Weyl asymptotics for magnetic Schrödinger operators and de Gennes' boundary condition
aa r X i v : . [ m a t h . SP ] J un WEYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGEROPERATORS AND DE GENNES' BOUNDARY CONDITIONAYMAN KACHMARAbstra t. This paper is on erned with the dis rete spe trum of theself-adjoint realization of the semi- lassi al S hrödinger operator with onstant magneti (cid:28)eld and asso iated with the de Gennes (Fourier/Robin)boundary ondition. We derive an asymptoti expansion of the numberof eigenvalues below the essential spe trum (Weyl-type asymptoti s).The methods of proof relies on results on erning the asymptoti be-havior of the (cid:28)rst eigenvalue obtained in a previous work [A. Ka hmar,J. Math. Phys. 47 (7) 072106 (2006)℄.1. Introdu tion and main resultsLet Ω ⊂ R be an open domain with regular and ompa t boundary.Given a smooth fun tion γ ∈ C ∞ ( ∂ Ω; R ) and a number α ≥ , we onsiderthe S hrödinger operator with magneti (cid:28)eld :(1.1) P α,γh, Ω = − ( h ∇ − iA ) , whose domain is, D (cid:16) P α,γh, Ω (cid:17) = (cid:8) u ∈ L (Ω) : ( h ∇ − iA ) j ∈ L (Ω) , j = 1 , , (1.2) ν · ( h ∇ − iA ) u + h α γ u = 0 on ∂ Ω (cid:9) . Here ν is the unit outward normal ve tor of the boundary ∂ Ω , A ∈ H (Ω; R ) is a ve tor (cid:28)eld and curl A is the magneti (cid:28)eld. Fun tions in the domain of P α,γh, Ω satisfy the de Gennes boundary ondition.The operator P α,γh, Ω arises from the analysis of the onset of super ondu -tivity for a super ondu tor pla ed adja ent to another materials. For thephysi al motivation and the mathemati al justi(cid:28) ation of onsidering thistype of boundary ondition and not the usual Neumann ondition ( γ ≡ ),we invite the interested reader to see the book of de Gennes [6℄ and the pa-pers [10, 11, 12, 13℄. We would like to mention that when γ ≡ , the operator P α,γh, Ω has been the subje t of many papers, see [4℄ and the referen es therein.Date: November 3, 2018.2000 Mathemati s Subje t Classi(cid:28) ation. Primary 81Q10, Se ondary 35J10, 35P15,82D55.Key words and phrases. Magneti S hrödinger operator, spe tral fun tion, dis retespe trum, semi lassi al analysis.This work has been partially supported by the European Resear h Network `Post-do toral Training Program in Mathemati al Analysis of Large Quantum Systems' with ontra t number HPRN-CT-2002-00277 and by the ESF S ienti(cid:28) Programme in Spe tralTheory and Partial Di(cid:27)erential Equations (SPECT).1 AYMAN KACHMARWe shall restri t ourselves with the ase of onstant magneti (cid:28)eld, namelywhen(1.3) curl A = 1 in Ω . It follows from the well-known inequality(1.4) Z Ω | ( h ∇ − iA ) u | d x ≥ h Z Ω | u | d x, ∀ u ∈ C ∞ (Ω) , and from a `magneti ' Persson's Lemma ( f. [16, 1℄), that the bottom ofthe essential spe trum of P α,γh, Ω is above h . Assuming that the boundary of Ω is smooth and ompa t, then it follows from [10℄ that (in the parameterregime α ≥ ), the operator P α,γh, Ω has dis rete spe trum below h . Thus,given b < , one is led to estimate the size of the dis rete spe trum below b h , i.e. we look for the asymptoti behavior of the number(1.5) N ( α, γ ; b h ) of eigenvalues of P α,γh, Ω (taking multipli ities into a ount) in luded in theinterval ]0 , b h ] .For the ase with non- onstant magneti (cid:28)eld and Neumann boundary on-dition, this problem has been analyzed by R. Frank [5℄ (related questions arealso treated in [2, 8, 18, 19℄). As we shall see, depending on the type of theboundary ondition, one an produ e mu h additional eigenvalues below theessential spe trum.To state the results on erning N ( α, γ ; b h ) , we need to introdu e some no-tation. Let us introdu e the smooth fun tions, whi h arise from the analysisof the model-operator in the half-plane (see [10, Se tion II℄),(1.6) R × R + ∋ ( γ, ξ ) µ ( γ, ξ ) , R ∋ γ Θ( γ ) , where µ ( γ, ξ ) = inf u ∈ B ( R + ) , u Z R + (cid:0) | u ′ ( t ) | + | ( t − ξ ) u ( t ) | (cid:1) d t + γ | u (0) | Z R + | u ( t ) | d t , Θ( γ ) = inf ξ ∈ R µ ( γ, ξ ) , and the spa e B ( R + ) onsists of fun tions in the spa e H ( R + ) ∩ L ( R + ; t d t ) .A tually, µ ( γ.ξ ) is the (cid:28)rst eigenvalue of the self-adjoint operator − ∂ t + ( t − ξ ) in L ( R + ) asso iated with the boundary ondition u ′ (0) = γ u (0) . The eigenvalues ofthis operator form an in reasing sequen e whi h we denote ( µ j ( γ, ξ )) j ∈ N , seeSubse tion 2.1 for more details.When γ = 0 , we write as in the usual ase (see [4℄) µ ( ξ ) := µ (0 , ξ ) , Θ := Θ(0) . Furthermore, we denote by,(1.7) C ( γ ) = 13 (cid:16) γ p Θ( γ ) + γ (cid:17) Θ ′ ( γ ) , C = C (0) . We are ready now to state our main results.EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 3Theorem 1.1. Assume that α > and Θ < b < . Then as h → ,(1.8) N ( α, γ ; b h ) = (cid:18) | ∂ Ω | π √ h (cid:12)(cid:12) { ξ ∈ R : µ ( ξ ) < b } (cid:12)(cid:12)(cid:19) (1 + o (1)) . On the other hand, if α = and Θ( γ ) < b < , then as h → , N ( α, γ ; b h ) = (1.9) π √ h Z ∂ Ω ∞ X j =1 (cid:12)(cid:12) { ξ ∈ R : µ j ( γ ( s ) , ξ ) < b } (cid:12)(cid:12) d s (1 + o (1)) . Here(1.10) γ = min s ∈ ∂ Ω γ ( s ) . If we suppose furthermore that γ ≥ , then (1.9) simpli(cid:28)es to N ( α, γ ; b h ) = (1.11) (cid:18) π √ h Z ∂ Ω (cid:12)(cid:12) { ξ ∈ R : µ ( γ ( s ) , ξ ) < b } (cid:12)(cid:12) d s (cid:19) (1 + o (1)) . By taking γ ≡ , we re over in Theorem 1.1 the result of R. Frank [5℄.We noti e that when α = and γ is onstant, we have additional eigenval-ues than the usual ase of Neumann boundary ondition if γ < and lesseigenvalues if γ > . This is natural as we apply the variational min-maxprin iple. However, when γ = 0 or α > , Theorem 1.1 fails to give a omparison with the Neumann ase, i.e. we have no more information aboutthe size of the di(cid:27)eren e: N ( α, γ ; b h ) − N (0; b h ) . This is at least a motivation for some of the next results, where we take b = b ( h ) asymptoti ally lose to Θ , ea h time with an appropriate s ale(this will over also the ase b = Θ ).Theorem 1.2. If < α < then for all a ∈ R , N (cid:16) α, γ ; h Θ + 3 aC h α + (cid:17) (1.12) = 1 π q h − α √ Θ (cid:18)Z ∂ Ω p ( a − γ ( s )) + d s (cid:19) (1 + o (1)) . In the parti ular ase when the fun tion γ is onstant, the leading or-der term in (1.12) will vanish when a is taken equal to γ . In this spe i(cid:28) regime, Theorem 1.5 (more pre isely the formula in (1.16)) will substituteTheorem 1.2.Theorem 1.3. Assume that α = 1 / . Let < ̺ < , ζ > , h > and ]0 , h ] ∋ h c ( h ) ∈ R + a fun tion su h that lim h → c ( h ) = ∞ . If c ( h ) h / ≤ | λ − Θ( γ ) | ≤ ζ h ̺ , ∀ h ∈ ]0 , h ] , AYMAN KACHMARthen we have the asymptoti formula, N ( α, γ ; hλ ) (1.13) = π Z ∂ Ω s [ λ − Θ( γ ( s ))] + h Θ ′ ( γ ( s )) p Θ( γ ( s )) + γ ( s ) d s ! (1 + o (1)) , where the fun tion Θ( · ) being introdu ed in (1.6).Remark 1.4. Theorem 1.3 be omes of parti ular interest when the fun tion γ has a unique non-degenerate minimum and λ = Θ( γ ) + ah β , for some a ∈ R + and β ∈ ]0 , [ . In this ase, we have in the support of [ λ − Θ( γ ( s ))] + , Θ( γ ( s )) = Θ( γ ) + c s + O ( h β/ ) , for an expli it onstant c > determined by the fun tions γ and Θ .Therefore, the asymptoti expansion (1.13) reads in this ase, for some ex-pli it onstant c > ,(1.14) N ( α, γ ; hλ ) = c a √ a h β − (1 + o (1)) . The next theorem deals with the regime where the s alar urvature be- omes e(cid:27)e tive in the asymptoti expansions.Theorem 1.5.