A simple criterion for the existence of nonreal eigenvalues for a class of 2D and 3D Pauli operators

Sambou, Diomba

doi:10.1016/j.laa.2017.04.004

“…Much less is known in the mathematically challenging and still physically relevant situations where V is allowed to be complex-valued, see [16,15] and references therein. Works concerning non-self-adjoint Pauli operators are much more sparse in the literature, see [26] and references therein. More results are available in the case of non-self-adjoint Dirac operators, see [8,11,6,25,7,12,9,14].…”

Section: Objectives and State Of The Artmentioning

confidence: 99%

Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers

Cossetti,

Fanelli,

Krejcirik

2019

Preprint

1

0

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By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

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“…Much less is known in the mathematically challenging and still physically relevant situations where V is allowed to be complex-valued, see [15,16] and references therein. Works concerning non-self-adjoint Pauli operators are much more sparse in the literature, see [36] and references therein. More results are available in the case of nonself-adjoint Dirac operators, see [5][6][7][8][10][11][12]14,35].…”

Section: Objectives and State Of The Artmentioning

confidence: 99%

“…Works concerning non-self-adjoint Pauli operators are much more sparse in the literature, see [36] and references therein. More results are available in the case of nonself-adjoint Dirac operators, see [5][6][7][8][10][11][12]14,35]. The paper [16] represents a first, physically satisfactory, application of the method of multipliers to spectral theory: the authors established sufficient conditions, which guarantee the total absence of eigenvalues of (1.1).…”

Section: Objectives and State Of The Artmentioning

confidence: 99%

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

Cossetti

¹

,

Fanelli

²

,

Krejčiřı́k

³

2020

Commun. Math. Phys.

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By developing the method of multipliers, we establish sufficient conditions on the magnetic field and the complex, matrix-valued electric potential, which guarantee that the corresponding system of Schrödinger operators has no point spectrum. In particular, this allows us to prove analogous results for Pauli operators under the same electromagnetic conditions and, in turn, as a consequence of the supersymmetric structure, also for magnetic Dirac operators.

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“…In the non-selfadjoint case, the study of the spectrum of D V was initiated by Cuenin et al [10] in the 1D case, followed by [8,11,18]. For the higher dimensional case, we refer to the works [9,16,20,38].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue bounds for non-selfadjoint Dirac operators

D’Ancona

¹

,

Fanelli

²

,

Schiavone

³

2021

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We prove that the eigenvalues of the n-dimensional massive Dirac operator $${\mathscr {D}}_0 + V$$ D 0 + V , $$n\ge 2$$ n ≥ 2 , perturbed by a potential V, possibly non-Hermitian, are contained in the union of two disjoint disks of the complex plane, provided V is sufficiently small with respect to the mixed norms $$L^1_{x_j} L^\infty _{{\widehat{x}}_j}$$ L x j 1 L x ^ j ∞ , for $$j\in \{1,\dots ,n\}$$ j ∈ { 1 , ⋯ , n } . In the massless case, we prove instead that the discrete spectrum is empty under the same smallness assumption on V, and in particular the spectrum coincides with the spectrum of the unperturbed operator: $$\sigma ({\mathscr {D}}_0+V)=\sigma ({\mathscr {D}}_0)={\mathbb {R}}$$ σ ( D 0 + V ) = σ ( D 0 ) = R . The main tools used are an abstract version of the Birman–Schwinger principle, which allows in particular to control embedded eigenvalues, and suitable resolvent estimates for the Schrödinger operator.

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A simple criterion for the existence of nonreal eigenvalues for a class of 2D and 3D Pauli operators

Cited by 10 publications

References 33 publications

Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers

Absence of eigenvalues of Dirac and Pauli Hamiltonians via the method of multipliers

Absence of Eigenvalues of Dirac and Pauli Hamiltonians via the Method of Multipliers

Eigenvalue bounds for non-selfadjoint Dirac operators

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