}(λ(t))=\frac{t}{2π}…
In this paper we prove a two-term asymptotic formula for for the spectral counting function for a 2D magnetic Schrödinger operator on a domain (with Dirichlet boundary conditions) in a semiclassical limit and with strong magnetic field. By scaling, this is equivalent to a thermodynamic limit of a 2D Fermi gas submitted to a constant external magnetic field. The original motivation comes from a paper by H. Kunz in which he studied, among other things, the boundary correction for the grand-canonical pressure and density of such a Fermi gas. Our main theorem yields a rigorous proof of the formula…
We present asymptotically sharp inequalities for the eigenvalues $μ_k$ of the Laplacian on a domain with Neumann boundary conditions, using the averaged variational principle introduced in \cite{HaSt14}. For the Riesz mean $R_1(z)$ of the eigenvalues we improve the known sharp semiclassical bound in terms of the volume of the domain with a second term with the best possible expected power of $z$. In addition, we obtain two-sided bounds for individual $μ_k$, which are semiclassically sharp. In a final section, we remark upon the Dirichlet case with the same methods.
We consider a magnetic Schrödinger operator $H^h=(-ih\nabla-\vec{A})^2$ with the Dirichlet boundary conditions in an open set $Ω\subset {\mathbb R}^3$, where $h>0$ is a small parameter. We suppose that the minimal value $b_0$ of the module $|\vec{B}|$ of the vector magnetic field $\vec{B}$ is strictly positive, and there exists a unique minimum point of $|\vec{B}|$, which is non-degenerate. The main result of the paper is upper estimates for the low-lying eigenvalues of the operator $H^h$ in the semiclassical limit. We also prove the existence of an arbitrary large number of spectral gaps in t…