(1) Assume that α = 1 . Then, for all a ∈ R , the following asymptoti expansion holds as h → , N (cid:16) , γ ; h Θ + aC h / (cid:17) = (1.15) π p h / √ Θ (cid:18)Z ∂ Ω p ( κ r ( s ) − γ ( s ) + a ) + d s (cid:19) (1 + o (1)) , where κ r is the s alar urvature of ∂ Ω .(2) If the fun tion γ is onstant, then for all α > / and a ∈ R , wehave the asymptoti expansion, N (cid:16) α, γ ; h Θ( h α − / γ ) + aC h / (cid:17) = (1.16) π p h / √ Θ (cid:18)Z ∂ Ω p ( κ r ( s ) + a ) + d s (cid:19) (1 + o (1)) . (3) If the fun tion γ is onstant and α = 1 / , then for all a ∈ R , wehave N (cid:16) α, γ ; h Θ( γ ) + aC ( γ ) h / (cid:17) = (1.17) γ p Θ( γ ) + γ π q h / p Θ( γ ) + γ (cid:18)Z ∂ Ω p ( κ r ( s ) + a ) + d s (cid:19) (1 + o (1)) . Here C ( γ ) has been de(cid:28)ned in (1.7).The proof of Theorems 1.1-1.5 is through areful estimates in the semi- lassi al regime of the quadrati form u q α,γh, Ω ( u ) = Z Ω | ( h ∇ − iA ) u | d x + h α Z ∂ Ω γ ( s ) | u ( s ) | d s . EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 5These estimates are essentially obtained in [7℄ when γ ≡ , then adaptedto situations involving the de Gennes boundary ondition in [10, 14℄. Weshall follow losely the arguments of [5℄ but we also require to use variousproperties of the fun tion γ Θ( γ ) established in [10℄.The paper is organized in the following way. Se tion 2 is devoted to theanalysis of the model operator in a half- ylinder when the fun tion γ is onstant. Se tion 3 extends the result obtained for the model ase in a half- ylinder for a general domain by whi h we prove Theorem 1.1. Se tion 4deals with model operators on weighted L spa es whi h serve in provingTheorems 1.2-1.5.2. Analysis of the model operator2.1. A family of one-dimensional di(cid:27)erential operators. Let us re allthe main results obtained in [9, 10℄ on erning a family of di(cid:27)erential oper-ators with Robin boundary ondition. Given ( γ, ξ ) ∈ R × R , we de(cid:28)ne thequadrati form,(2.1) B ( R + ) ∋ u q [ γ, ξ ]( u ) = Z R + (cid:0) | u ′ ( t ) | + | ( t − ξ ) u ( t ) | (cid:1) dt + γ | u (0) | , where, for a positive integer k ∈ N and a given interval I ⊆ R , the spa e B k ( I ) is de(cid:28)ned by :(2.2) B k ( I ) = { u ∈ H k ( I ); t j u ( t ) ∈ L ( I ) , ∀ j = 1 , · · · , k } . By Friedri hs Theorem, we an asso iate to the quadrati form (2.1) a selfadjoint operator L [ γ, ξ ] with domain, D ( L [ γ, ξ ]) = { u ∈ B ( R + ); u ′ (0) = γu (0) } , and asso iated to the di(cid:27)erential operator,(2.3) L [ γ, ξ ] = − ∂ t + ( t − ξ ) . We denote by { µ j ( γ, ξ ) } + ∞ j =1 the in reasing sequen e of eigenvalues of L [ γ, ξ ] .When γ = 0 we write,(2.4) µ j ( ξ ) := µ j (0 , ξ ) , ∀ j ∈ N , L N [ ξ ] := L [0 , ξ ] . We also denote by { µ Dj ( ξ ) } + ∞ j =1 the in reasing sequen e of eigenvalues of theDiri hlet realization of − ∂ t + ( t − ξ ) .By the min-max prin iple, we have,(2.5) µ ( γ, ξ ) = inf u ∈ B ( R + ) ,u =0 q [ γ, ξ ]( u ) k u k L ( R + ) . Let us denote by ϕ γ,ξ the positive (and L -normalized) (cid:28)rst eigenfun tion of L [ γ, ξ ] . It is proved in [10℄ that the fun tions ( γ, ξ ) µ ( γ, ξ ) , ( γ, ξ ) ϕ γ,ξ ∈ L ( R + ) are regular (i.e. of lass C ∞ ), and we have the following formulae, ∂ ξ µ ( γ, ξ ) = − (cid:0) µ ( γ, ξ ) − ξ + γ (cid:1) | ϕ γ,ξ (0) | , (2.6) ∂ γ µ ( γ, ξ ) = | ϕ γ,ξ (0) | . (2.7) AYMAN KACHMARNoti e that (2.7) will yield that the fun tion ( γ, ξ ) ϕ γ,ξ (0) is also regular of lass C ∞ .We de(cid:28)ne the fun tion :(2.8) Θ( γ ) = inf ξ ∈ R µ ( γ, ξ ) . It is a result of [3℄ that there exists a unique ξ ( γ ) > su h that,(2.9) Θ( γ ) = µ ( γ, ξ ( γ )) , Θ( γ ) < , and ξ ( γ ) satis(cid:28)es ( f. [10℄),(2.10) ξ ( γ ) = Θ( γ ) + γ . Moreover, the fun tion Θ( γ ) is of lass C ∞ and satis(cid:28)es,(2.11) Θ ′ ( γ ) = | ϕ γ (0) | , where ϕ γ is the positive (and L -normalized) eigenfun tion asso iated to Θ( γ ) :(2.12) ϕ γ = ϕ γ,ξ ( γ ) . When γ = 0 , we write,(2.13) Θ := Θ(0) , ξ := ξ (0) . It is a onsequen e of (2.11) that the onstant C introdu ed in (1.7) an bede(cid:28)ned by the alternative manner,(2.14) C := | ϕ (0) | . Let us re all an important onsequen e of standard Sturm-Liouville theory( f. [5, Lemma 2.1℄).Lemma 2.1. For all ξ ∈ R , we have µ ( ξ ) > µ D ( ξ ) > . Let us also introdu e,(2.15) Θ k ( γ ) = inf ξ ∈ R µ k ( γ, ξ ) , ∀ k ∈ N . Another onsequen e of Sturm-Liouville theory that we shall need is thefollowing result on Θ ( γ ) .Lemma 2.2. For any γ ∈ R , we have, Θ ( γ ) > Θ( γ ) . EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 7Proof. Let us introdu e the ontinuous fun tion f ( γ ) = Θ ( γ ) − Θ( γ ) . Usingthe min-max prin iple, it follows from (2.9) and Lemma 2.1 that f (0) > .It is then su(cid:30) ient to prove that the fun tion f never vanish. Suppose by ontradi tion that there is some γ = 0 su h that Θ ( γ ) = Θ( γ ) . By thesame method used in [10℄ one is able to prove that there exists ξ ( γ ) > su h that Θ ( γ ) = µ ( γ , ξ ( γ )) , and that ξ ( γ ) = Θ ( γ ) + γ . Therefore we get ξ ( γ ) = ξ ( γ ) .Now, by Sturm-Liouville theory, the eigenvalues of the operator L [ γ , ξ ( γ )] are all simple, whereas, by the above, we get a degenerate eigenvalue µ ( γ , ξ ( γ )) = Θ( γ ) = µ ( γ , ξ ( γ )) , whi h is the desired ontradi tion. (cid:3) One more useful result in Sturm-Liouville theory is the following.Lemma 2.3. Let γ ∈ R − and k ∈ N . Then Θ k ( γ ) < k + 1 and for all b ∈ ]Θ k ( γ ) , k + 1[ , the equation µ k ( γ, ξ ) = b has exa tly two solutions ξ k, − ( γ, b ) and ξ k, + ( γ, b ) . Moreover, { ξ ∈ R : µ k ( γ, ξ ) < b } = ] − ξ k, − ( γ, b ) , ξ k, + ( γ, b )[ . Proof. We an study the variations of the fun tion ξ µ k ( γ, ξ ) using ex-a tly the same method of [10, 14, 3℄. We obtain that the fun tion ξ µ k ( γ, ξ ) attains a unique non-degenerate minimum at the point ξ k ( γ ) = p Θ k ( γ ) + γ , and