arXiv subjects

Arne Jensen

Publications and source records attributed to Arne Jensen.

At least 19 recordsLinked to original sources

arXiv2026

On the directional growth of the resolvent norm

Let $A$ be a closed densely defined operator on a separable Hilbert space $\mathcal{H}$. Assume the resolvent set $\rho(A)$ is non-empty. For $z,z'\in\rho(A)$ let $[z,z']$ denote the straight line segment from $z$ to $z'$. For each $z\in\rho(A)$ we classify the behavior of the resolvent norm $\zeta\mapsto\lVert R_A(\zeta) \rVert$ near $z$. Either there are $z'\in\rho(A)$, $z'\neq z$, $[z,z']\subset\rho(A)$, such that $\lVert R_A(\zeta) \rVert \geq \lVert R_A(z) \rVert + C\lvert \zeta-z \rvert^\delta$ for $\zeta\in[z,z']$ with $\delta=1$ or $\delta=2$, or the function $\zeta\mapsto\lVert R_A(\zeta) \rVert$ has a global minimum at $\zeta=z$.

math.SP↗

arXiv2025

A modified Fermi Golden Rule at threshold for 3D magnetic Schr\"odinger operators

In this paper we consider three-dimensional Schr\"odinger operators with a simple threshold eigenvalue. We show, under certain assumptions, that when a small magnetic field is introduced, this eigenvalue turns into a resonance in the time-dependent sense. We find the leading term in the asymptotic expansion of the imaginary part of the resonance and discuss the principal differences with respect to resonances induced by weak electric fields obtained previously in the literature.

math-ph↗

arXiv2024

Discrete approximations to Dirichlet and Neumann Laplacians on a half-space and norm resolvent convergence

We extend recent results on discrete approximations of the Laplacian in $\mathbf{R}^d$ with norm resolvent convergence to the corresponding results for Dirichlet and Neumann Laplacians on a half-space. The resolvents of the discrete Dirichlet/Neumann Laplacians are embedded into the continuum using natural discretization and embedding operators. Norm resolvent convergence to their continuous counterparts is proven with a quadratic rate in the mesh size. These results generalize with a limited rate to also include operators with a real, bounded, and Hölder continuous potential, as well as certain functions of the Dirichlet/Neumann Laplacians, including any positive real power. Note (Nov 27, 2024): A corrigendum has been added to the end of the PDF.

math.FA↗

arXiv2024

Resolvent expansions of 3D magnetic Schroedinger operators and Pauli operators

We obtain asymptotic resolvent expansions at the threshold of the essential spectrum for magnetic Schrödinger and Pauli operators in dimension three. These operators are treated as perturbations of the Laplace operator in $L^2(\mathbb{R}^3)$ and $L^2(\mathbb{R}^3;\mathbb{C}^2)$, respectively. The main novelty of our approach is to show that the relative perturbations, which are first order differential operators, can be factorized in suitably chosen auxiliary spaces. This allows us to derive the desired asymptotic expansions of the resolvents around zero. We then calculate their leading and sub-leading terms explicitly. Analogous factorization schemes for more general perturbations, including e.g.~finite rank perturbations, are discussed as well.

math.SP↗

arXiv2022

Discrete approximations to Dirac operators and norm resolvent convergence

We consider continuous Dirac operators defined on $\mathbf{R}^d$, $d\in\{1,2,3\}$, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

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arXiv2021

Norm resolvent convergence of discretized Fourier multipliers

We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance.

math.FA↗

arXiv2020

A solvable model of the breakdown of the adiabatic approximation

Let $L\geq0$ and $0<\varepsilon\ll1$. Consider the following time-dependent family of $1D$ Schrödinger equations with scaled and translated harmonic oscillator potentials $ i\varepsilon\partial_t u_{\varepsilon}=-\tfrac12\partial_x^2u_{\varepsilon}+V(t,x)u_{\varepsilon}$, $u_{\varepsilon}(-L-1,x)=π^{-1/4}\exp(-x^2/2) $, where $ V(t,x)= (t+L)^2x^2/2$, $t<-L$, $ V(t,x)= 0$, $-L\leq t \leq L$, and $ V(t,x)=(t-L)^2x^2/2$, $t>L$. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions we show that the adiabatic theorem breaks down as $\varepsilon\to 0$. For the case $L=0$ complete results are obtained. The survival probability of the ground state $π^{-1/4}\exp(-x^2/2)$ at microscopic time $t=1/\varepsilon$ is $1/\sqrt{2}+O(\varepsilon)$. For $L>0$ the framework for further computations and preliminary results are given.

