Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

Bruneau, Vincent; Miranda, Pablo; Raikov, Georgi

doi:10.1142/s0129055x14500032

Cited by 18 publications

(45 citation statements)

References 17 publications

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“…After completing this paper, we discovered the paper "Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians", by V. Bruneau, P. Miranda and G. Raikov [6]. Their Corollary 2.4, part (i), is similar to our Theorem 5.1.…”

Section: Acknowledgementsmentioning

confidence: 57%

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

Dombrowski

Hislop

Soccorsi

2014

Asymptotic Analysis

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We study two-dimensional magnetic Schrödinger operators with a magnetic field that is equal to b > 0 for x > 0 and −b for x < 0. This magnetic Schrödinger operator exhibits a magnetic barrier at x = 0. The unperturbed system is invariant with respect to translations in the y-direction. As a result, the Schrödinger operator admits a direct integral decomposition. We analyze the band functions of the fiber operators as functions of the wave number and establish their asymptotic behavior. Because the fiber operators are reflection symmetric, the band functions may be classified as odd or even. The odd band functions have a unique absolute minimum. We calculate the effective mass at the minimum and prove that it is positive. The even band functions are monotone decreasing. We prove that the eigenvalues of an Airy operator, respectively, harmonic oscillator operator, describe the asymptotic behavior of the band functions for large negative, respectively positive, wave numbers. We prove a Mourre estimate for perturbations of the magnetic Schrödinger operator and establish the existence of absolutely continuous spectrum in certain energy intervals. We prove lower bounds on magnetic edge currents for states with energies in the same intervals. We also prove that these lower bounds imply stable lower bounds for the asymptotic currents. Because of the unique, non-degenerate minimum of the first band function, we prove that a perturbation by a slowly decaying negative potential creates an infinite number of eigenvalues accumulating at the bottom of the essential spectrum from below. We establish the asymptotic behavior of the eigenvalue counting function for these infinitely-many eigenvalues below the bottom of the essential spectrum.

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Section: Acknowledgementsmentioning

confidence: 57%

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

Dombrowski

Hislop

Soccorsi

2014

Asymptotic Analysis

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“…From a general point of view, it is known that the extrema of the band functions play a significant role in the description of the spectral properties of fibered operators(see [15]). In the particular case where these extrema are reached and non-degenerate, there is a well known procedure to obtain effective Hamiltonians that allows to describe the distribution of eigenvalues (as in [33,7]) and the singularities of the SSF (see [5]).…”

Section: Introduction Motivationsmentioning

confidence: 99%

Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

Bruneau

Miranda

2018

Journal of Functional Analysis

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We consider the Schrödinger operator with constant magnetic field defined on the halfplane with a Dirichlet boundary condition, H 0 , and a decaying electric perturbation V . We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of H 0 , by studying the Spectral Shift Function (SSF) associated to the pair (H 0 + V, H 0 ). For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of V is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.

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“…Second order. We have that w 1 ϕ 1 " Opα 2 1 q, therefore we shall give priority to the term w 2 ϕ 0 since α 2 1 " opα 2 q, see (5). These considerations bring us to solve the equation ph 0μ 0 qϕ 2 " pµ 2´w2 qϕ 0 , and as above we get µ 2 " xw 2 ϕ 0 , ϕ 0 y and ϕ 2 " ph 0´E0 q´1pµ 2´w2 qϕ 0 .…”

Section: 1mentioning

confidence: 92%

“…We will assume: (5) aq D x 0 P R, @p P N, @x ě x 0 , b ppq pxq ‰ 0. bq b 1 pxq " opb`´bpxqq, and @p P N, b pp`1q pxq " opb ppq pxqq; x Ñ`8. cq pb 1 q 2 pxq " opb 2 pxqq; x Ñ`8.…”

Section: Asymptotics Of the Band Functionsmentioning

confidence: 99%

Spectrum of the Iwatsuka Hamiltonian at thresholds

Miranda

Popoff

2018

Journal of Mathematical Analysis and Applications

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ABSTRACT. We consider the bi-dimensional Schrödinger operator with unidirectionally constant magnetic field, H 0 , sometimes known as the "Iwatsuka Hamiltonian". This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given threshold in the spectrum of H 0 . In addition, for non-negative electric perturbations V we study the spectral density of H 0˘V , by considering the Spectral Shift Function associated to the operator pair pH 0V , H 0 q. We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grows to infinity.

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Dirichlet and Neumann eigenvalues for half-plane magnetic Hamiltonians

Cited by 18 publications

References 17 publications

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

Edge currents and eigenvalue estimates for magnetic barrier Schrödinger operators

Threshold singularities of the spectral shift function for a half-plane magnetic Hamiltonian

Spectrum of the Iwatsuka Hamiltonian at thresholds

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