We consider the spectrum of a two-dimensional Pauli operator with a compactly supported electric potential and a variable magnetic field with a positive mean value. The rate of accumulation of eigenvalues to zero is described in terms of the logarithmic capacity of the support of the electric potential. A connection between these eigenvalues and orthogonal polynomials in complex domains is established.
We consider the 3D Pauli operator with nonconstant magnetic field B of constant direction, perturbed by a symmetric matrix-valued electric potential V whose coefficients decay fast enough at infinity. We investigate the low-energy asymptotics of the corresponding spectral shift function. As a corollary, for generic negative V, we obtain a generalized Levinson formula, relating the low-energy asymptotics of the eigenvalue counting function and of the scattering phase of the perturbed operator.
We study the asymptotic behavior, as Planck's constant $\hbar\to 0$, of the number of discrete eigenvalues and the Riesz means of Pauli and Dirac operators with a magnetic field $μ\mathbf{B}(x)$ and an electric field. The magnetic field strength $μ$ is allowed to tend to infinity as $\hbar\to 0$. Two main types of results are established: in the first $μ\hbar\le constant$ as $\hbar\to 0$, with magnetic fields of arbitrary direction; the second results are uniform with respect to $μ\ge 0$ but the magnetic fields have constant direction. The results on the Pauli operator complement recent work o…
We obtain explicit upper and lower bounds on the norms of the spectral projections of the non-self-adjoint harmonic oscillator. Some of our results apply to a variety of other families of orthogonal polynomials.
We investigate the behaviour of the eigenvalues of two-dimensional Pauli operators with nonconstant magnetic fields perturbed by a sign-indefinite decaying electric potential V. We prove new eigenvalues asymptotics.
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…
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}…
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 two-dimensional Pauli operator perturbed by a weakly coupled, attractive potential. We show that besides the eigenvalues arising from the Aharonov-Casher zero modes there are two or one (depending on whether the flux of the magnetic field is integer or not) additional eigenvalues for arbitrarily small coupling and we calculate their asymptotics in the weak coupling limit.
We establish the asymptotic zero distribution for polynomials generated by a four-term recurrence relation with varying recurrence coefficients having a particular limiting behavior. The proof is based on ratio asymptotics for these polynomials. We can apply this result to three examples of multiple orthogonal polynomials, in particular Jacobi-Pineiro, Laguerre I and the example associated with Macdonald functions. We also discuss an application to Toeplitz matrices.
First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $ϕ, ϕ(0)=1$, such that $\|(Bϕ)'\| \leq C n$, where $C$ is an absolute constant. Then we apply this result to a certain problem of finding the asymptotic of orthogonal polynomials.
We study solutions of three-term recurrence relations whose $N$-step transfer matrices belong to the uniform Stolz class. In particular, we derive the first order of their uniform asymptotics. For orthonormal polynomials we show more. Namely, we find the constructive formula for the density of their orthogonality measure in terms of Turán determinants and we determine their exact asymptotic behavior. We treat both bounded and unbounded cases in a uniform manner.
We consider Dirac, Pauli and Schrödinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of Bögli results [Bög17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the …
We continue studying polynomials generated by the Szegő recursion when a finite number of Verblunsky coefficients lie outside the closed unit disk. We prove some asymptotic results for the corresponding orthogonal polynomials and then translate them to the real line to obtain the Szegő asymptotics for the resulting polynomials. The latter polynomials give rise to a non-symmetric tridiagonal matrix but it is a finite-rank perturbation of a symmetric Jacobi matrix.
The main aim of this short paper is to advertize the Koosis theorem in the mathematical community, especially among those who study orthogonal polynomials. We (try to) do this by proving a new theorem about asymptotics of orthogonal polynomials for which the Koosis theorem seems to be the most natural tool. Namely, we consider the case when a Szegö measure on the unit circumference is perturbed by an arbitrary measure inside the unit disk and an arbitrary Blaschke sequence of point masses outside the unit disk.
Let $μ$ be a measure from Szegő class on the unit circle $\mathbb T$ and let $\{f_n\}$ be the family of Schur functions generated by $μ$. In this paper, we prove a version of the classical Szegő's formula which controls the oscillation of $f_n$ on $\mathbb T$ for all $n \ge 0$. Then, we focus on an analog of Lusin's conjecture for polynomials $\{φ_n\}$ orthogonal with respect to measure $μ$ and prove that pointwise convergence of $\{|φ_n|\}$ almost everywhere on $\mathbb T$ is equivalent to a certain condition on zeroes of $φ_n$.