math-ph↗

arXiv2020

Continuum limit for lattice Schrödinger operators

We study the behavior of solutions of the Helmholtz equation $(- Δ_{disc,h} - E)u_h = f_h$ on a periodic lattice as the mesh size $h$ tends to 0. Projecting to the eigenspace of a characteristic root $λ_h(ξ)$ and using a gauge transformation associated with the Dirac point, we show that the gauge transformed solution $u_h$ converges to that for the equation $(P(D_x) - E)v = g$ for a continuous model on ${\bf R}^d$, where $λ_h(ξ) \to P(ξ)$. For the case of the hexagonal and related lattices, {in a suitable energy region}, it converges to that for the Dirac equation. For the case of the square lattice, triangular lattice, {hexagonal lattice (in another energy region)} and subdivision of a square lattice, one can add a scalar potential, and the solution of the lattice Schr{ö}dinger equation $( - Δ_{disc,h} +V_{disc,h} - E)u_h = f_h$ converges to that of the continuum Schr{ö}dinger equation $(P(D_x) + V(x) -E)u = f$.

math-ph↗

arXiv2020

Hypergeometric expression for the resolvent of the discrete Laplacian in low dimensions

We present an explicit formula for the resolvent of the discrete Laplacian on the square lattice, and compute its asymptotic expansions around thresholds in low dimensions. As a by-product we obtain a closed formula for the fundamental solution to the discrete Laplacian. For the proofs we express the resolvent in a general dimension in terms of the Appell--Lauricella hypergeometric function of type $C$ outside a disk encircling the spectrum. In low dimensions it reduces to a generalized hypergeometric function, for which certain transformation formulas are available for the desired expansions.

math-ph↗

arXiv2019

Branching form of the resolvent at threshold for multi-dimensional discrete Laplacians

We consider the discrete Laplacian on $\mathbb Z^d$, and compute asymptotic expansions of its resolvent around thresholds embedded in continuous spectrum as well as those at end points. We prove that the resolvent has a square-root branching if $d$ is odd, and a logarithm branching if $d$ is even, and, moreover, obtain explicit expressions for these branching parts involving the Lauricella hypergeometric function. In order to analyze a non-degenerate threshold of general form we use an elementary step-by-step expansion procedure, less dependent on special functions.

math-ph↗

arXiv2018

A local directional growth estimate of the resolvent norm

We study the resolvent norm of a certain class of closed linear operators on a Hilbert space, including unbounded operators with compact resolvent. It is shown that for any point in the resolvent set there exist directions in which the norm grows at least quadratically with the distance from this point. This provides a new proof not using the maximum principle that the resolvent norm of the considered class cannot have local maxima. Finally, we give new criteria for the existence of local non-degenerate minima of the resolvent norm and provide examples of (un)bounded non-normal operators having this property.

math.SP↗

arXiv2018

Resolvent expansion for the Schrödinger operator on a graph with infinite rays

We consider the Schrödinger operator on a combinatorial graph consisting of a finite graph and a finite number of discrete half-lines, all jointed together, and compute an asymptotic expansion of its resolvent around the threshold $0$. Precise expressions are obtained for the first few coefficients of the expansion in terms of the generalized eigenfunctions. This result justifies the classification of threshold types solely by growth properties of the generalized eigenfunctions. By choosing an appropriate free operator a priori possessing no zero eigenvalue or zero resonance we can simplify the expansion procedure as much as that on the single discrete half-line.

math.SP↗

arXiv2018

Instability of resonances under Stark perturbations

Let $H^{\varepsilon}=-\frac{d^2}{dx^2}+\varepsilon x +V$, $\varepsilon\geq0$, on $L^2(\mathbf{R})$. Let $V=\sum_{k=1}^Nc_k|ψ_k\rangle\langleψ_k|$ be a rank $N$ operator, where the $ψ_k\in L^2(\mathbf{R})$ are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if $ζ_n$, ${\rm Im}ζ_n<0$, are resonances of $H^{\varepsilon_n}$ for a sequence $\varepsilon_n\downarrow0$ as $n\to\infty$ and $ζ_n\toζ_0$ as $n\to\infty$, ${\rm Im}ζ_0<0$, then $ζ_0$ is \emph{not} a resonance of $H^0$.

