Essential Spectrum and Exponential Decay Estimates of Solutions of Elliptic Systems of Partial Differential Equations. Applications to Schrödinger and Dirac Operators

Rabinovitch, Vladimir; Roch, Steffen

doi:10.1515/gmj.2008.333

Cited by 15 publications

(8 citation statements)

References 20 publications

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“…Here (27) follows from (2). Finally, we need the following crucial estimate stating that R A (z) stays bounded after conjugation with suitable exponential weights, e F , acting as multiplication operators in H .…”

Section: Miscellaneous Results On Spectral Projectionsmentioning

confidence: 99%

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On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields

Matte

Stockmeyer

2009

Integr. equ. oper. theory

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We consider a relativistic no-pair model of a hydrogenic atom in a classical, exterior magnetic field. First, we prove that the corresponding Hamiltonian is semi-bounded below, for all coupling constants less than or equal to the critical one known for the Brown-Ravenhall model, i.e., for vanishing magnetic fields. We give conditions ensuring that its essential spectrum equals [1, ∞) and that there exist infinitely many eigenvalues below 1. (The rest energy of the electron is 1 in our units.) Assuming that the magnetic vector potential is smooth and that all its partial derivatives increase subexponentially, we finally show that an eigenfunction corresponding to an eigenvalue λ < 1 is smooth away from the nucleus and that its partial derivatives of any order decay pointwise exponentially with any rate a < √ 1 − λ 2 , for λ ∈ [0, 1), and a < 1, for λ < 0. Mathematics Subject Classification (2000). Primary 81Q10; Secondary 47B25.

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Section: Miscellaneous Results On Spectral Projectionsmentioning

confidence: 99%

“…Although such bounds are essentially well-known [4,16,34] it seems illustrative to include them as a remark here. For a general scheme to study the exponential decay of solutions of an elliptic system of partial differential equations we refer to [26,27].…”

Section: Introductionmentioning

confidence: 99%

On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields

Matte

Stockmeyer

2009

Integr. equ. oper. theory

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“…The identity (32) implies that (D ′g w µ − λI)(D ′g w µ + λI) is the diagonal matrix diag (F, F ) with…”

Section: The Essential Spectrummentioning

confidence: 99%

“…Exponential estimates of solutions of pseudodifferential equations are considered in [15,19,20,23,26,27]. Note also the our recent papers [32,33] where we proposed a new approach to exponential estimates for partial differential and pseudodifferential operators, based on the limit operators method (see [34]).…”

Section: Introductionmentioning

confidence: 99%

Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

Rabinovich¹,

Roch²

2009

J. Phys. A: Math. Theor.

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This paper is devoted to estimates of the exponential decay of eigenfunctions of difference operators on the lattice Z n which are discrete analogs of the Schrödinger, Dirac and square-root Klein-Gordon operators. Our investigation of the essential spectra and the exponential decay of eigenfunctions of the discrete spectra is based on the calculus of the so-called pseudodifference operators (i.e., pseudodifferential operators on the group Z n with analytic symbols, as developed in the paper [29]), and the limit operators method (see [34] and the references cited there). We obtain a description of the location of the essential spectra and estimates of the eigenfunctions of the discrete spectra of the main lattice operators of quantum mechanics, namely: matrix Schrödinger operators on Z n , Dirac operators on Z 3 , and square root Klein-Gordon operators on Z n .

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“…Applications of results of this paper allow us to obtain an effective description of the essential spectra of the operators  ,a,W, in terms of their limit operators. Earlier, the limit operators have been used for the study of the Fredholm property of different problems of operator theory and mathematical physics (see other studies [22][23][24][25][26][27] ).…”

Section: Introductionmentioning

confidence: 99%

Magnetic Schrödinger operators with delta‐type potentials

Rabinovich

2020

Math Methods in App Sciences

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We consider the magnetic anisotropic Schrödinger operator on where is the vector potential of the magnetic field and W(x) is the scalar potential of the electric field. We assume that aj and ϱij are real‐valued functions belonging to the space of bounded with first derivatives on functions, whereas is a complex‐valued electric potential. Let Σ be a C2‐hypersurface in dividing on two open domains Ω± with common boundary Σ. We assume that Σ is a closed C2‐hypersurface or unbounded hypersurface of bounded geometry. We consider the magnetic Schrödinger operator with singular potentials Ws with supports on Σ. We associate with an unbounded operator in generated by Hϱ,a, W with domain in H2(Ω+) ⊕ H2(Ω−) consisting of functions satisfying interaction conditions on Σ. We study the self‐adjointness of the operator and its Fredholm properties. Moreover, we consider the spectral problem and reduce this problem to a spectral problem for some boundary pseudodifferential operators on Σ. We describe the domains in where problem (2) has the discrete spectrum.

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Essential Spectrum and Exponential Decay Estimates of Solutions of Elliptic Systems of Partial Differential Equations. Applications to Schrödinger and Dirac Operators

Cited by 15 publications

References 20 publications

On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields

On the Eigenfunctions of No-Pair Operators in Classical Magnetic Fields

Essential spectra and exponential estimates of eigenfunctions of lattice operators of quantum mechanics

Magnetic Schrödinger operators with delta‐type potentials

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