Un exemple de champ magnetique dans Rν

Dufresnoy, Alain

doi:10.1215/s0012-7094-83-05035-4

Cited by 19 publications

(26 citation statements)

References 2 publications

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“…This means that a magnetic field can only improve the situation from our point of view. Papers by J. Avron, I. Herbst and B. Simon [1], Y. Colin de Verdière [4], A. Dufresnoy [6] and A. Iwatsuka [14] provide some quantitative results which show that even in case V = 0 the magnetic field can make the spectrum discrete. (This situation is called magnetic bottle.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Kondratiev

Maz′ya

Shubin

2005

Communications in Partial Differential Equations

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We establish necessary and sufficient conditions for the discreteness of spectrum and strict positivity of magnetic Schrödinger operators with a positive scalar potential. They are expressed in terms of Wiener's capacity and the local energy of the magnetic field. The conditions for the discreteness of spectrum depend, in particular, on a functional parameter which is a decreasing function of one variable whose argument is the normalized local energy of the magnetic field. This function enters the negligibility condition of sets for the scalar potential. We give a description for the range of all admissible functions which is precise in a certain sense.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

“…Other, more effective sufficient conditions (which do not include µ 0 ) and related results (in particular, asymptotics of eigenvalues under appropriate conditions) can be found in [4,6,8,11,12,13,14,15,20,23,29,32,34].…”

Section: Remark 13mentioning

confidence: 99%

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Kondratiev

Maz′ya

Shubin

2005

Communications in Partial Differential Equations

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“…The power 3 2 in (1.3) is not a coincidence as it is known to be optimal (with little-o replaced by a sufficiently small constant in (1.3)) with respect to the separation property in the self-adjoint case [14,15,8] (see also [11], [20] in the magnetic case).…”

Section: About the Powermentioning

confidence: 99%

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

et al. 2017

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We consider non-self-adjoint electromagnetic Schrödinger operators on arbitrary open sets with complex scalar potentials whose real part is not necessarily bounded from below. Under a suitable sufficient condition on the electromagnetic potential, we introduce a Dirichlet realisation as a closed densely defined operator with non-empty resolvent set and show that the eigenfunctions corresponding to discrete eigenvalues satisfy an Agmon-type exponential decay.

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“…This means that a magnetic field can only improve the situation from our point of view. Papers by J. Avron, I. Herbst and B. Simon [1], A. Dufresnoy [8] and A. Iwatsuka [18] provide some quantitative results which show that even in case V = 0 the magnetic field can make the spectrum discrete. In this paper we will improve the results of the above mentioned papers.…”

Section: )mentioning

confidence: 99%

Discreteness of Spectrum for the Magnetic Schrödinger Operators

Kondratiev

Shubin

2002

Communications in Partial Differential Equations

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We consider a magnetic Schrödinger operator H in R n or on a Riemannian manifold M of bounded geometry. Sufficient conditions for the spectrum of H to be discrete are given in terms of behavior at infinity for some effective potentials V ef f which are expressed through electric and magnetic fields. These conditions can be formulated in the form V ef f (x) → +∞ as x → ∞. They generalize the classical result by K.Friedrichs (1934), and include earlier results of J. Avron, I. Herbst and B. Simon (1978), A. Dufresnoy (1983) and A. which were obtained in the absence of an electric field. More precise sufficient conditions can be formulated in terms of the Wiener capacity and extend earlier work by A.M. Molchanov (1953) and V. Kondrat'ev and M. Shubin (1999) who considered the case of the operator without a magnetic field. These conditions become necessary and sufficient in case there is no magnetic field and the electric potential is semi-bounded below.

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Un exemple de champ magnetique dans Rν

Cited by 19 publications

References 2 publications

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Discreteness of Spectrum and Strict Positivity Criteria for Magnetic Schrödinger Operators

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

Discreteness of Spectrum for the Magnetic Schrödinger Operators

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