Interpolation inequalities and spectral estimates for magnetic operators

We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring estimates for the principal eigenvalue of magnetic Schr{\"o}dinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate.


Introduction and main results
In dimensions d = 2 and d = 3, let us consider the magnetic Laplacian defined via a magnetic potential A by The magnetic field is B = curl A. The quadratic form associated with −∆ A is given by R d |∇ A ψ| 2 d x and well defined for all functions in the space We shall consider the following spectral gap inequality (1.1) Let us notice that Λ depends only on B = curl A. Throughout this paper, we shall assume that there is equality in (1.1) for some function in H 1 A (R d ). If B is a constant magnetic field, we recall that Λ[B] = |B|. If d = 2, the spectrum of −∆ A is the countable set {(2 j + 1) |B| : j ∈ N}, the eigenspaces are of infinity dimension and called the Landau levels. The eigenspace corresponding to the lowest level ( j = 0) is called the Lowest Landau Level and will be considered in Section 5.4.
The main result of this paper is to establish lower bounds for the optimal constants µ B , ν B and ξ B in the case of general magnetic fields (respectively in Propositions 3.1, 3.4 and in Section 3.5) and in the case of two-dimensional constant magnetic fields (respectively in Propositions 4.2, 4.3 and 4.5). Upper estimates, theoretical and numerical, are also given in Section 5.
The magnetic interpolation inequalities have interesting applications to optimal spectral estimates for the magnetic Schrödinger operators Let us denote by λ A,φ its principal eigenvalue, and by α B : (0, +∞) → (−Λ, +∞) the inverse function of α → µ B (α). We denote by φ − := (φ − |φ|)/2 the negative part of φ. By duality as we shall see in Section 2, Theorem 1.1 has a counterpart, which is a result on magnetic Keller-Lieb-Thirring estimates. Corollary 1.2. With these notations, let us assume that A satisfies the same hypotheses as in Theorem 1.1. Then we have: (1.7) The function α B satisfies (ii) For any q = p/(2 − p) ∈ (1, +∞) and any potential W ≥ 0 such that (1.8) (iii) For any γ > 0 and any potential W ≥ 0 such that e −W /γ ∈ L 1 (R d ), For general potentials changing sign, a more general estimate is proved in Proposition 2.1. A first result without magnetic field was obtained by Keller in the one-dimensional case in [16], before being rediscovered and extended to the sum of all negative eigenvalues in any dimension by Lieb and Thirring in [19]. In the meantime, an estimate similar to (1.9) was established in [13] which, by duality, provides a proof of the logarithmic Sobolev inequality given by Gross in [14]. In the Euclidean framework without magnetic fields, scalings provide a scale invariant form of the inequality, which is stronger (see [26,11]) but was already known as the Blachmann-Stam inequality and goes back at least to [23]: see [25,24] for an historical account. Many papers have been devoted to the issue of estimating the optimal constants for the so-called Lieb-Thirring inequalities: see for instance [18,9,10] for estimates on the Euclidean space, [6,7] in the case of compact manifolds, and [8] for non-compact manifolds (infinite cylinders). As far as we know, no systematic study as in Theorem 1.1 nor as in Corollary 1.2 has been done so far in the presence of a magnetic field, although many partial results have been previously obtained using, e.g., the diamagnetic inequality.
Section 2 is devoted to the duality between Theorem 1.1 and Corollary 1.2. Most of our paper is devoted to estimates of the best constants in (1.3), (1.4) and (1.5), which also provide estimates of the best constants in (1.7), (1.8) and (1.9). In Section 3 we prove lower estimates in the case of a general magnetic field and establish Theorem 1.1. Sharper estimates are obtained in Section 4 for a constant magnetic field in dimension two. Section 5 is devoted to upper bounds and the numerical computation of various upper and lower bounds (constant magnetic field, dimension two). Our theoretical estimates are remarkably accurate for the values of p and d that we have considered numerically, using radial functions. This is why we conclude this paper by a numerical investigation of the stability of a radial optimal function.

