Magnetic Schrodinger Operator Research Articles

All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schrodinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.

Eigenvalue estimates for magnetic Schrodinger operators in a waveguide

We present an upper estimate for the number of negative eigenvalues below the essential spectrum for the magnetic Schrodinger operator with Aharonov-Bohm magnetic field in a strip.

Quantitative Weighted Estimates for Some Singular Integrals Related to Critical Functions

Let $$(X, d, \mu )$$ be a space of homogeneous type with a metric d and a doubling measure $$\mu $$ . Assume that $$\rho $$ is a critical function on X which has an associated class of weights containing the Muckenhoupt weights as a proper subset. In this paper, we prove the quantitative weighted estimates for certain singular integrals corresponding to the new class of weights. It is important to note that the assumptions on the kernels of these singular integrals do not have any regularity conditions. Our applications include the spectral multipliers and the Riesz transforms associated to Schrodinger operators in various settings, ranging from the magnetic Schrodinger operators in Euclidean spaces to the Laguerre operators.

Feynman path integrals for magnetic Schrödinger operators on infinite weighted graphs

We prove a Feynman path integral formula for the unitary group exp(—itLυ,θ), t ≥ 0, associated with a discrete magnetic Schrodinger operator Lυ,θ on a large class of weighted infinite graphs. As a consequence, we get a new Kato-Simon estimate $$\left| {\exp \left({- it{L_{v,\theta}}} \right)\left({x,y} \right)} \right| \le \exp \left({- t{L_{- \deg ,0}}} \right)\left({x,y} \right),$$ which controls the unitary group uniformly in the potentials in terms of a Schrodinger semigroup, where the potential deg is the weighted degree function of the graph.

Semiclassical approximation of the magnetic Schrödinger operator on a strip : dynamics and spectrum

In the semiclassical regime (i.e., ϵ↘0), we study the effect of a slowly varying potential V(ϵt,ϵz) on the magnetic Schrodinger operator P=Dx2+(Dz+μx)2 on a strip [−a,a]×ℝz. The potential V(t,z) is assumed to be smooth. We derive the semiclassical dynamics and we describe the asymptotic structure of the spectrum and the resonances of the operator P+V(ϵt,ϵz) for ϵ small enough. All our results depend on the eigenvalues corresponding to Dx2+(μx+k)2 on L2([−a,a]) with Dirichlet boundary condition.

A note on the surjectivity of operators on vector bundles over discrete spaces

In this note we give a short and self-contained proof for a criterion of Eidelheit on the solvability of linear equations in infinitely many variables. We use this criterion to study the surjectivity of magnetic Schrodinger operators on bundles over graphs.

Averaging and Spectral Bands for The 2-D Magnetic Schrödinger Operator with Growing and One-Direction Periodic Potential

In this paper, we consider the spectral problem for the magnetic Schrodinger operator on the 2-D plane (x1, x2) with the constant magnetic field normal to this plane and with the potential V having the form of a harmonic oscillator in the direction x1 and periodic with respect to variable x2. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large, this allows us to make the averaging and to reduce the original problem to a spectral problem for a 1-D Schrodinger operator with effective periodic potential. Then we use semiclassical analysis to construct the band spectrum of this reduced operator, as well as that of the original 2-D magnetic Schroodinger operator.

Discrete Magnetic Bottles on Quasi-Linear Graphs

We study discrete magnetic Schrodinger operators on graphs that are obtained by adding decorations to the linear graph $$\mathbb {N}$$ . We construct examples in which the magnetic field destroys the essential spectrum. The methods rely on spectral comparison theorems for quadratic forms and a careful study of an explicit Schrodinger operator on a finite graph.

Absolute continuity of the magnetic Schrodinger operator with periodic potential

We consider the magnetic Schr\odinger operator coupled with two different potentials. One of them is a harmonic oscillator and the other is a periodic potential. We give some periodic potential classes for which the operator has purely absolutely continuous spectrum. We also prove that for strong magnetic field or large coupling constant, there are open gaps in the spectrum and we give a lower bound on their number.

Determine a Magnetic Schrödinger Operator with a Bounded Magnetic Potential from Partial Data in a Slab

We study an inverse boundary value problem with partial data in an infinite slab in $$\mathbb {R}^{n}$$ , $$n\ge 3$$ , for the magnetic Schrodinger operator with bounded magnetic potential and electric potential. We show that the magnetic field and the electric potential can be uniquely determined, when the Dirichlet and Neumann data are given on either different boundary hyperplanes or on the same boundary hyperplanes of the slab. These generalize the results in Krupchyk et al. (Commun Math Phys 312:87–126, 2012), where the same uniqueness results were established when the magnetic potential is Lipschitz continuous. The proof is based on the complex geometric optics solutions constructed in Krupchyk and Uhlmann (Commun Math Phys 327:993–1009, 2014), which are special solutions to the magnetic Schrodinger equation with $$L^{\infty }$$ magnetic and electric potentials in a bounded domain.

