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Estimates for the Lowest Eigenvalue of Magnetic Laplacians T. Ekholm∗, H. Kovaˇr´ık†, F. Portmann‡ 5 1 0 2 n Abstract a J We prove various estimates for the first eigenvalue of the magnetic Dirichlet Laplacian on a bounded domain in two dimensions. When 2 2 the magnetic field is constant, we give lower and upper bounds in terms of geometric quantities of the domain. We furthermore prove a ] lower bound for the first magnetic Neumann eigenvalue in the case of P constant field. S . h t 1 Introduction a m [ Let Ω be a bounded open domain in R2 and B L∞(R2) a real-valued ∈ loc function, the magnetic field. To B we associate a vector potential A 1 ∈ v L∞(Ω) such that B = curlA = ∂1A2 ∂2A1 in Ω, see Section 2 for an − 4 explicit construction of A. The magnetic Dirichlet Laplacian on Ω, 5 5 HD := ( i +A)2 (1.1) 5 Ω,B − ∇ 0 . is then defined through the Friedrichs extension of the quadratic form 1 0 5 hD [u] := ( i +A)u(x) 2 dx, 1 Ω,A | − ∇ | : ZΩ v i on C∞(Ω). Altogether, there is a huge amount of literature dealing with X c spectral properties of the operator HD on bounded as well as unbounded r Ω,B a domains in R2. We refer to the [AHS, CFKS] for an introduction on Schro¨dinger operators with magnetic fields. Various estimates for sums and Riesz means of eigenvalues of HD on bounded domains were established Ω,B in [ELV, KW, LLR, LS]. Hardy-type inequalities for hD were studied in Ω,A [BLS, LW, W]. For a version of the well-known Faber-Krahn inequality in the case of constant magnetic field we refer to [Er]. ∗[email protected], RoyalInstituteof Technology KTH, Sweden †[email protected],Università degli studidi Brescia, Italy ‡Corresponding Author;[email protected], Universityof Copenhagen, Denmark 1 The main object of interest in this note will be the quantity hD [u] λ (Ω,B) := infspecHD = inf Ω,A 0. 1 Ω,B u∈C∞(Ω) u 2 ≥ c k k Since Ω is bounded, our conditions on B imply that the form domain of HD is H1(Ω) and λ (Ω,B) is indeed the lowest eigenvalue of HD . There Ω,B 0 1 Ω,B existtwowell-known lower boundsforλ (Ω,B). Byacommutator estimate, 1 see e.g. [AHS], one obtains hD [u] B(x) u(x)2dx. (1.2) Ω,A ≥ ± | | ZΩ For a constant magnetic field B(x) = B , inequality (1.2) yields 0 λ (Ω,B ) B . (1.3) 1 0 0 ≥ ± The pointwise diamagnetic inequality (see for example [LL, Theorem 7.21]) ( i +A)u(x) u(x) , for a.e. x Ω, (1.4) | − ∇ | ≥ |∇| || ∈ on the other hand tells us that hD [u] u 2 v 2 inf Ω,A inf Ω|∇| || = inf Ω|∇ | u∈H01(Ω) kuk2 ≥ u∈H01(Ω) R kuk2 v∈H01(Ω) R kvk2 v≥0 v 2 inf Ω|∇ | . ≥ v∈H01(Ω) R kvk2 This implies that λ (Ω,B) λ (Ω,0). (1.5) 1 1 ≥ Under very weak regularity conditions on B it was shown in [He] that inequality (1.5) is in fact strict; λ (Ω,B)> λ (Ω,0). 