Magnetic interpolation inequalities and Keller-Lieb-Thirring inequalities: duality and a generalization
Let us prove Corollary 1.2 as a consequence of Theorem 1.1. Details on duality will be provided in the proof and in the subsequent comments.
Proof of Corollary 1.2. Consider first Case (i) with q > d /2. Using the definition of the negative part of V and Hölder's inequality with 1/q + 2/p = 1, we know that In Case (ii), by Hölder's inequality with exponents 2/(2 − p) and 2/p, with β = 1/ W −1 q , which proves (1.8). In Case (iii), let us consider for a given function ψ ∈ H 1 A (R d ) such that ψ 2 = 1 and mimimize this functional with respect to the potential W , so that where the last inequality is given by (1.5). Minimizing F [ψ,W ] with respect to W under the condition ψ 2 = 1 establishes (1.9). It is straightforward that the equality case is given by the equality case in (1.5) when there is a function ψ for which this equality holds.
In Case (iii) of Theorem 1.1 and Corollary 1.2, the duality relation of (1.5) and (1.9) is a straightforward consequence of the convexity inequality A similar observation can be done in Cases (i) or (ii). If q = p/(p − 2) ∈ (d /2, +∞), i.e., in Case (i), for an arbitrary negative potential V and an arbitrary function By minimizing with respect to either V or ψ, we reduce the inequality to (1.3) or (1.7), and in both cases V = − |ψ| p−2 is optimal. The two estimates are henceforth dual of each other, which is reflected by the fact that p/2 and q are Hölder for any positive potential W and any ψ ∈ H 1 A (R d ). Again a minimization with respect to either W or ψ reduces the inequality to (1.4) or (1.8), which are also dual of each other. With these observations, it is clear that Theorem 1.1 can be proved as a consequence of Corollary 1.2: the two results are actually equivalent.
The restriction to a negative potential V or to its negative part (resp. to a positive potential W ) is artificial in the sense that we can put the threshold at an arbitrary level λ. Let us consider a general potential φ on R d . We can first rewrite (2.1) in a more general setting as . Here u q,+ is a new notation which stands for This makes sense of course if µ is finite and well defined which, for instance, requires that A similar estimate holds in the range p ∈ (1, 2). Let λ ≤ infess x∈R d φ(x). Then we have We can collect these estimates in the following result.
These estimates hold for any λ ∈ R such that all above norms are well defined, with the additional condition that φ ≥ λ a.e. in Case (ii).
Notice that weaker conditions than φ ≥ λ a.e. can be given, like, for instance, Details are left to the reader. In Corollary 1.2, Case (iii) does not involve a threshold at level λ = 0 and one can notice that the estimate (1.

Lower estimates: general magnetic field
In this section, we consider a general magnetic field in dimension d = 2 or 3. We establish lower estimates of the best constants in (1.3), (1.4) and (1.5) before proving Theorem 1.1.

Preliminaries: interpolation inequalities without magnetic field
Assume that p > 2 and let C p denote the optimal constant defined in (1.2), that is, the best constant in the Gagliardo-Nirenberg inequality By scaling, if we test (3.1) by u · /λ , we find that An optimization on λ > 0 shows that the best constant in the scale-invariant inequality is given by Next, let us consider the case p ∈ (1, 2) and the corresponding Gagliardo-Nirenberg inequality where, compared to the case p > 2, the positions of the norms u 2 2 and u 2 p have been exchanged. A scaling similar to the one of (3.2) shows that, for any λ > 0, By optimizing on λ > 0, we obtain the scale-invariant inequality Optimal functions for (3.5) or (3.6) have compact support according to, e.g., [1,4,5,21]. See Section 5.2 for more details.
The logarithmic Sobolev inequality corresponds to the limit case p = 2. Let us consider (3.2) written with λ 2 = 1 p−2 , i.e., By passing to the limit as p → 2, we recover the Euclidean logarithmic Sobolev inequality with optimal constant in case γ = 1/2. The general case corresponding to any γ > 0, that is follows by a simple scaling argument. It was proved in [3] that there is equality in the above inequality if and only if, up to a translation and a multiplication by a constant, ψ(x) = e −γ |x| 2 /4 . As a consequence, we obtain that the limit of C p as p → 2 + is 1 and In other words, this means that , where C p denotes the optimal constant in (3.1) and S p is given by (3.4). Proof. Let t ∈ [0, 1]. From the diamagnetic inequality and from (1.1) and (3.2) applied with λ = α+Λ t 1−t , we deduce that for any ψ ∈ H 1 A . Finally we can optimize the quantity , the maximum is achieved at t = 0, which proves the second inequality.
By duality the estimates of Proposition 3.1 provide a lower estimate for the best constant in the Keller-Lieb-Thirring estimate (1.7).