analogous formulae to (2.6)-(2.7) ontinue to hold for ( γ, ξ ) µ k ( γ, ξ ) . Moreover, lim ξ →−∞ µ k ( γ, ξ ) = ∞ and lim ξ →∞ µ k ( γ, ξ ) = 2 k + 1 .For instan e, the restri tions of the fun tion ξ µ k ( γ, ξ ) to the intervals ] − ∞ , ξ k ( γ )[ and ] ξ k ( γ ) , ∞ [ are invertible. (cid:3) It is a result of the variational min-max prin iple that the fun tion γ Θ k ( γ ) is ontinuous, see [10, Proposition 2.5℄ for the ase k = 1 . Thus theset(2.16) U k = { ( γ, b ) ∈ R × R : Θ k ( γ ) < b < k + 1 } is open in R . Lemma 2.4. The fun tions U k ∋ ( γ, b ) ξ k, ± ( γ, b ) admit ontinuous extensions R × ] − ∞ , k + 1[ ξ k, ± ( γ, b ) . Proof. Using the regularity of µ k ( γ, ξ ) , the impli it fun tion theorem appliedto U k × R ∋ ( γ, b, ξ ) µ k ( γ, ξ ) − b near ( γ , b , ξ k, ± ( γ , b )) (for an arbitrary point ( γ , b ) ∈ U k ) permits todedu e that the fun tions U k ∋ ( γ, b ) ξ k, ± ( γ, b ) are C . AYMAN KACHMARWe then de(cid:28)ne the following ontinuous extensions of ξ k, ± , ξ k, ± ( γ, b ) = (cid:26) ξ k, ± ( γ, b ) , if Θ k ( γ ) < b < k + 1 ,ξ k ( γ ) , if Θ k ( γ ) ≥ b , where ξ k ( γ ) is the unique non-degenerate minimum of ξ µ k ( γ, ξ ) . (cid:3) The next lemma justi(cid:28)es that the sum on the right hand side of (1.9) isindeed (cid:28)nite.Lemma 2.5. For ea h
M > and b ∈ ]0 , , there exists a onstant C > su h that, for all γ ∈ ] − M, M [ and b ∈ ]Θ( γ ) , b [ , we have ∞ X j =1 (cid:12)(cid:12) { ξ ∈ R : µ j ( γ, ξ ) < b } (cid:12)(cid:12) ≤ C. Proof. Let us noti e that for all j ≥ , { ξ ∈ R : µ j ( γ, ξ ) < b } ⊂ { ξ ∈ R : µ ( γ, ξ ) < b } , and for all γ ∈ ] − M, M [ (using the monotoni ity of η µ ( η, ξ ) ), { ξ ∈ R : µ ( γ, ξ ) < b } ⊂ { ξ ∈ R : µ ( − M, ξ ) < b } . Consequently, there exists a onstant f M > su h that { ξ ∈ R : µ j ( γ, ξ ) < b } ⊂ [ − f M , f M ] , ∀ b ≤ b , ∀ γ ∈ ] − M, M [ . Sin e the fun tions ξ µ j ( γ, ξ ) ( j ∈ N ) are regular, we introdu e onstants ( ξ j ( M )) j ∈ N ⊂ [ − f M , f M ] by µ j ( − M, ξ j ( M )) = min ξ ∈ [ − f M, f M ] µ j ( − M, ξ ) . We laim that(2.17) lim j →∞ µ j ( − M, ξ j ( M )) = ∞ . On e this laim is proved, we get the result of the lemma, sin e by mono-toni ity µ j ( γ, ξ ) ≥ µ ( − M, ξ ) ∀ γ ≥ − M , ∀ ξ ∈ R . Let us assume by ontradi tion that the laim (2.17) were false. Then wemay (cid:28)nd a onstant M > and a subsequen e ( j n ) su h that(2.18) µ j n ( − M, ξ j n ( M )) ≤ M , ∀ n ∈ N . Sin e − f M ≤ ξ j n ( M ) ≤ f M for all n , we get a subsequen e, denoted again by ξ j n ( M ) , su h that lim n →∞ ξ j n ( M ) = ζ ( M ) ∈ [ − f M , f M ] . It is quiet easy, by omparing the orresponding quadrati forms, to provethe existen e of a onstant
C > su h that, for all ε ∈ ]0 , [ and n ∈ N , wehave the estimate µ j n ( − M, ξ j n ( M )) ≥ (1 − ε ) µ j n ( − M, ζ ( M )) (2.19) − C (cid:0) ε + ε − | ξ j n ( M ) − ζ ( M ) | (cid:1) . EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 9We shall provide some details on erning the above estimate, but we wouldlike (cid:28)rst to a hieve the proof of the lemma. Noti e that, sin e the operator L [ − M, ζ ( M )] has ompa t resolvent, then lim n →∞ µ j n ( − M, ζ ( M )) = ∞ . Upon hoosing ε = | ξ j n ( M ) − ζ ( M ) | , we get from (2.19) that lim n →∞ µ j n ( − M, ξ j n ( M )) = ∞ , ontradi ting thus (2.18).We on lude by some wards on erning the proof of (2.19). Noti e that, fora normalized L -fun tion u , we have by Cau hy-S hwarz inequality: (cid:12)(cid:12)(cid:12)(cid:12)Z ∞ ( ζ − ξ j n ( M ))( t − ζ ) | u | d t (cid:12)(cid:12)(cid:12)(cid:12) ≤ | ζ − ξ j n ( M ) | × k ( t − ζ ) u k L ( R + ) ≤ ε k ( t − ζ ) u k L ( R + ) + ε − | ζ − ξ j n ( M ) | , for any ε > . On the other hand, writing ( t − ξ j n ( M )) = ( t − ζ ) + ( ξ − ξ j n ( M )) + 2( ζ − ξ j n ( M ))( t − ζ ) , we get the following omparison of the quadrati forms q [ − M, ξ j n ( M )]( u ) ≥ q [ − M, ζ ]( u ) − ε k ( t − ζ ) u k L ( R + ) − ε − | ζ − ξ j n ( M ) | , where q [ − M, · ] has been introdu ed in (2.1). Noti ing that for ε ∈ ]0 , [ , − M − ε ≥ − M , the appli ation of the min-max prin iple permits then to on- lude the desired bound (2.19). (cid:3) Remark 2.6. On e the asymptoti expansion (1.9) is proved, the formula(1.11) be omes a onsequen e of Lemma 2.1.The next lemma will play a ru ial role in establishing the main results ofthis paper.Lemma 2.7. The fun tion S : R × ] − ∞ , ∋ ( γ, b ) ∞ X j =1 |{ ξ ∈ R : µ j ( γ, ξ ) < b }| is lo ally uniformly ontinuous.Proof. Let b ∈ ]0 , and m > . It is su(cid:30) ient to establish,(2.20) sup | γ |≤ m, b ≤ b |S ( γ + τ, b + δ ) − S ( γ, b ) | ! → τ, δ ) → . Let τ = 1 − b > . By monotoni ity, for all τ, δ ∈ [ − τ , τ ] , the followingholds { ξ ∈ R : µ j ( γ + τ, ξ ) < b + δ } ⊂ { ξ ∈ R : µ ( − m − τ , ξ ) < b + τ } , ∀ j ∈ N . Therefore, we may (cid:28)nd a onstant
M > depending only on m and b su hthat(2.21) { ξ ∈ R : µ j ( γ + τ, ξ ) < b + δ } ⊂ [ − M, M ] , ∀ τ, δ ∈ [ − τ , τ ] , ∀ j ∈ N . ξ j ( M ) as in the proof of Lemma 2.5, i.e. ∀ ξ ∈ [ − M, M ] , ∀ τ ∈ [ − τ , τ ] , µ j ( γ + τ, ξ ) ≥ µ j ( − m − τ , ξ j ( M )) , we get as in (2.17): lim j →∞ µ j ( − m − τ , ξ j ( M )) = ∞ . Hen e, we may (cid:28)nd j ≥ depending only on m and b su h that µ j ( − m − , ξ j ( M )) ≥ b + 2 τ ∀ j ≥ j , and onsequently, for | τ | ≤ τ , | δ | ≤ τ , we get ∞ X j =1 |{ ξ ∈ R : µ j ( γ + τ, ξ ) < b + δ }| = j X j =1 |{ ξ ∈ R : µ j ( γ + τ, ξ ) < b + δ }| . Therefore, we deal only with a (cid:28)nite sum of j terms, j being indepen-dent from τ , δ , γ and b . So given k ∈ { , · · · , j } and setting S k ( γ, b ) = |{ ξ ∈ R : µ k ( γ, ξ ) < b }| , it is su(cid:30) ient to show that(2.22) lim ( τ,δ ) → | τ | + | δ |≤ τ sup | γ |≤ m, b ≤ b |S k ( γ + τ, b + δ ) − S k ( γ, b ) | ! = 0 . The above formula is only a dire t onsequen e of Lemmas 2.3 and 2.4. (cid:3) P α,γh, Ω S = − ( h ∇ − iA ) , where Ω S is the half- ylinder Ω S =]0 , S [ × ]0 , ∞ [ ,S > and γ ∈ R are onstants. The magneti potential A is taken in the anoni al way(2.23) A ( s, t ) = ( − t, , ∀ ( s, t ) ∈ [0 , S ] × [0 , ∞ [ . Fun tions in the domain of P α,γh, Ω S satisfy the periodi onditions u (0 , · ) = u ( S, · ) on R + , and the de Gennes boundary ondition at t = 0 , h ∂ t u (cid:12)(cid:12) t =0 = h α γu (cid:12)(cid:12) t =0 . We shall from now on use the following notation. For a self-adjoint oper-ator T and a real number λ < inf σ ess ( T ) , we denote by N ( λ, T ) the numberof eigenvalues of T ( ounted with multipli ity) in luded in ] − ∞ , λ ] .Lemma 2.8. For ea h M > , there exists a onstant C > su h that, forall b ∈ ] − ∞ , , α ≥ , S > , γ ∈ [ − M, M ] and h ∈ ]0 , , we have, (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:16) b