1 1 Let us briefly discuss the Neumann case. The quadratic form corresponding to the magnetic Neumann Laplacian HN is given by Ω,B hN [u] := ( i +A)u(x)2dx, Ω,A | − ∇ | ZΩ and the form domain is now H1(Ω). Again, µ (Ω,B) := infspecHN 1 Ω,B is the first eigenvalue of HN , provided Ω is sufficiently regular. The esti- Ω,B mate (1.5) remains valid in the Neumann case (since (1.4) holds a.e.), and gives: µ (Ω,B) µ (Ω,0) = 0. (1.6) 1 1 ≥ The corresponding estimate (1.2) (resp. (1.3)) is a priori not available due to the different boundary conditions. A lot of attention has been paid to the asymptotic behavior of µ (Ω,B) for large values of the magnetic field, 1 see e.g. [Bo, FH1, LP, Ra, Si]. 2 1.1 Overview of the Main Results A natural question which arises in this context is whether estimates (1.2), (1.3)and(1.5)can beimproved byaddingapositiveterm totheirrighthand sides. Itisclearthattheircombinationscannotbeachieved bysimpleaddition; already for the constant magnetic field any lower bound of the type λ (Ω,B ) λ (Ω,0)+cB , (1.7) 1 0 1 0 ≥ with c > 0 independent of B , must fail. Indeed, since the eigenfunction of 0 HD = ∆D relative to the eigenvalue λ (Ω,0) may be chosen real-valued, Ω,0 − Ω 1 analytic perturbation theory yields λ (Ω,B ) = λ (Ω,0)+ (B2), B 0. (1.8) 1 0 1 O 0 0 → This clearly contradicts (1.7) for B small enough. 0 The main results of our paper are the following. In Section 2 we give quantitative lower bounds on the quadratic form hD [u] B(x) u(x)2dx, Ω,A ∓ | | ZΩ denoted by estimates of the first type. Estimates for the difference λ (Ω,B) λ (Ω,0), 1 1 − referred to as estimates of the second type, are studied in Section 3. In both cases, particular attention will be devoted to the case of constant magnetic field. Last but not least, we will also establish a lower bound of the second type for the lowest eigenvalue of the magnetic Neumann Laplacian in the case of constant magnetic field in Section 3.2. Notation: Given x Ω and r > 0, we denote by B(x,r) the open disc of radius r ∈ centered in x. We also introduce the distance function δ(x) := dist(x,∂Ω), and the in-radius of Ω, R := supδ(x). in x∈Ω Finally, given a positive real number x we denote by [x] its integer part. 3 2 Estimates of the First Type In this section we will derive lower bounds on the forms hD [u] B(x) u(x)2dx. Ω,A ∓ | | ZΩ Instead of introducing a vector potential A associated to B, we decide link bothquantities through aso called super potential, an approach that is well- known in the study of the Pauli operator, see e.g. [EV]. For our magnetic fieldshowever, thisapproachisequivalent withthestandarddefinitiongiven in the introduction. Let r > 0 be such that Ω B(0,r). For any B L∞(R2) let ⊂ ∈ loc (B) := Ψ : R2 R : ∆Ψ = B in B(0,r) F → be the family of super (cid:8)potentials associated to B. Note th(cid:9)at (B) is not F empty. Indeed, the function 1 Ψ (x) = log x y B(y)dy, x R2, 0 2π ZB(0,r) | − | ∈ which is well defined in view of the regularity of B, solves B(x) x B(0,r), ∈ ∆Ψ (x) = (2.1) 0  0 elsewhere,  in the distributional sense. Since B ∈ L∞loc(R2), standard regularity theory implies that Ψ W2,p(B(0,r)) for every 1 p < , see [GT, Thm. 9.9]. 