math-ph↗

arXiv2017

On the adiabatic theorem when eigenvalues dive into the continuum

We consider a reduced two-channel model of an atom consisting of a quantum dot coupled to an open scattering channel described by a three-dimensional Laplacian. We are interested in the survival probability of a bound state when the dot energy varies smoothly and adiabatically in time. The initial state corresponds to a discrete eigenvalue which dives into the continuous spectrum and re-emerges from it as the dot energy is varied in time and finally returns to its initial value. Our main result is that for a large class of couplings, the survival probability of this bound state vanishes in the adiabatic limit. At the end of the paper we present a short outlook on how our method may be extended to cover other classes of Hamiltonians; details will be given elsewhere.

math-ph↗

arXiv2017

Resolvent expansions for the Schrödinger operator on the discrete half-line

Simplified models of transport in mesoscopic systems are often based on a small sample connected to a finite number of leads. The leads are often modelled using the Laplacian on the discrete half-line $\mathbb N$. Detailed studies of the transport near thresholds require detailed information on the resolvent of the Laplacian on the discrete half-line. This paper presents a complete study of threshold resonance states and resolvent expansions at a threshold for the Schrödinger operator on the discrete half-line $\mathbb N$ with a general boundary condition. A precise description of the expansion coefficients reveals their exact correspondence to the generalized eigenspaces, or the threshold types. The presentation of the paper is adapted from that of Ito-Jensen [Rev.\ Math.\ Phys.\ {\bf 27} (2015), 1550002 (45 pages)], implementing the expansion scheme of Jensen-Nenciu [Rev.\ Math.\ Phys.\ \textbf{13} (2001), 717--754, \textbf{16} (2004), 675--677] in its full generality.

math-ph↗

arXiv2014

Metastable states when the Fermi Golden Rule constant vanishes

Resonances appearing by perturbation of embedded non-degenerate eigenvalues are studied in the case when the Fermi Golden Rule constant vanishes. Under appropriate smoothness properties for the resolvent of the unperturbed Hamiltonian, it is proved that the first order Rayleigh-Schrödinger expansion exists. The corresponding metastable states are constructed using this truncated expansion. We show that their exponential decay law has both the decay rate and the error term of order $\varepsilon^4$, where $\varepsilon$ is the perturbation strength.

math-ph↗

arXiv2013

A complete classification of threshold properties for one-dimensional discrete Schrödinger operators

We consider the discrete one-dimensional Schrödinger operator $H=H_0+V$, where $(H_0x)[n]=-(x[n+1]+x[n-1]-2x[n])$ and $V$ is a self-adjoint operator on $\ell^2(\mathbb{Z})$ with a decay property given by $V$ extending to a compact operator from $\ell^{\infty,-β}(\mathbb{Z})$ to $\ell^{1,β}(\mathbb{Z})$ for some $β\geq1$. We give a complete description of the solutions to $Hx=0$, and $Hx=4x$, $x\in\ell^{\infty,-β}(\mathbb{Z})$. Using this description we give asymptotic expansions of the resolvent of $H$ at the two thresholds $0$ and $4$. One of the main results is a precise correspondence between the solutions to $Hx=0$ and the leading coefficients in the asymptotic expansion of the resolvent around $0$. For the resolvent expansion we implement the expansion scheme of Jensen-Nenciu \cite{JN0, JN1} in the full generality.

math-ph↗

arXiv2012

Memory effects in non-interacting mesoscopic transport

Consider a quantum dot coupled to two semi-infinite one-dimensional leads at thermal equilibrium. We turn on adiabatically a bias between the leads such that there exists exactly one discrete eigenvalue both at the beginning and at the end of the switching procedure. It is shown that the expectation on the final bound state strongly depends on the history of the switching procedure. On the contrary, the contribution to the final steady-state corresponding to the continuous spectrum has no memory, and only depends on the initial and final values of the bias.

math-ph↗