On the Bound States of Magnetic Laplacians on Wedges

This paper is mainly inspired by the conjecture about the existence of bound states for magnetic Neumann Laplacians on planar wedges of any aperture Φ ∈ (0, π). So far, a proof was only obtained for apertures Φ ≲ 0.511π. The conviction in the validity of this conjecture for apertures Φ ≳ 0.511π mainly relied on numerical computations. In this paper we succeed to prove the existence of bound states for any aperture Φ ≲ 0.583π using a variational argument with suitably chosen test functions. Employing some more involved test functions and combining a variational argument with computer assistance, we extend this interval up to any aperture Φ ≲ 0.595π. Moreover, we analyse the same question for closely related problems concerning magnetic Robin Laplacians on wedges and for magnetic Schrodinger operators in the plane with δ-interactions supported on broken lines.

Magnetic Schrödinger Operator from the Point of View of Noncommutative Geometry

We give an interpretation of the magnetic Schrodinger operator in terms of noncommutative geometry. In particular, spectral properties of this operator are reformulated in terms of C*-algebras. Using this reformulation, one can employ the machinery of noncommutative geometry, such as Hochschild cohomology, to study the properties of the magnetic Schrodinger operator. We show how this idea can be applied to the integer quantum Hall effect.

An inverse problem for the magnetic Schrödinger operator on Riemannian manifolds from partial boundary data

We consider the inverse problem of recovering the magnetic and potential term of a magnetic Schrodinger operator on certain compact Riemannian manifolds with boundary from partial Dirichlet and Neumann data on suitable subsets of the boundary. The uniqueness proof relies on proving a suitable Carleman estimate for functions which vanish only on a part of boundary and constructing complex geometric optics solutions which vanish on a part of the boundary.

Stability estimates for a magnetic Schrödinger operator with partial data

In this paper we study local stability estimates for a magnetic Schrodinger operator with partial data on an open bounded set in dimension \begin{document}$ n≥3$\end{document} . This is the corresponding stability estimates for the identifiability result obtained by Bukhgeim and Uhlmann [ 2 ] in the presence of a magnetic field and when the measurements for the Dirichlet-Neumann map are taken on a neighborhood of the illuminated region of the boundary for functions supported on a neighborhood of the shadow region. We obtain log log-estimates for the magnetic fields and log log log-estimates for the electric potentials.

Inverse Scattering for the Magnetic Schrödinger Operator on Surfaces with Euclidean Ends

We prove a fixed frequency inverse scattering result for the magnetic Schrodinger operator (or connection Laplacian) on surfaces with Euclidean ends. We show that, under suitable decaying conditions, the scattering matrix for the operator determines both the gauge class of the connection and the zeroth order potential.

The reflection principle and Calderón problems with partial data

We propose a geometric approach towards for solving the partial data Calderon problem for the magnetic Schrodinger operator in dimension two. This allowed us to clarify some of the ideas in the previous works while simultaneously consider the problem on general Riemann surfaces with boundary.

On the Aharonov–Bohm Operators with Varying Poles: The Boundary Behavior of Eigenvalues

We consider a magnetic Schrodinger operator with magnetic field concentrated at one point (the pole) of a domain and half integer circulation, and we focus on the behavior of Dirichlet eigenvalues as functions of the pole. Although the magnetic field vanishes almost everywhere, it is well known that it affects the operator at the spectral level (the Aharonov–Bohm effect, Phys Rev (2) 115:485–491, 1959). Moreover, the numerical computations performed in (Bonnaillie-Noel et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010) show a rather complex behavior of the eigenvalues as the pole varies in a planar domain. In this paper, in continuation of the analysis started in (Bonnaillie-Noel et al., Anal PDE 7(6):1365–1395, 2014; Noris and Terracini, Indiana Univ Math J 59(4):1361–1403, 2010), we analyze the relation between the variation of the eigenvalue and the nodal structure of the associated eigenfunctions. We deal with planar domains with Dirichlet boundary conditions and we focus on the case when the singular pole approaches the boundary of the domain: then, the operator loses its singular character and the k-th magnetic eigenvalue converges to that of the standard Laplacian. We can predict both the rate of convergence and whether the convergence happens from above or from below, in relation with the number of nodal lines of the k-th eigenfunction of the Laplacian. The proof relies on the variational characterization of eigenvalues, together with a detailed asymptotic analysis of the eigenfunctions, based on an Almgren-type frequency formula for magnetic eigenfunctions and on the blow-up technique.

Topological damping of Aharonov-Bohm effect: quantum graphs and vertex conditions

The magnetic Schrodinger operator was studied on a figure 8-shaped graph. It is shown that for specially chosen vertex conditions, the spectrum of the magnetic operator is independent of the flux t ...

One-Dimensional Interpolation Inequalities, Carlson–Landau Inequalities, and Magnetic Schrödinger Operators

In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring inequalities for magnetic Schrodinger operators on multi-dimensional cylinders.

A Feynman–Kac–Itô formula for magnetic Schrödinger operators on graphs

In this paper we prove a Feynman–Kac–Ito formula for magnetic Schrodinger operators on arbitrary weighted graphs. To do so, we have to provide a natural and general framework both on the operator theoretic and the probabilistic side of the equation. On the operator side we identify a very general class of potentials that allows the definition of magnetic Schrodinger operators. On the probabilistic side, we introduce an appropriate notion of stochastic line integrals with respect to magnetic potentials. Apart from linking the world of discrete magnetic operators with the probabilistic world through the Feynman–Kac–Ito formula, the insights from this paper gained on both sides should be of an independent interest. As applications of the Feynman–Kac–Ito formula, we prove a Kato inequality, a Golden–Thompson inequality and an explicit representation of the quadratic form domains corresponding to a large class of potentials.