h, P α,γh, Ω S (cid:17) − S h − / π ∞ X j =1 (cid:12)(cid:12)(cid:12) { ξ ∈ R : µ j ( h α − / γ, ξ ) ≤ b } (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C. EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 11Proof. By separation of variables ( f. [17℄) and a s aling we may de ompose P α,γh, Ω S as a dire t sum: M n ∈ Z h (cid:18) − d d τ + (2 πnh / S − + τ ) (cid:19) in M n ∈ Z L ( R + ) , with the boundary ondition u ′ (0) = h α − / γ u (0) at τ = 0 .Consequently we obtain:(2.24) σ (cid:16) P α,γh, Ω S (cid:17) = [ j ∈ N n hµ j ( h α − / γ, πh / S − n ) : n ∈ N o , and ea h eigenvalue is of multipli ity .Thus, putting f j ( ξ ) = { ξ ∈ R : µ j ( h α − / γ,ξ ) and γ ∈ R are onstants. The magneti potential A was de(cid:28)nedin (2.23).Fun tions in the domain of P α,γh, Ω S,T satisfy the de Gennes ondition h∂ t u = h α γ u at t = 0 and Diri hlet ondition on the other sides of the boundary.The next lemma gives a omparison between the ounting fun tion of P α,γh, Ω S,T and that of P α,γh, Ω S .Lemma 2.9. There exists a onstant c > su h that, ∀ S > , ∀ T > , ∀ γ ∈ R , ∀ δ ∈ ]0 , S/ , ∀ b ∈ ] − ∞ , , we have, N (cid:16) b h − c h ( δ − + T − ) , P α,γh, Ω S − δ ) (cid:17) ≤ N (cid:16) b h, P α,γh, Ω S,T (cid:17) ≤ N (cid:16) b h, P α,γh, Ω S (cid:17) . Proof. Sin e the extension by zero of a fun tion in the form domain of P α,γh, Ω S,T is in luded in that of P α,γh, Ω S , and the values of the quadrati forms oin ide for su h a fun tion, we get the upper bound of the lemma by asimple appli ation of the variational prin iple.We turn now to the lower bound. The argument is like the one used in [2, 5℄but we explain it be ause it illustrates in a simple ase the arguments of thispaper.Let us introdu e two partitions of unity ( ϕ δi ) and ( ψ Tj ) su h that: (cid:16) ϕ δ (cid:17) + (cid:16) ϕ δ (cid:17) = 1 in [0 , S − δ )] , (cid:0) ψ T (cid:1) + (cid:0) ψ T (cid:1) = 1 in R + , supp ϕ δ ⊂ [0 , S ]supp ϕ δ ⊂ [ S − δ, S − δ )] X i =1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:16) ϕ δi (cid:17) ′ (cid:12)(cid:12)(cid:12)(cid:12) ≤ c δ − & supp ψ T ⊂ [ T / , ∞ [supp ψ T ⊂ [0 , T ] X i =0 (cid:12)(cid:12)(cid:12)(cid:0) ψ Ti (cid:1) ′ (cid:12)(cid:12)(cid:12) ≤ c T − , where c > is a onstant independent from S, T and δ .Upon putting χ δ,Ti ( s, t ) = ϕ δi ( s ) ψ T ( t ) ( i = 1 , , χ δ,T ( s, t ) = ψ T ( t ) , we get a partition of unity of Ω S − δ ) =]0 , S − δ )[ × R + .Let us take a fun tion u in the form domain of P α,γh, Ω S − δ ) . Then, by the IMSde omposition formula: Z Ω S − δ ) | ( h ∇ − iA ) u | d x = X i =0 Z Ω S − δ ) | ( h ∇ − iA ) χ δ,Ti u | d x − h X i =0 (cid:13)(cid:13)(cid:13) |∇ χ δ,Ti | u (cid:13)(cid:13)(cid:13) L (Ω S,T ) ≥ X i =0 Z Ω S − δ ) | ( h ∇ − iA ) χ δ,Ti u | d x − c ( δ − + T − ) h k u k L (Ω S,T ) . Noti e that χ δ,T u is in the form domain of P α,γh, Ω S,T and χ δ,T u is in thatof P α,γh, ] S − δ, S − δ )[ × ]0 ,T [ (this last operator, thanks to translational invarian ewith respe t to s , is unitary equivalent to P α,γh, Ω S,T ). Also, χ δ,T u is in theEYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 13form domain of the P Dh, Ω S − δ ) , the Diri hlet realization of − ( h ∇ − iA ) in Ω S − δ ) .Sin e the form domain of P α,γh, Ω S − δ ) an be viewed in1 F D (cid:16) P α,γh, Ω S,T (cid:17) ⊕ F D (cid:16) P α,γh, ] S − δ, S − δ )[ × ]0 ,T [ (cid:17) ⊕ F D (cid:16) P Dh, Ω S − δ ) (cid:17) via the isometry u (cid:16) χ δ,T u, χ δ,T u, χ δ,T u (cid:17) , we get upon applying the vari-ational prin iple (see [17, Se tion XII.15℄), N (cid:16) b h − c h ( δ − + T − ) , P α,γh, Ω S − δ ) (cid:17) (2.25) ≤ N (cid:16) b h, P α,γh, Ω S − δ ) (cid:17) + N (cid:16) b h, P Dh, Ω S − δ ) (cid:17) . Noti e that, by (1.4), the operator P Dh, Ω S − δ ) has no spe trum below b h when b < . Hen e, N (cid:16) b h, P Dh, Ω S − δ ) (cid:17) = 0 . Coming ba k to (2.25), we get the lower bound stated in the lemma. (cid:3)
3. Proof of Theorem 1.1We ome ba k to the ase of a general smooth domain Ω whose boundaryis ompa t. We introdu e the following quadrati forms: q α,γh, Ω ( u ) = Z Ω | ( h ∇ − iA ) u | d x + h α Z ∂ Ω γ ( s ) | u ( s ) | d s, (3.1) q h, Ω ( u ) = Z Ω | ( h ∇ − iA ) u | d x, (3.2)de(cid:28)ned for fun tions in the magneti Sobolev spa e:(3.3) H h,A (Ω) = { u ∈ L (Ω) : ( h ∇ − iA ) u ∈ L (Ω) } . Here, we re all that A ∈ C (Ω) is su h that curl A = 1 in Ω . We shall re all in the appendix a standard oordinate transformation validin a su(cid:30) iently thin neighborhood of the boundary: Φ t : Ω( t ) ∋ x ( s ( x ) , t ( x )) ∈ [0 , | ∂ Ω | [ × [0 , t ] , where for t > , Ω( t ) is the tubular neighborhood of ∂ Ω : Ω( t ) = { x ∈ Ω : dist( x, ∂ Ω) < t } . Let us mention that t ( x ) = dist( x, ∂ Ω) measures the distan e to the bound-ary and s ( x ) measures the urvilinear distan e in ∂ Ω .Using the oordinate transformation Φ t , we asso iate to any fun tion u ∈ L (Ω) , a fun tion e u de(cid:28)ned in [0 , | ∂ Ω | [ × [0 , t ] by,(3.4) e u ( s, t ) = u (Φ − t ( s, t )) . A , F D ( A ) denotes its form domain.4 AYMAN KACHMARThe next lemma states a standard approximation of the quadrati form q α,γh, Ω ( u ) by the anoni al one in the half-plane, provided that the fun tion u is supported near the boundary.Lemma 3.1. There exists a onstant C > , and for all S ∈ [0 , | ∂ Ω | [ , S ∈ ] S , | ∂ Ω | [ , there exists a fun tion φ ∈ C ([ S , S ] × [0 , t ]; R ) su h that, for all e S ∈ [ S , S ] , T ∈ ]0 , t [ , ε ∈ [ CT, Ct ] , and for all u ∈ H h,A (Ω) satisfying supp e u ⊂ [ S , S ] × [0 , T ] , one has the following estimate, (1 − ε ) q α, b γ h, Ω (cid:16) e iφ/h e u (cid:17) − Cε − (cid:0) (( S − S ) + T ) + h (cid:1) k e u k L (Ω ) ≤ q α,γh, Ω ( u ) ≤ (1 + ε ) q α, e γ h, Ω (cid:16) e iφ/h e u (cid:17) + Cε − (cid:0) (( S − S ) + T ) + h (cid:1) k e u k L (Ω ) . Here Ω = [ S , S ] × [0 , T ] , e γ = γ ( e S ) + C ( S − S )1 + ε , b γ = γ ( e S ) − C ( S − S )1 − ε ,and the fun tion e u is asso iated to u by (3.4).Proof. Noti e that when γ ≡ , the result follows from [5, Lemma 3.5℄,whi h reads expli itly in the form: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) q α, h, Ω ( u ) − Z [ S ,S ] × [0 ,T ] | ( h ∇ − iA ) e iφ/h e u | d s d t (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ε Z [ S ,S ] × [0 ,T ] | ( h ∇ − iA ) e iφ/h e u | d s d t + Cε − (cid:0) ( T + ( S − S ) ) + h (cid:1) k e u k L . Sin e e u restri ted to the boundary is supported in [ S , S ] , we get as animmediate onsequen e the following two-sided estimate for non-zero γ : (1 + ε ) − q α,γh, Ω ( u ) ≤ Z [ S ,S ] × [0 ,T ] | ( h ∇ − iA ) e iφ/h e u | d s d t + h α