0 ∈ ≤ ∞ Moreover, for any Ψ (B) the difference Ψ Ψ is a harmonic function 0 ∈ F − in B(0,r). Hence for any Ψ (B) and any p [1, ) we have Ψ ∈ F ∈ ∞ ∈ W2,p(B(0,r)). By Sobolev’s embedding theorem it then follows that Ψ is continuous on B(0,r), so we may define the oscillation of Ψ over Ω; osc(Ω,Ψ) = supΨ(x) inf Ψ(x). x∈Ω −x∈Ω Accordingly we set D(Ω,B) := inf osc(Ω,Ψ). Ψ∈F(B) Note also that a vector field A: B(0,r) R2 defined by → A := ( ∂ Ψ,∂ Ψ), Ψ (B), 2 1 − ∈ F belongs to W1,p(B(0,r)) for every 1 p < , in view of the regularity of ≤ ∞ Ψ, and satisfies curlA(x) = B(x) in B(0,r) in the distributional sense. Hence by the Sobolev embedding theorem we have A L∞(B(0,r)) and furthermore divA = 0 almost everywhere on ∈ B(0,r). 4 Theorem 2.1. Let Ω R2 be a bounded open domain and suppose that ⊂ B L∞(R2). Then ∈ loc hD [u] B(x) u(x)2dx+e−2D(Ω,B)λ (Ω,0) u(x)2dx (2.2) Ω,A ≥ ± | | 1 | | ZΩ ZΩ holds true for all u C∞(Ω). ∈ c Proof. We first prove inequality (2.2) with the plus sign on the right hand side. To do so, we pick any Ψ (B) and perform the ground state sub- ∈ F stitution u(x) =: v(x)e−Ψ(x) and obtain, after a relatively lengthy (but straightforward) calculation, hD [u] B(x) u(x)2dx = e−2Ψ ( i∂ ∂ )v 2dx. (2.3) Ω,A − | | | − 1− 2 | ZΩ ZΩ Next, we have e−2Ψ ( i∂1 ∂2)v 2dx e−2supx∈ΩΨ(x) ( i∂1 ∂2)v 2dx | − − | ≥ | − − | ZΩ ZΩ = e−2supx∈ΩΨ(x) v 2dx, |∇ | ZΩ where in the last step we used that ((∂ v)∂ v¯ (∂ v¯)∂ v) dx = 0. 1 2 1 2 − ZΩ It then follows that hD [u] B(x) u(x)2dx e−2supx∈ΩΨ(x) v 2dx Ω,A − | | ≥ |∇ | ZΩ ZΩ e−2supx∈ΩΨ(x)λ1(Ω,0) v 2dx ≥ | | ZΩ e−2osc(Ω,Ψ)λ (Ω,0) u2dx. 1 ≥ | | ZΩ To prove the corresponding lower bound with the minus sign in front of B on the right hand side, we note that the substitution u(x) =: w(x)eΨ(x) gives hD [u]+ B(x) u(x)2dx = e2Ψ ( i∂ +∂ )w 2dx. Ω,A | | | − 1 2 | ZΩ ZΩ Moreover, since osc(Ω, Ψ) =osc(Ω,Ψ), the same procedure as above gives − an identical lower bound. To complete the proof of (2.2) it now suffices to optimize the right hand side with respect to Ψ (B), keeping in mind ∈ F that the spectral properties of hD only depend on B. Ω,A 5 2.1 Estimates for the Constant Magnetic Field In this section we consider the case of a constant magnetic field: B(x) = B > 0. 0 Clearly, all Ψ (B ) are smooth, and optimizing estimate (2.2) amounts 0 ∈ F to minimizing the oscillation of B Ψ, where Ψ satisfies 0 ∆Ψ=e1 in Ωe. The optimal Ψ depends very muech on the geometry of Ω, and we start with a rather general result. Wepickaneypointx Ωandarotation R(x ,θ) SO(2), parametrized 0 0 ∈ ∈ by an angle θ [0,2π) and center of rotation x . Set 0 ∈ ℓ(Ω,x ,θ):= sup x inf x , (2.4) 0 2 2 x∈R(x0,θ)Ω −x∈R(x0,θ)Ω the maximal x -distance of the rotated set R(x ,θ)Ω. The quantity ℓ(Ω) is 2 0 then defined as follows: ℓ(Ω):= inf ℓ(Ω,x ,θ). (2.5) 0 θ∈[0,2π) It is easily seen that ℓ(Ω) is independent of the choice of x Ω and finite, 0 ∈ since Ω is bounded. Theorem 2.2. Let Ω R2 a bounded open domain. Then ⊂ λ1(Ω,B0) B0+e−B40 ℓ(Ω)2λ1(Ω,0). (2.6) ≥ Proof. In view of estimate (2.2) we have λ (Ω,B ) B +e−2osc(Ω,Ψ)λ (Ω,0), Ψ (B ). (2.7) 1 0 0 1 0 ≥ ∀ ∈ F The rotational symmetry of the problem allows us to assume that Ω has been rotated such that ℓ(Ω) = supx inf x . 