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Eigenvalue asymptotics for magnetic fields and degenerate potentials Franc¸oise Truc

37 pages
Eigenvalue asymptotics for magnetic fields and degenerate potentials Franc¸oise Truc Abstract. We present various asymptotic estimates of the counting function of eigenvalues for Schrodinger operators in the case where the Weyl formula does not apply. The situations treated seem to establish a similarity between mag- netic bottles (magnetic fields growing at infinity) and degenerate potentials, and this impression is reinforced by an explicit study in classical mechanics, where the classical Hamiltonian induced by an axially symmetric magnetic bottle can be seen as a perturbation of the Hamiltonian derived from an operator with a degenerate potential. Table of contents 1. Introduction 2 2. Degenerate potentials 5 2.1. The Tauberian approach 5 2.2. The min-max approach 7 3. Magnetic bottles 9 3.1. General setting 9 3.2. The Euclidean case 10 3.3. The hyperbolic half-plane 13 3.4. Geometrically finite hyperbolic surfaces 19 4. A Neumann problem with magnetic field 21 4.1. A problem arising from super-conductivity 21 4.2. The spectrum in the case of the half-space, for a constant field and for h=1 22 4.3. Non-Weyl-type asymptotics when the field is nearly tangent to the boundary 23 5. A problem of magnetic bottle in classical mechanics 24 5.1. The Lorentz equation 24 5.2. Adiabatic invariants 25 Received by the editors April 2O, 2009. 2000 Mathematics Subject Classification.

  • ginzburg-landau func- tional associated

  • field

  • symmetric magnetic

  • counting func

  • constant field

  • neumann realization

  • homogeneous potential

  • universal constant


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Eigenvalue asymptotics for magnetic fields and degenerate potentials
¸ Francoise Truc
Abstract.We present various asymptotic estimates of the counting function of eigenvaluesforSchro¨dingeroperatorsinthecasewheretheWeylformuladoes not apply. The situations treated seem to establish a similarity between mag-netic bottles (magnetic fields growing at infinity) and degenerate potentials, and this impression is reinforced by an explicit study in classical mechanics, where the classical Hamiltonian induced by an axially symmetric magnetic bottle can be seen as a perturbation of the Hamiltonian derived from an operator with a degenerate potential.
Table of contents
1. Introduction 2 2. Degenerate potentials 5 2.1. The Tauberian approach 5 2.2. The min-max approach 7 3. Magnetic bottles 9 3.1. General setting 9 3.2. The Euclidean case 10 3.3. The hyperbolic half-plane 13 3.4. Geometrically finite hyperbolic surfaces 19 4. A Neumann problem with magnetic field 21 4.1. A problem arising from super-conductivity 21 4.2. The spectrum in the case of the half-space, for a constant field and forh=1 22 4.3. Non-Weyl-type asymptotics when the field is nearly tangent to the boundary 23 5. A problem of magnetic bottle in classical mechanics 24 5.1. The Lorentz equation 24 5.2. Adiabatic invariants 25
Received by the editors April 2O, 2009. 2000Mathematics Subject Classification.Primary 35P20. Key words and phrases.ampxiM-npielircnlasy,Weyticsmptodo¨rhcS,eporegni,rstoraSctpem,ru magnetic bottles.
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Francoise Truc ¸
5.3. Bounded trajectories 6. Open problems and conclusion References
1. Introduction
26 31 34
In this review are presented several results of spectral analysis, based for most of them on the min-max variational principle . These are mainly ”non-Weyl-type” asymptoticsforsomenongenericSchr¨odingeroperators.Intheappropriate setup, the Weyl formula describes the asymptotic relationship between the number of eigenvalues less than some fixed valueλand the volume, in phase space, of trajectories with energy less thanλfor the corresponding classical problem. To be more precise, let us consider a continuous positive-valued potentialVonRm, and let us make the following assumption forV(x) : (1.1)V(x)+when|x| →+(we call such aV(x) a non degenerate potential). Then for any value of the pa-rameterhin ]01], the operatorHh=h2Δ +Vdefined onL2(Rm) is essentially self-adjoint and has a compact resolvent [50]. Moreover, denoting byN(λ Hh) the number of eigenvalues less than some fixed valueλ, we get the following semi-classical asymptotic behaviour, whenh0 :
(1.2)N(λ Hh)hm(2π)mvmZRmλV(x))m+2dx  ( In this so-called semi-classical Weyl asymptotic formula,vmdenotes the vol-ume of the unit ball inRm, and byW+we mean that we take the positive part of W. If we takeh= 1 in the previous formula we get the asymptotics for large energies of the operatorH1=Δ +V: (1.3)N(λ H1)λ+(2π)mvmZm(λV(x))+m2dx  R The right-hand side of the formula (1.2) can be seen more generically as the volume, in phase space, of the set{(x ξ)H(x ξ)λ}, whereH(x ξ) =ξ2+V(x) is the principal symbol ofHhand the Hamiltonian of the associated dynamics . Anaturelquestionisthenthefollowing:whatcanbesaidofaSchro¨dinger operator which has a discrete spectrum but does not verify the non-degeneracy condition (1.1) ? In that case the volume of{(x ξ) ξ2+V(x)λ}may happen to be infinite, so that the formula (1.2) becomes irrelevant. This is the case for instance for the following potential ( inR2) V(x y) = (1 +x2)y2( Figure 1) . The problems presented below discuss precisely this question for various situations, and the estimates obtained will be called non-Weyl-type asymptotics.
Eigenvalue asymptotics for magnetic fields
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and degenerate potentials
-2 -1 0x
Figure 1.The potentialV(x y) = (1 +x2)y2
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First we recall the results obtained in ([39], [41]), for a class of degenerate potentials in the sense we previously defined. These potentials can be seen as a generalization of the preceding example ; they are of the form V(x) =f(y)g(z) x= (y z)Rn×RpfC(Rn;R+)gC(Rp;R+) ghomogeneous of degreea. ThenweconsiderSchr¨odingeroperatorswithmagneticeldHh(A) = ((h∇− iA))2. One can call them degenerate in the sense that the principal symbol of Hh(A), which isH(x ξ) = (ξA(x))2, annihilates on a non compact mani-fold ofT(Rm). If the magnetic fieldB=dAis such that the counting func-tionN(λ Hh(Acan look for some alternative to Weyl)) can be defined, then we formula. In particular, when the magnetic fieldB=dAsatisfies some so-called magnetic bottles conditions :
(1.4)
kB(x)k →+quand|x| →+
Hh(Aself-adjoint and has a compact resolvent on) is essentially L2(Rm) [3]. The spectralasymptoticsforlargeenergieswerecomputedbyY.ColindeVerdie`re[6]. Here are discussed the semi-classical version of this result [60], and the case of magneticbottlesinthehyperboliccontext([42]forthePoincare´half-plane,[43] for geometrically finite hyperbolic surfaces). These non-Weyl-type asymptotics can be seen as the expression of an integrated density of states on the whole space. For a constant magnetic fieldB=Pjr=1bjdxjdyj b1b2br>0, the