ε Z [ S ,S ] γ ( s ) | e u ( s, | d s + C (1 + ε ) − ε − (cid:0) ( T + ( S − S ) ) + h (cid:1) k e u k L , (3.5)and (1 − ε ) − q α,γh, Ω ( u ) ≥ Z [ S ,S ] × [0 ,T ] | ( h ∇ − iA ) e iφ/h e u | d s d t + h α − ε Z [ S ,S ] γ ( s ) | e u ( s, | d s − C (1 − ε ) − ε − (cid:0) ( T + ( S − S ) ) + h (cid:1) k e u k L . (3.6)EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 15The idea now is to approximate γ by a onstant in a simple manner withoutneeding an estimate of the boundary integral. A tually, Taylor's formulaapplied to the fun tion γ near e S leads to the estimate | γ ( s ) − γ ( e S ) | ≤ C ( S − S ) , ∀ s ∈ [ S , S ] , where the onstant C > is possibly repla ed by a larger one.Having this estimate in hand, we get: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z [ S ,S ] γ ( s ) | e u ( s, | d s − γ ( e S ) Z [ S ,S ] | e u ( s, | d s (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C ( S − S ) Z [ S ,S ] | e u ( s, | d s . Re alling the de(cid:28)nition of e γ and b γ in Lemma 3.1 (they a tually depend on ε , S , S and hen e a ount to all possible errors), we infer dire tly from theprevious estimate, (1 − ε ) b γ Z [ S ,S ] | e u ( s, | d s ≤ Z [ S ,S ] γ ( s ) | e u ( s, | d s ≤ (1 + ε ) e γ Z [ S ,S ] | e u ( s, | d s . (3.7)Substituting the lower and upper bound of (3.7) in (3.5) and (3.6) respe -tively, and re alling the hypothesis that e u ( s, is supported in [ S , S ] , weobtain the desired estimates of the lemma. (cid:3) We shall divide a thin neighborhood of ∂ Ω into many small sub-domains,and in ea h sub-domain, we shall apply Lemma 3.1 to approximate the qua-drati form. This will yield a two-sided estimate of N ( λ, P α,γh, Ω ) in terms ofthe spe tral ounting fun tions of model operators on half- ylinders.Let us put N = [ h − / ] , the greatest positive integer below h − / .Let S = | ∂ Ω | N , s n = nS, n ∈ { , , . . . , N } , and we emphasize that these quantities depend on h . We put further Ω S =]0 , S [ × R . With these notations, the proof of Theorem 1.1 is given by the followingproposition.Proposition 3.2. Let b ∈ ] − ∞ , . There exist onstants C, h > su hthat for all h ∈ ]0 , h ] , δ ∈ ]0 , S/ , e S n ∈ [ s n − , s n ] , n ∈ { , , . . . , N } , N X n =1 N (cid:16) hb − Ch δ − , P α, e γ n h, Ω S − δ ) (cid:17) ≤ N (cid:16) hb , P α,γh, Ω (cid:17) ≤ N X n =1 N (cid:16) hb + Ch δ − , P α, b γ n h, Ω S +2 δ (cid:17) . Here e γ n = γ ( e S n ) + CS h / and b γ n = γ ( e S n ) − CS − h / .Before proving Proposition 3.2, let us see how it serves for obtaining the on lusion of Theorem 1.1.Proof of Theorem 1.1. We keep the notation introdu ed for the statementof Proposition 3.2. The proof is in two steps.Step 1.Let us establish the asymptoti formula (as h → ): h / N (cid:16) b h, P α,γh, Ω (cid:17) = (3.8) π Z ∂ Ω ∞ X j =1 (cid:12)(cid:12) { ξ ∈ R : µ j ( h α − / γ ( s ) , ξ ) < b } (cid:12)(cid:12) d s + o (1) , where b ∈ ] − ∞ , .From the lower bound in Proposition 3.2, we get upon applying Lemma 2.8,a onstant e C > su h that h / N (cid:16) b h, P α,γh, Ω (cid:17) ≥ π N X n =1 ( S − δ ) ∞ X j =1 (cid:12)(cid:12)(cid:12) { ξ ∈ R : µ j ( h α − / e γ n , ξ ) < b − Chδ − } (cid:12)(cid:12)(cid:12) − e Ch / . Sin e α ≥ / and ∂ Ω is bounded, it is a result of Lemmas 2.5 and 2.7 thatthere exist onstants C > and h > together with a fun tion ]0 , h ] ∋ h ǫ ( h ) tending to as h → su h that, for all h ∈ ]0 , h ] and n ∈ { , , . . . , N } ,we have (provided that hδ − is su(cid:30) iently small), S (cid:16) h α − / e γ n , b (cid:17) ≤ C , (cid:12)(cid:12)(cid:12) S (cid:16) h α − / e γ n , b − Chδ − (cid:17) − S (cid:16) h α − / γ n , b (cid:17)(cid:12)(cid:12)(cid:12) ≤ ǫ ( h ) , where the fun tion S is introdu ed in 2.7, and γ n = (1 + h / ) e γ n − CS = γ ( e S n ) .Re alling that S = | ∂ Ω | /N and N = [ h − / ] , we get upon hoosing δ = 1 /N , h / N (cid:16) b h, P α,γh, Ω (cid:17) ≥ π N X n =1 ∞ X j =1 S (cid:12)(cid:12)(cid:12) { ξ ∈ R : µ j ( h α − / γ n , ξ ) < b } (cid:12)(cid:12)(cid:12) − ǫ ( h ) − Ch / , EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 17where the leading order term on the right hand side is a Riemann sum. Weget then the following lower bound, h / N (cid:16) b h, P α,γh, Ω (cid:17) ≥ π Z ∂ Ω ∞ X j =1 (cid:12)(cid:12)(cid:12) { ξ ∈ R : µ j ( h α − / γ ( s ) , ξ ) < b } (cid:12)(cid:12)(cid:12) d s + o (1) . This is the lower bound in (3.8). In a similar manner we obtain an upperbound.Step 2.If α = 1 / , the asymptoti formula (3.8) is just the on lusion of Theo-rem 1.1.We turn to the ase when α > / . Again, it results from Lemma 2.7 theexisten e of a onstant h and a fun tion ]0 , h ] ∋ h ǫ ( h ) tending to as h → su h that for all h ∈ ]0 , h ] and s ∈ ∂ Ω , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) S (cid:16) h α − / γ ( s ) , b (cid:17) − ∞ X j =1 |{ ξ ∈ R : µ j ( ξ ) < b | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ǫ ( h ) , where µ j ( ξ ) = µ j (0 , ξ ) . Moreover, by Lemma 2.1, it holds that ∞ X j =2 |{ ξ ∈ R : µ j ( ξ ) < b }| = 0 . Now we an infer from (3.8) the asymptoti formula announ ed in (1.8). (cid:3)
Proof of Proposition 3.2. Let us establish the lower bound. Let P α,γN,h, Ω bethe restri tion of the operator P α,γh, Ω for fun tions u in D ( P α,γh, Ω ) that vanisheson the set { x ∈ Ω : t ( x ) ≥ T } ∪ N [ n =1 { x ∈ Ω : 0 ≤ t ( x ) ≤ T, s ( x ) = s n } , where T > is to be spe i(cid:28)ed later. The important remark is that the spe -trum of P α,γh, Ω is below that of P α,γN,h, Ω .Let us take a fun tion u in the form of domain of P α,γN,h, Ω . Applying Lemma 3.1with T = h / and ε = h / , we get the estimate, q α,γh, Ω ( u ) ≤ (1 + h / ) N X n =1 (cid:18) q α, e γ n h, Ω n (cid:16) e − iφ n /h e u (cid:17) + Ch / (cid:13)(cid:13)(cid:13) e − iφ n /h e u (cid:13)(cid:13)(cid:13) L (Ω n ) (cid:19) , where Ω n =] s n − , s n [ × ]0 , T [ and e γ n = γ ( e S n )+ CS h / . Then, by the variationalprin iple, we obtain (re all that the spe trum of P α,γh, Ω is below that of P α,γN,h, Ω ), N (cid:16) λ, P α,γh, Ω (cid:17) ≥ N X n =1 N λ − Ch / h / , P α, e γ n h, Ω S,T ! , where Ω S,T =]0 , S [ × ]0 , T [ . Applying Lemma 2.9, this is su(cid:30) ient to on- lude the lower bound announ ed in Proposition 3.2.The upper bound is derived by introdu ing a partition of unity atta hed8 AYMAN KACHMARto the sub-domains Ω n and by using the IMS de omposition formula. Theanalysis is similar to that presented for the lower bound above and also tothat in the proof of Lemma 2.9, so we omit the proof. For the details, werefer to [5, Proposition 3.6℄. (cid:3)