2 2 x∈Ω −x∈Ω Let α := inf x and β := sup x . We then chose the super potential x∈Ω 2 x∈Ω 2 B Ψ(x ,x )= 0(x a)2, (2.8) 1 2 2 2 − whereaisafreeparameter. Observethatwemayassumethattheentireline between α and β is contained in Ω, because the oscillation of a function over 6 adomaincanonlyincreaseasthedomainisincreased(andΨisgloballywell- defined). Next, we calculate osc(Ω,Ψ) and minimize the result with respect to a. A direct calculation shows that the bestchoice is a = (α+β)/2, which gives B B osc(Ω,Ψ) 0 (β α)2 = 0 ℓ(Ω)2. ≤ 8 − 8 This in combination with (2.7) implies (2.6). Remark 2.3. It was shown in [Er, Er2] that for any ε > 0 there exists a constant C(ε) such that λ1(B(0,R),B0) ≥B0+ CR(ε2)e−B0(21+ε)R2, (2.9) Together with the Faber-Krahn inequality [Er] this yields λ (Ω,B ) λ (B(0,R),B ) 1 0 1 0 ≥ ≥ B0+ CR(ε2)e−B0(12+ε)R2, ε > 0, (2.10) whereRissuchthat B(0,R) = Ω . Itisclearthat(2.6)isanimprovement | | | | of the estimate (2.10) for domains that are geometrically very far from the disc, as for example very wide rectangles or thin ellipses. Proposition 2.4. Let Ω R2 be any bounded convex open domain, then ⊂ 2R ℓ(Ω) 3R . (2.11) in in ≤ ≤ Proof. Let B Ω be a disc of radius R contained in Ω. Independently of in ⊂ Ω being convex or not we have ℓ(Ω,x ,θ) 2R , θ [0,2π), x Ω. (2.12) 0 in 0 ≥ ∀ ∈ ∈ This follows directly from (2.4); for any θ [0,2π) and x Ω we have that 0 ∈ ∈ ℓ(Ω,x ,θ) is larger or equal to the length of the intersection of Ω with the 0 vertical line passing through the center of B. The latter is obviously larger or equal to 2R , hence equation (2.12). in It remains to prove the second inequality of (2.11). Let B Ω be a ⊂ disc of radius R . Assume first that ∂B ∂Ω contains at least two distinct in ∩ points P and P and that the vectors OP and OP are linearly dependent. 1 2 1 2 By convexity, Ω is contained in an infinite rectangle of height 2R , and in ℓ(Ω) 3R . in ≤ P 1 O P 2 7 Assumenowthat∂B ∂Ωisdistributedinsuchaway thatthereisaclosed, ∩ connected set Γ ∂B of length πR with the property that the distance in ⊂ ρ(x):= inf x y , x Γ, y∈∂Ω| − | ∈ is positive. u P O Γ 1 ∂Ω P 2 Since ρ is continuous and Γ is closed there is an ε such that ρ(x) ε > 0, ≥ for all x Γ. Hence we can move the disc B a distance ε/2 much in the ∈ direction of u, such that B becomes a proper subset of Ω. This contradicts that the inner radius of Ω is R . in Assumethat∂B ∂Ωcontains atleast threepoints P ,P andP . They 1 2 3 ∩ must be distributed in such a way that there is no such Γ as above. Since Ω is convex, it is contained in a triangle given by the tangent lines of the intersection points. P 3 P O 1 P 2 For a triangle, ℓ(T) is given by the smallest height, which is maximized for the equilateral triangle. Hence ℓ(Ω) 3R . in ≤ Notethatthesecondtermontherighthandsideof (2.6)decaysexponen- tially fast to zero as B tends to infinity. This was in fact already observed 0 in [FH2, Remark 1.4.3], where the authors observed that λ (Ω,B ) 1 0 = 1+ (exp( αB )), B , (2.13) 0 0 B O − → ∞ 0 and α is a positive constant. The optimal value of α is in general