Corollary 3.2.
Under the assumptions of Proposition 3.1, for any q = p/(p − 2) ∈ (d /2, +∞) and any potential V such that in V − ∈ L q (R d ), we have Proof. With p = 2 q q−1 , the estimates of Proposition 3.1 on α → µ B (α) provide estimates on its inverse µ → α B (µ) which go as follows: The result is then a consequence of Corollary 1.2.
By duality the estimates of Proposition 3.4 provide a lower estimate for the best constant in the Keller-Lieb-Thirring estimate (1.8).

Proof of Theorem 1.1
Proof of Theorem 1.1. Let us consider Case (i): p ∈ (2, 2 * ). The positivity of the function µ B is a consequence of Proposition 3.1 while the concavity follows from the definition of α → µ B (α) as the infimum on H 1 A (R d ) of an affine function of α. The estimate as α → (−Λ) + is easily obtained by considering as test function the function ψ ∈ H 1 A (R d ) for which there is equality in (1.1). We know from Proposition 3.1 that Interpolation inequalities and spectral estimates for magnetic operators 11 To prove the equality, we take as test function for µ B (α) the function v α := v( α ·), with α > 0, where the radial function v realizes the equality in (3.1). The function v is smooth, positive everywhere and decays like e −|x| as |x| → +∞. Notice that v α realizes the equality in (3.2) and there is a constant The result follows from α The proof of (ii) is very similar to that of (i). The positivity of the function ν B is a consequence of Proposition 3.4 while the concavity follows from the definition of β → ν B (β). From Proposition 3.4, we know that To prove the equality, for any β > 0, we take as test function for ν B (β) the function where the radial function w realizes the equality in (3.5), so that w β realizes the equality in (3.6). The function w has compact support and can be estimated from above and from below, up to a multiplicative constant, by the characteristic function of centered balls. The same computation as above shows that . The result follows from and (1.6) with σ = β p 2 p+d (2−p) . The case p = 2 is much simpler. As a straightforward consequence of the Euclidean logarithmic Sobolev inequality (3.7) and of the diamagnetic inequality (3.9), we know that As a consequence, we deduce the existence of a concave function ξ B in Inequality (1.9), such that Note that the r.h.s. is negative for γ large. The function w γ (x) = (γ/π) d /4 e − γ 2 |x| 2 is optimal in (3.7) and can be used as a test function in (1.5) in the regime as γ → +∞. Using the fact that w γ 2 = 1, ∇w γ 2 = d γ and we get that, for some positive constant c, → 0 as γ → +∞ according to (1.6). This establishes that ξ B γ is equal to d 2 γ log π e 2 /γ at leading order as γ → +∞.