4. Curvature effe ts4.1. A family of ordinary di(cid:27)erential operators on a weighted L spa e.A (cid:28)ner approximation of the quadrati form (3.1) leads to the analysis of afamily of ordinary di(cid:27)erential operators on a weighted L spa e that takesinto a ount the urvature e(cid:27)e ts of the boundary. We shall re all in thisse tion the main results for the lowest eigenvalue problem on erning thisfamily of operators. These results were obtained in [7℄ for the Neumann prob-lem and then generalized in [10℄ for situations involving de Gennes' boundary ondition.Let us introdu e, for te hni al reasons that will be lari(cid:28)ed later, a positiveparameter δ ∈ ] , [ . Let us also onsider parameters h > and β ∈ R su hthat | β | h δ < . We de(cid:28)ne the family of quadrati forms (indexed by ξ ∈ R ) q α,ηh,β,ξ ( u ) = Z h δ − / " | u ′ ( t ) | + (1 + 2 βh / t ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) t − ξ − βh / t (cid:19) u ( t ) (cid:12)(cid:12)(cid:12)(cid:12) × (1 − βh / t ) dt + h α − / η | u (0) | , (4.1)de(cid:28)ned for fun tions u in the spa e :(4.2) D (cid:16) q α,ηh,β,ξ (cid:17) = n u ∈ H (cid:16) ]0 , h δ − / [ (cid:17) : u (cid:16) h δ − / (cid:17) = 0 o . Let us denote by H α,ηh,β,ξ the self-adjoint realization asso iated to the quadrati form (4.1) by Friedri h's' theorem. Let us denote also by (cid:16) µ j (cid:16) H α,ηh,β,ξ (cid:17)(cid:17) j thein reasing sequen e of eigenvalues of H α,ηh,β,ξ .For ea h α ≥ and η ∈ R we introdu e the positive numbers : d (cid:18) , η (cid:19) = ξ ( η )Θ ′ ( η ) , d ( α, η ) = ξ Θ ′ (0) (cid:18) α > (cid:19) ,d (cid:18) , η (cid:19) = 13 ( ηξ ( η ) + 1) Θ ′ ( η ) , d ( α, η ) = 13 Θ ′ (0) (cid:18) α > (cid:19) . (4.3)The result on erning the lowest eigenvalue of H α,ηh,β,ξ in the next theoremhas been proved in [10℄.Theorem 4.1. Suppose that δ ∈ ] , [ and α ≥ . Let e η = h α − / η , ρ = (cid:26) δ − , if α = min( δ − , α − ) , if α > , and 0 < ρ < ρ . EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 19For ea h
M > and ζ > , there exists a onstant ζ > , and for ea h ζ ≥ ζ , there exist positive onstants C, h and a fun tion ]0 , h ] ∋ h ǫ ( h ) ∈ R + with lim h → ǫ ( h ) = 0 su h that, ∀ η, β ∈ ] − M, M [ , ∀ h ∈ ]0 , h ] , the following assertions hold : • If | ξ − ξ ( e η ) | ≤ ζh ρ , then (cid:12)(cid:12)(cid:12) µ (cid:16) H α,η,Dh,β,ξ (cid:17) − n Θ ( e η ) + d ( α, η )( ξ − ξ ( e η )) − d ( α, η ) βh / o(cid:12)(cid:12)(cid:12) (4.4) ≤ C h h / | ξ − ξ ( e η ) | + h δ +1 / + h / ǫ ( h ) i , and(4.5) µ (cid:16) H α,η,Dh,β,ξ (cid:17) ≥ Θ ( e η ) + ζ h ρ . • If | ξ − ξ ( e η ) | ≥ ζh ρ , then(4.6) µ (cid:16) H α,η,Dh,β,ξ (cid:17) ≥ Θ ( e η ) + ζ h ρ . Here, the parameters d ( α, η ) and d ( α, η ) has been introdu ed in (4.3).Proof. The existen e of ζ so that the lower bound (4.6) holds for | ξ − ξ ( e η ) | ≥ ζ h ρ has been established in [10, Lemma V.8℄. Now, for ζ ≥ ζ ,(4.6) obviously holds. Under the hypothesis | ξ − ξ ( e η ) | ≤ ζh ρ , the existen eof the onstants C , h and the estimate (4.4) have been established in [10,Lemma V.8 & V.9℄. So we only need to establish (4.5).We start with the ase α = . It results from the min-max prin iple (see[10℄ or [14, Lemma 4.2.1℄ for details), (cid:12)(cid:12)(cid:12) µ (cid:16) H α,η,Dh,β,ξ (cid:17) − µ (cid:16) H α,η,Dh, ,ξ (cid:17)(cid:12)(cid:12)(cid:12) ≤ e Ch δ − (cid:16) µ (cid:16) H α,η,Dh, ,ξ (cid:17)(cid:17) , where the onstant e C depends only on M .It results again from the min-max prin iple, µ (cid:16) H α,η,Dh, ,ξ (cid:17) ≥ µ ( L [ η, ξ ]) , where L [ η, ξ ] is the operator introdu ed in (2.3).We get then the following lower bound, µ (cid:16) H α,η,Dh,β,ξ (cid:17) ≥ (cid:16) − e Ch δ − (cid:17) µ ( L [ η, ξ ]) − e Ch δ − ≥ (cid:16) − e Ch δ − (cid:17) Θ ( η ) − e Ch δ − ≥ Θ( η ) + ζ h ρ . Here, we re all the de(cid:28)nition of Θ ( η ) in (2.15). Let us also point out thatthe (cid:28)nal on lusion above follows by Lemma 2.2 upon taking h ∈ ]0 , h ] with h hosen so small that ζ h ρ + 2 e Ch δ − (Θ ( η ) + 1) < (Θ ( η ) − Θ( η )) .When α > , the result follows from the above argument upon using the ontinuity of our spe tral fun tions with respe t to small perturbations. (cid:3) h > , δ > , S > β ∈ R s . t . | β | h δ < . We denote by e L α,ηh,β,S the self adjoint operator in L (cid:16) ]0 , S [ × ]0 , h δ [ ; (1 − βt ) d s d t (cid:17) asso iated with the quadrati form e Q α,ηh,β,S ( u ) = Z S Z h δ | h∂ t u | + (1 + 2 βt ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) h∂ s + t − β t (cid:19) u (cid:12)(cid:12)(cid:12)(cid:12) ! × (1 − βt ) d s d t + h α η Z S | u ( s, | d s, de(cid:28)ned for fun tions u in the form domain D (cid:16) e Q α,ηh,β,S (cid:17) = { u ∈ H (cid:16) ]0 , S [ × ]0 , h δ [ (cid:17) : u ( · , h δ ) = 0 , u (0 , · ) = u ( S, · ) } . We re all again the notation that for a self adjoint operator A and a num-ber λ < inf σ ess ( A ) , we denote by N ( λ, A ) the number of eigenvalues of A ( ounted with multipli ity) below λ .Proposition 4.2. With the notation and hypotheses of Theorem 4.1, let ζ > , e h > and λ = λ ( h ) su h that(4.7) | λ − Θ( e η ) | < ζ h ρ , ∀ h ∈ ]0 , e h ] . Then, for ea h
M > , there exist onstants C > and h > and a fun tion ]0 , h ] ∋ h ǫ ( h ) ∈ R + with lim h → ǫ ( h ) = 0 su h that, for all h ∈ ]0 , h ] and S, η, β ∈ ] − M, M [ , (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N (cid:16) hλ, e L α,ηh,β,S (cid:17) − h − / Sπ p d ( α, η ) q(cid:0) d ( α, η ) β + h − / [ λ − Θ( e η )] (cid:1) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ h − / Sπ p d ( α, η ) q(cid:0) d ( α, η ) β + h − / [ λ − Θ( e η )] (cid:1) + ǫ ( h ) . (4.8)Proof. By separation of variables and by performing the s aling τ = h − / t ,we de ompose e L α,ηh,β,S as a dire t sum, M n ∈ Z h H α,ηh,β, πnh / S − in M n ∈ N L (cid:16) ]0 , h δ − / [; (1 − βh / t ) dt (cid:17) . Consequently, by Theorem 4.1 and the hypothesis λ − Θ( e η ) < ζ h ρ , weobtain, N (cid:16) hλ, e L α,ηh,β,S (cid:17) = Card (cid:16) { n ∈ Z ; µ (cid:16) H α,ηh,β, πnh / S − (cid:17) ≤ λ } (cid:17) . Again, Theorem 4.1 yields the existen e of a positive onstant ζ (that wemay hoose su(cid:30) iently large as we wish) and a fun tion h e ǫ ( h ) su h that,EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 21upon de(cid:28)ning the subsets S ± = (cid:26) n ∈ Z : (cid:12)(cid:12)(cid:12) πnh / S − − ξ ( e η ) (cid:12)(cid:12)(cid:12) ≤ ζh ρ , Θ( e η ) + d ( α, η ) (cid:16) πnh / S − − ξ ( e η ) (cid:17) − d ( α, η ) βh / ± h / e ǫ ( h ) < λ (cid:27) , one gets the in lusion,(4.9) S + ⊂ (cid:26) n ∈ Z ; µ (cid:16) H α,ηh,β, πnh / S − (cid:17) ≤ λ (cid:27) ⊂ S − . Therefore, we dedu e that