We consider 2- and 3-dimensional Schrödinger or generalized Schrödinger-Pauli operators with the non-degenerating magnetic field in the open domain under certain non-degeneracy assumptions we derive pointwise spectral asymptotics. We also consider asymptotics of some related expressions (see below). For all asymptotics loops rather than periodic trajectories play important role.
Our goal is to find asymptotic formulas for orthonormal polynomials $P_{n}(z)$ with the recurrence coefficients slowly stabilizing as $n\to\infty$. To that end, we develop spectral theory of Jacobi operators with long-range coefficients and study the corresponding second order difference equation. We suggest an Ansatz for its solutions playing the role of the semiclassical Green-Liouville Ansatz for solutions of the Schrödinger equation. The formulas obtained for $P_{n}(z)$ as $n\to\infty$ generalize the classical Bernstein-Szegö asymptotic formulas.
We consider two-dimensional Pauli and Dirac operators with a polynomially vanishing magnetic field. The main results of the paper provide resolvent expansions of these operators in the vicinity of their thresholds. It is proved that the nature of these expansions is fully determined by the flux of the magnetic field. The most important novelty of the proof is a comparison between the spatial asymptotics of the zero modes obtained in two different manners. The result of this matching allows to compute explicitly all the singular terms in the associated resolvent expansions. As an application we…
Orthogonal polynomials $P_{n}(λ)$ are oscillating functions of $n$ as $n\to\infty$ for $λ$ in the absolutely continuous spectrum of the corresponding Jacobi operator $J$. We show that, irrespective of any specific assumptions on coefficients of the operator $J$, amplitude and phase factors in asymptotic formulas for $P_{n}(λ)$ are linked by certain universal relations found in the paper. Our approach relies on a study of operators diagonalizing Jacobi operators. Diagonalizing operators are constructed in terms of orthogonal polynomials $P_{n}(λ)$. They act from the space $L^2 (\Bbb R)$ of func…
In this paper, we present a new method via the transfer matrix approach to obtain asymptotic formulae of orthogonal polynomials with asymptotically identical coefficients of bounded variation. We make use of the hyperbolicity of the recurrence matrices and employ Kooman's Theorem to diagonalize them simultaneously. The method introduced in this paper allows one to consider products of matrices such that entries of consecutive matrices are of bounded variation. Finally, we apply the asymptotic formulae obtained to solve the point mass problem on the real line when the measure is essentially sup…
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…
Our goal is to find an asymptotic behavior as $n\to\infty$ of the orthogonal polynomials $P_{n}(z)$ defined by Jacobi recurrence coefficients $a_{n}$ (off-diagonal terms) and $ b_{n}$ (diagonal terms). We consider the case $a_{n}\to\infty$, $b_{n}\to\infty$ in such a way that $\sum a_{n}^{-1}<\infty$ $($that is, the Carleman condition is violated$)$ and $γ_{n}:=2^{-1}b_{n} (a_{n}a_{n-1})^{-1/2} \to γ$ as $n\to\infty$. In the case $|γ| \neq 1$ asymptotic formulas for $P_{n}(z)$ are known; they depend crucially on the sign of $| γ|-1$. We study the critical case $| γ|=1$. The formulas obtained a…
We use a non-linear characterization of orthonormal polynomials due to Saff in order to show that the behavior of orthonormal polynomials is determined only by its leading coefficient and its normalization. Several applications of this equivalence are also discussed. One of our main results is that for regular measures on the closed unit disk - including, but not limited to the unit circle - one has ratio asymptotics along a sequence of asymptotic density 1.
There is a vast theory of the asymptotic behavior of orthogonal polynomials with respect to a measure on $\mathbb{R}$ and its applications to Jacobi matrices. That theory has an obvious affine invariance and a very special role for $\infty$. We extend aspects of this theory in the setting of rational functions with poles on $\overline{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$, obtaining a formulation which allows multiple poles and proving an invariance with respect to $\overline{\mathbb{R}}$-preserving Möbius transformations. We obtain a characterization of Stahl--Totik regularity of a GMP ma…