Lower estimates: constant magnetic field in dimension two
In the particular case when the magnetic field is constant, of strength B > 0, and d = 2, we can improve the lower estimates of the last section. In this section we assume that B = (0, B ) and choose the gauge so that (4.1)

A preliminary result
The next result follows from [20, proof of and equality holds with ψ = u e i S and u > 0 if and only if Proof. For every c ∈ [0, 1], Interpolation inequalities and spectral estimates for magnetic operators 13 An expansion of the square shows that with equality only if c |∇u| = |A + ∇S| u. Next we obtain that 2 |∇u| |A + ∇S| u = |∇u 2 | |A + ∇S| ≥ ∇u 2 ⊥ · (A + ∇S) , where ∇u 2 ⊥ := −∂ 2 u 2 , ∂ 1 u 2 , and there is equality if and only if for some γ. Since c |∇u| = |A+∇S| u, we have γ = 2 u 2 /c. Integration by parts yields

Case p ∈ (1, 2)
Now let us turn our attention to the case p ∈ (1, 2). The strategy of the proof of Proposition 4.2 applies: for any c ∈ (0, 1), for any β > 0, by applying (3.6) with λ 4/p = β/(1 − c 2 ), we obtain The function c → c B + C p β p/2 (1 − c 2 ) 1−p/2 is positive in [0, 1] and its derivative is positive at 0 + , and negative in a neighborhood of 1 − . The maximum is achieved at the unique point c * ∈ (0, 1) given by This establishes the following result.

Proposition 4.3. Consider a constant magnetic field with field strength B in two
dimensions. Given any p ∈ (1, 2), and any β > 0, we have with c * given by (4.5).
Proof. By using (3.8) with d = 2 and (4.4), we see that for any η > 0, we can pass to the limit as p → 2 + and establish (4.6) by setting γ = η B .
It turns out that the above magnetic logarithmic Sobolev inequality is optimal. To identify the minimizers, we observe that the magnetic Schrödinger operator is not invariant under the standard translations. For any b = (b 1 , b 2 ) ∈ R 2 , Notice that in the semi-classical regime corresponding to a limit of the magnetic field B such that 1/(2 η) = Λ = Λ[B] → 0, we recover the classical logarithmic Sobolev inequality (3.7) without magnetic field.

An upper estimate and some numerical results
In this section, we assume that d = 2, consider a constant magnetic field, establish a theoretical upper bound, and numerically compute the difference with the lower bounds of Sections 3 and 4.

Numerical estimates based on Euler-Lagrange equations
Instead of a Gaussian test function, one can numerically compute the minimum of Q (p) α in the class C 0 by solving the corresponding Euler-Lagrange equation. Case (i). Assume that p ∈ (2, +∞). The equation is  [15] and [22]. Numerically, we solve the ODE on a finite interval, which induces a numerical error: the interval has to be chosen large enough, so that the computed value is a good upper approximation of µ EL (α). Case (ii). Assume that p ∈ (1, 2). A radial minimizer of Q and have therefore to solve the reduced problem among nonnegative functions in H 1 ((0, +∞), r d r ) such that +∞ 0 |v| p r d r < +∞. Notice that the compact support principle applies according to, e.g., [1,4,5,21], since p − 1 < 1 so that the nonlinearity in the right hand side of (5.2) is non-Lipschitz. Numerically, we can therefore solve (5.2) using a shooting method, with a shooting parameter a = v(0) > 0 that has to be adjusted to provide a nonnegative solution with compact support, which minimizes +∞ 0 |v| p r d r . The set of solutions is then parametrized by the parameter ν > 0, while β is recovered by the above integral condition. In other words, we approximate ν → β B (ν) and recover β → ν B (β) as the inverse of β B . Since we compute the size of the support of the approximated solution, there is no numerical error due to finite size truncation.

Asymptotic regimes
where µ 0 (α) = ψ 0 p−2 p = Q (p) α [ψ 0 ]. The linear stability of ψ 0 with respect to perturbations in (C 1 ) can be recast as the eigenvalue problem The numerical results for d = 2, B = 1 and p = 3 of Fig. 5 suggest that Q (p) α is linearly stable for α > −B , not too large. This indicates that µ EL is a good candidate for computing the exact value of µ B for arbitrary values of B 's.