Card S + ≤ N (cid:16) hλ, e L α,ηh,β,S (cid:17) ≤ Card S − . On the other hand, thanks to (4.7), we may hoose ζ > su(cid:30) iently largeso that Θ( e η ) + d ( α, η ) (cid:16) πnh / S − − ξ ( e η ) (cid:17) − d ( α, η ) βh / ± h / e ǫ ( h ) < λ = ⇒ (cid:12)(cid:12)(cid:12) πnh / S − − ξ ( e η ) (cid:12)(cid:12)(cid:12) ≤ ζh ρ . With this hoi e of ζ , one an rewrite S ± in the following equivalent form S ± = (cid:26) n ∈ Z : d ( α, η ) (cid:16) πnS − − h − / ξ ( e η ) (cid:17) ≤ h − / (cid:16) d ( α, η ) β + h − / [ λ − Θ( e η )] ± e ǫ ( h ) (cid:17) + (cid:27) , from whi h one obtains a positive fun tion ǫ ( h ) ≪ su h that (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Card S ± − h − / Sπ p d ( α, η ) q(cid:0) d ( α, η ) β + h − / [ λ − Θ( e η )] (cid:1) + (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ h − / Sπ p d ( α, η ) q(cid:0) d ( α, η ) β + h − / [ λ − Θ( e η )] (cid:1) + ǫ ( h ) , when S varies in a bounded interval ] − M, M [ . This (cid:28)nishes the proof ofthe proposition. (cid:3) β , η , α ≥ , h > , δ ∈ ] , [ and S . Let us onsider the operator L α,ηh,β,S obtained from e L α,ηh,β,S by imposing additional Diri hlet boundary onditionsat s ∈ { , S } , i.e. L α,ηh,β,η : D ( L α,ηh,β,S ) ∋ u e L α,ηh,β,S u with D ( L α,ηh,β,S ) = { u ∈ D ( e L α,ηh,β,η ) : u (0 , · ) = u ( S, · ) = 0 } . L α,ηh,β,S is the self adjoint operator in L (cid:0) ]0 , S [ × ]0 , h δ [; (1 − βh δ ) d s d t (cid:1) asso iated with the quadrati form, Q α,ηh,β,S ( u ) = Z S Z h δ | h∂ t u | + (1 + 2 βt ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18) h∂ s + t − β t (cid:19) u (cid:12)(cid:12)(cid:12)(cid:12) ! × (1 − βt ) d s d t + h α η Z S | u ( s, | d s, de(cid:28)ned for fun tions u in the form domain D (cid:16) Q α,ηh,β,S (cid:17) = { u ∈ H (cid:16) ]0 , S [ × ]0 , h δ [ (cid:17) : u ( · , h δ ) = u (0 , · ) = u ( S, · ) = 0 } . Using the same reasoning as that for the proof of Lemma 2.9, we get in thenext lemma an estimate of the spe tral ounting fun tion of the operator L α,ηh,β,S .Lemma 4.3. For ea h M > , there exist onstants C > and h > su hthat, for all β, η, S ∈ ] − M, M [ , ε ∈ ]0 , S/ , λ ∈ R , h ∈ ]0 , h [ , one has the estimate : N (cid:16) λ − Cε − h , e L α,ηh,β, S − ε ) (cid:17) ≤ N (cid:16) λ, L α,ηh,β,S (cid:17) ≤ N (cid:16) λ, e L α,ηh,β,S (cid:17) . Ω whose boundary is ompa t.An energy estimate.Let us re all the notation that κ r denotes the s alar urvature of the bound-ary ∂ Ω . As was (cid:28)rst noti ed in [7℄, sin e the magneti (cid:28)eld is onstant, thequadrati form (3.2) an be estimated with a high pre ision by showing thein(cid:29)uen e of the s alar urvature. This is a tually the ontent of the nextlemma, whi h we quote from [5, Lemma 4.7℄. Before stating the estimate,let us re all that to a given fun tion u ∈ H (Ω) , we asso iate by means ofboundary oordinates a fun tion e u , see (3.4).Lemma 4.4. Let δ ∈ ] , [ . There exists a onstant C > , and for all S ∈ [0 , | ∂ Ω | [ , e S ∈ ]0 , S [ , there exists a fun tion φ ∈ C ([0 , S ] × [0 , Ch δ ]; R ) su h that, for all ε ∈ [ Ch δ , , and for all u ∈ H h,A (Ω) satisfying supp e u ⊂ [0 , S ] × [0 , Ch δ ] , one has the following estimate, (cid:12)(cid:12)(cid:12) q h, Ω ( u ) − Q h, e κ,S (cid:16) e iφ/h e u (cid:17)(cid:12)(cid:12)(cid:12) ≤ C (cid:18) h δ S Q h, e κ,S (cid:16) e iφ/h e u (cid:17) + ( h δ + Sh δ ) (cid:13)(cid:13)(cid:13) e iφ/h e u (cid:13)(cid:13)(cid:13) L (cid:19) . Here e κ = κ r ( e S ) , and the fun tion e u is asso iated to u by (3.4).EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 23Let us mention that we omit α and η from the notation in Subse tion 4.3when η ≡ .Estimates of the ounting fun tion.As in Se tion 3, we introdu e a partition of a thin neighborhood of ∂ Ω :Given N ∈ N (that will be hosen later as a fun tion of h ) su h that N = N ( h ) ≫ h → , we put S = | ∂ Ω | N , s n = nS, κ n = κ r ( s n ) , n ∈ { , , · · · , N } . By this way, we are able to estimate the spe tral ounting fun tion of theoperator P α,γh, Ω by those of the operators L α,γh,κ n ,S .Proposition 4.5. Let δ ∈ ] , [ , γ = min x ∈ ∂ Ω γ ( x ) and λ = λ ( h ) su h that(4.10) (cid:12)(cid:12)(cid:12) λ − Θ (cid:16) h α − / γ (cid:17)(cid:12)(cid:12)(cid:12) ≪ h → . There exist onstants
C > and h > su h that, for all h ∈ ]0 , h [ , ε ∈ ]0 , S/ , e S n ∈ [ s n − , s n ] , n ∈ { , · · · , N } , one has the estimate N X n =1 N (cid:16) hλ − C ( Sh δ + ε − h ) , e L α, e γ n h, e κ n , S + ε ) (cid:17) ≤ N (cid:16) hλ, P α,γh, Ω (cid:17) ≤ N X n =1 N (cid:16) hλ + C ( Sh δ + ε − h ) , e L α, b γ n h, e κ n ,S +2 ε (cid:17) . Here e κ n = κ r ( e S n ) , e γ n = γ ( e S n ) + CS Ch δ S , b γ n = γ ( e S n ) − CS − Ch δ S .
The proof is exa tly as that of Proposition 3.2 and we omit it. Let us onlymention the main points. Thanks to a partition of unity asso iated withthe intervals [ s n − , s n ] and the variational prin iple, the result follows fromlo al estimates of the quadrati form. For this sake, the pro edure onsistsof the implementation of Lemma 4.4, bounding γ ( s ) from above and belowrespe tively by (1 + Ch δ S ) e γ n and (1 − Ch δ S ) b γ n in ea h [ s n − , s n ] (thusgetting errors of order S ) and (cid:28)nally of the appli ation of Lemma 4.3.An asymptoti formula of the ounting fun tion.Let us take onstants ζ > , c > , δ ∈ ] , [ , and let us re all that weintrodu e a parameter ρ su h that : < ρ < δ −
14 if α = 12 , < ρ < min (cid:18) δ − , α − (cid:19) if α > . We take e h > and λ = λ ( h ) su h that,(4.11) c h / ≤ (cid:12)(cid:12)(cid:12) λ − Θ (cid:16) h α − / γ (cid:17)(cid:12)(cid:12)(cid:12) < ζ h ρ , ∀ h ∈ ]0 , e h ] . γ = min x ∈ ∂ Ω γ ( x ) and γ ∈ C ∞ ( ∂ Ω; R ) is the fun tion in de Gennes' boundary ondition thatwe impose on fun tions in the domain of the operator P α,γh, Ω , see (1.2).Theorem 4.6. Let δ ∈ ] , [ . With the above notations, we have the fol-lowing asymptoti formula as h → , N (cid:16) hλ, P α,γh, Ω (cid:17) = Z ∂ Ω h − / π p d ( α, γ ( s )) × s(cid:18) d ( α, γ ( s )) κ r ( s ) + h − / (cid:2) λ − Θ (cid:0) h α − / γ ( s ) (cid:1)(cid:3) (cid:19) + d s ! (cid:0) o (1) (cid:1) . Here, for a given ( α, η ) ∈ R × R , the parameters d ( α, η ) and d ( α, η ) havebeen introdu ed in (4.3).Proof. The proof is similar to that of the asymptoti formula (3.8). Therewe have shown how to establish a lower bound, so we show here how toestablish an upper bound.For ea h n ∈ { , · · · , N } , let us introdu e the fun tion (see Lemma 4.3) : f n ( δ, S, ε ) = d ( α, b γ n ) e κ n + h − / h λ + C ( Sh δ − + ε − h ) − Θ (cid:16) h α − / b γ n (cid:17)i . Here b γ n and e κ n are given by Proposition 4.5. Then, ombining Proposi-tions 4.2 and 4.5, we get,(4.12) N (cid:16) hλ, P α,γh, Ω (cid:17) ≤ N X n =1 h − / ( S + 2 ε ) π p d ( α, b γ n ) p [ f n ( δ, S, ε )] + ! (cid:0) ǫ ( h ) (cid:1) , where the fun tion ǫ is independent of N and satis(cid:28)es lim h → ǫ ( h ) = 0 . We re all also that S = | ∂ Ω | N . We make the following hoi e of ε ∈ ]0 , S/ : ε = S ς with ς > . Then we pose the following ondition on S as h → , Sh δ − + S − − ς h ≪ (cid:12)(cid:12)(cid:12) λ − Θ (cid:16) h α − / b γ n (cid:17)(cid:12)(cid:12)(cid:12) , ∀ n ∈ { , · · · , N } . By the hypothesis in (4.11), it su(cid:30) es to hoose S in the following way : Sh δ − + S − − ς h ≪ h / ( h → . This yields, h ς ) ≪ S ≪ h − δ ) and we noti e that a hoi e of ς > su h that ς ) > (cid:18) − δ (cid:19) EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 25is possible only if δ ∈ ] , [ (this will yield when α = that ρ ∈ ] , [ , ρ being introdu ed in Theorem 4.1). With this hoi e, the upper bound (4.12)be omes (for a possibly di(cid:27)erent e ǫ ( h ) ≪ ),(4.13) N (cid:16) hλ, P α,γh, Ω (cid:17) ≤ (cid:0) e ǫ ( h ) (cid:1) N X n =1 h − / Sπ p d ( α, b γ n ) p [ g n ( λ, α )] + , with g n ( λ, α ) = d ( α, b γ n ) e κ n + h − / h λ − Θ (cid:16) h α − / b γ n (cid:17)i . Repla ing b γ n by γ n = γ ( e S n ) in (4.13) will yield an error of the order O ( S ) ,and the sum on the right hand side of (4.13) be omes a Riemann sum. Wetherefore on lude the following upper bound N (cid:16) hλ, P α,γh, Ω (cid:17) ≤ (cid:0) e ǫ ( h ) (cid:1) Z ∂ Ω h − / π p d ( α, γ ( s )) p [ g ( λ, α ; s )] + d s, with g ( λ, α ; s ) = d ( α, γ ( s )) e κ ( s ) + h − / h λ − Θ (cid:16) h α − / γ ( s ) (cid:17)i . By a similar argument, we get a lower bound. (cid:3)
Remark 4.7. When relaxing the hypotheses of Theorem 4.6 by allowing λ to satisfy ( ompare with (4.11)): (cid:12)(cid:12)(cid:12) λ − Θ (cid:16) h α − / γ (cid:17)(cid:12)(cid:12)(cid:12) = o ( h / ) as h → , the result for the ounting fun tion be omes (as an be he ked by adjustingthe proof of Theorem 4.6) N (cid:16) hλ, P α,γh, Ω (cid:17) = Z ∂ Ω h − / π p d ( α, γ ( s )) s(cid:18) d ( α, γ ( s )) κ r ( s ) (cid:19) + d s ! (cid:0) o (1) (cid:1) . Proof of Theorem 1.2.We re all in this ase that < α < and that λ ( h ) = Θ + 3 aC h α − with a ∈ R \ { γ } . Here C > is the universal onstant introdu ed in (1.7).In this spe i(cid:28) regime, (4.11) is veri(cid:28)ed when making a hoi e of ρ ∈ ]0 , min( δ − , α − )[ .The leading order term of the integrand in the asymptoti formula of Theo-rem 4.6 is, up to a multipli ation by a positive onstant, q h − / (cid:2) λ − Θ (cid:0) h α − / γ ( s ) (cid:1)(cid:3) + . We write by using the asymptoti expansion of Θ( · ) given by Taylor's formula(see (2.11)-(2.14)) : Θ (cid:16) h α − / γ ( s ) (cid:17) = Θ + 3 C γ ( s ) h α − / + O ( h α − ) , ( h → . N (cid:16) hλ, P α,γh, Ω (cid:17) = h ( α − ) π √ ξ Z ∂ Ω q(cid:0) a − γ ( s ) (cid:1) + d s ! (1 + o (1)) . When a = γ , we may en ounter the regime of Remark 4.7, hen e by usingthe result of that remark and noti ing that when < α < h − / ≪ h ( α − ) ≪ h − / as h → , we re over the asymptoti expansion announ ed in Theorem 1.3 in the present ase. (cid:3) Proof of Theorem 1.3.In this ase α = and h / ≪ | λ − Θ( γ ) | ≤ ζ h ̺ with 0 < ̺ < . Taking ρ = ̺/ , then we may hoose δ ∈ ] , [ su h that (4.11) is satis(cid:28)ed.Thus, the asymptoti formula of Theorem 4.6 is still valid in this regime, andthe leading order term of the integrand is, up to a multipli ative onstant, h − / π s [ λ − Θ ( γ ( s ))] + d (cid:0) , γ ( s ) (cid:1) . This proves the theorem. (cid:3)
Proof of Theorem 1.5.Again, the proof follows from Theorem 4.6 and the properties of the fun tion Θ( · ) . (cid:3) A knowledgementsThe author would like to express his thanks to R. Frank for the fruitfuldis ussions around the subje t, and also to B. Hel(cid:27)er for his helpful remarks.He wishes also to thank the anonymous referees for their areful reading ofthe paper, for pointing out many orre tions and for their many helpfulsuggestions. Appendix A. Boundary oordinatesWe re all now the de(cid:28)nition of the standard oordinates that straightensa portion of the boundary ∂ Ω . Given t > , let us introdu e the followingneighborhood of the boundary,(A.1) N t = { x ∈ R ; dist( x, ∂ Ω) < t } . As the boundary is smooth, let s ∈ ] − | ∂ Ω | , | ∂ Ω | ] M ( s ) ∈ ∂ Ω be a regularparametrization of ∂ Ω that satis(cid:28)es : s is the oriented `ar length' between M (0) and M ( s ) .T ( s ) := M ′ ( s ) is a unit tangent ve tor to ∂ Ω at the point M ( s ) . The orientation is positive, i.e. det( T ( s ) , ν ( s )) = 1 . EYL ASYMPTOTICS FOR MAGNETIC SCHRÖDINGER OPERATORS 27We re all that ν ( s ) is the unit outward normal of ∂ Ω at the point M ( s ) . Thes alar urvature κ r is now de(cid:28)ned by :(A.2) T ′ ( s ) = κ r ( s ) ν ( s ) . When t is su(cid:30) iently small, the map :(A.3) Φ : ] − | ∂ Ω | / , | ∂ Ω | / × ] − t , t [ ∋ ( s, t ) M ( s ) − tν ( s ) ∈ N t , is a di(cid:27)eomorphism. For x ∈ N t , we write,(A.4) Φ − ( x ) := ( s ( x ) , t ( x )) , where t ( x ) = dist( x, ∂ Ω) if x ∈ Ω and t ( x ) = − dist( x, ∂ Ω) if x Ω . The Ja obin of the transformation Φ − is equal to,(A.5) a ( s, t ) = det (cid:0) D Φ − (cid:1) = 1 − tκ r ( s ) . To a ve tor (cid:28)eld A = ( A , A ) ∈ H ( R ; R ) , we asso iate the ve tor (cid:28)eld ˜ A = ( ˜ A , ˜ A ) ∈ H (] − | ∂ Ω | / , | ∂ Ω | / × ] − t , t [; R ) by the following relations :(A.6) ˜ A ( s, t ) = (1 − tκ r ( s )) ~A (Φ( s, t )) · M ′ ( s ) , ˜ A ( s, t ) = ~A (Φ( s, t )) · ν ( s ) . We get then the following hange of variable formulae.Proposition A.1. Let u ∈ H A ( R ) be supported in N t . Writing e u ( s, t ) = u (Φ( s, t )) , then we have :(A.7) Z Ω | ( ∇ − iA ) u | dx = Z | ∂ Ω | − | ∂ Ω | Z t h | ( ∂ s − i ˜ A ) e u | + a − | ( ∂ t − i ˜ A ) e u | i a dsdt, (A.8) Z Ω c | ( ∇ − iA ) u | dx = Z | ∂ Ω | − | ∂ Ω | Z − t h | ( ∂ s − i ˜ A ) e u | + a − | ( ∂ t − i ˜ A ) e u | i a dsdt, and(A.9) Z R | u ( x ) | dx = Z | ∂ Ω | − | ∂ Ω | Z t − t | e u ( s, t ) | a dsdt. We have also the relation : ( ∂ x A − ∂ x A ) dx ∧ dx = (cid:16) ∂ s ˜ A − ∂ t ˜ A (cid:17) a − ds ∧ dt, whi h gives, curl ( x ,x ) A = (1 − tκ r ( s )) − curl ( s,t ) ˜ A. We give in the next proposition a standard hoi e of gauge.Proposition A.2. Consider a ve tor (cid:28)eld A = ( A , A ) ∈ C ( R ; R ) su hthat curl A = 1 in R . x ∈ ∂ Ω , there exist a neighborhood V x ⊂ N t of x and asmooth real-valued fun tion φ x su h that the ve tor (cid:28)eld A new := A − ∇ φ x satis(cid:28)es in V x :(A.10) ˜ A new = 0 , and,(A.11) ˜ A new = − t (cid:18) − t κ r ( s ) (cid:19) , with ˜ A new = ( ˜ A new , ˜ A new ) ∼∼