unknown, however it was conjectured in [FH2, Remark 1.4.3] that α is proportional to R2 . This is in agreement with Proposition 2.4 and the following result. in Proposition 2.5. Let Ω R2 be a bounded open domain and suppose that ⊂ B R2 4. Then 0 in ≥ λ1(Ω,B0)≤ B0+eB02Ri2n e−B20R2in. (2.14) 8 Proof. In view of (2.3), any v H1(Ω) satisfies ∈ 0 e−2Ψ ( i∂ ∂ )v 2dx λ (Ω,B ) B + Ω | − 1− 2 | . (2.15) 1 0 ≤ 0 e−2Ψ v 2dx R Ω | | Without loss of generality we may assuRme that the largest disc contained in Ω is centered in the origin. Hence B(0,R ) Ω. We choose the super in ⊂ potential in the form B Ψ(x) = 0 x 2, 4 | | and apply inequality (2.15) with 1 x R 1 , | | ≤ in− RinB0  v(x) = R B (R x ) R 1 < x < R ,  in 0 in−| | in− RinB0 | | in   0 elsewhere.     Obviously v H1(Ω). Note also that since v is real valued, we have ∈ 0 ( i∂ ∂ )v 2 = v 2. Performing both integrations in (2.15) in polar 1 2 | − − | |∇ | coordinates and taking into account the condition B R2 4, we find that 0 in ≥ e−2Ψ|(−i∂1 −∂2)v|2dx = 2πB02Ri2n Rin 1 e−B20 r2rdr ZΩ ZRin−RinB0 = 2πB0Ri2n e−B20 R2in exp 1− 2B21R2 −1 (cid:18) (cid:18) 0 in(cid:19) (cid:19) ≤ 2π(e−1)B0Ri2n e−B20R2in and 1 e−2Ψ v 2dx 2π Rin−RinB0 e−B20 r2rdr 2π (1 e−1) | | ≥ ≥ B − ZΩ Z0 0 which proves (2.14). In the special case when Ω is a disc we have Proposition 2.6. Let Ω = B(0,R) and let B R2 4. Then 0 ≥ 1+ Bj02R,12 e−12B0R2 ≤ λ1(B(0B,R),B0) ≤ 1+eB0R2 e−21B0R2, (2.16) 0 0 where j 2.405 is the first zero of the Bessel function J . 0,1 0 ≃ 9 Proof. The upper bound is given by Proposition 2.5. For the lower bound we use Theorem 2.1 with the super potential Ψ(x) = B0 x 2, which is in 4 | | this case better suited to the geometry of the domain than the one used in Proposition 2.4. Hence B R2 osc(B(0,R),Ψ) = 0 , 4 and since λ(B(0,R),0) = j02,1, the lower bound in (2.16) follows from (2.2). R2 Remark 2.7. In [HM, Prop. 4.4] it was stated that λ1(B(0,R),B0) 1+ 23/2 B0 Re−12B0R2, B0 . (2.17) B ∼ √π → ∞ 0 p B. Helffer however pointed out to us that the argument used to establish the above was slightly flawed – the above is only true if taken as a lower bound. However, from Proposition 2.6 we easily see that log[λ (B(0,R),B ) B ] 1 1 0 0 lim − = , B0→∞ B0R2 −2 which confirms, up to a pre-factor, the asymptotic (2.17) stated in [HM, Prop. 4.4]. Note also that the lower bound in (2.16) improves qualitatively the lower bound (2.9), since it allows us to pass to the limit ε 0. → 3 Estimates of the Second Type 3.1 Dirichlet boundary conditions In this section we are going to establish a lower bound on the difference λ (Ω,B) λ (Ω,0). Given a point x Ω, we introduce the function 1 1 − ∈ 1 Φ(r,x) := B(x)dx, 0 r δ(x), (3.1) 2π ZB(x,r) ≤ ≤ thefluxthroughB(x,r). Thenextresultshowsthatassoonasthemagnetic field is not identically zero in Ω, the difference λ (Ω,B) λ (Ω,0) is strictly 1 1 − positive. Theorem 3.1. Let Ω be a bounded open domain in R2 and B L∞(R2). ∈ loc If B is not identically zero in Ω, then there exists y Ω such that ∈ λ (Ω,B) λ (Ω,0) D(y,B), (3.2) 1 1 − ≥ where D(y,B)> 0 is given by (3.9). 10