MAGNETIC LIEB–THIRRING INEQUALITIES ON THE TORUS

. In this paper we prove Lieb–Thirring inequalities for magnetic Schr¨odinger operators on the torus, where the constants in the inequalities depend on the magnetic ﬂux.


INTRODUCTION
Lieb-Thirring inequalities have important applications in mathematical physics, analysis, dynamical systems, attractors, to mention a few.A current state of the art of many aspects of the theory is presented in [7].
In certain applications Lieb-Thirring inequalities are considered on a compact manifold (e. g., torus, sphere [11]).In this case one has to impose the zero mean orthogonality condition.However, in the case of a torus the corresponding constants in the Lieb-Thirring inequalities depend on the aspect ratios of the periods, for example, on the 2D torus the rate of growth of the constants is proportional to the aspect ratio.
On the other hand, on the torus T d with arbitrary periods it is possible to obtain bounds for the Lieb-Thirring constants that are independent of the ratios of the periods, provided that we impose a stronger orthogonality condition that the functions must have zero average over the shortest period uniformly with respect to the remaining variables [10].
In this work we prove Lieb-Thirring inequalities on the torus for the magnetic Laplacian.The introduction of the magnetic potential not only removes the orthogonality condition but makes it possible to obtain bounds for the constants that are independent of the periods of the torus (more precisely, depend only on the corresponding magnetic fluxes).
In this paper, when obtaining the constant in the Lieb-Thirring inequality we use a combination of the result obtained in [8] and also adopting the proof from [7] to the case of the magnetic operator on the torus.Surprisingly both such independent estimates play important and noninterchangeable roles depending on the magnetic fluxes.
In conclusion of this brief introduction we point out that magnetic interpolation inequalities both in R d , and in the periodic case received much attention over the last years, see [3,4,13] and the references therein.
We now describe our main result.Let T d = T d (L) be the d-dimensional torus with periods L 1 , . . ., L d .Let us consider the eigenvalue problem for the magnetic Schrödinger operator H in L 2 (T d ): where is the real-valued magnetic vector potential in the "diagonal" case when a j (x) = a j (x j ).For each j we define the magnetic flux and assume that α j ∈ Z for all j.Then we have the following result.
Let γ ≥ 1.Then the following bound holds for the γ-moments of the negative eigenvalues of operator (1.1): where Here L cl γ,d is the semiclassical constant (3.4), and The expressions for K 1 (α) and K 2 (α) are as follows: (1.5) In Sections 2 and 3 we consider the one dimensional case, where this theorem is proved in the equivalent dual formulation in terms of orthonormal systems in the scalar case and the matrix case, respectively.We point out that this theorem with K(α) ≤ K 1 (α) was proved in [8] and the proof was based on the magnetic interpolation inequality (2.3) (whose proof is briefly recalled in Section 2).In the 1D scalar case this inequality immediately gives the result by the method of [5], while the inequality in the essential matrix case was proved in [8] (see also [2] for the starting point of this approach).
The bound for the constant K(α) ≤ K 2 (α) was proved in the 1D scalar case in [9].The proof in the matrix case is given in Theorem 3.1.Then the inequalities for orthonormal systems are equivalently reformulated in Theorem 3.2 in terms of estimates for the negative trace and for higher-order Riesz means of negative eigenvalues in Corollary 3.1.Finally, Theorem 1.1 is proved in Section 4 by using the lifting argument with respect to dimensions [12].The fact that the magnetic potential is of the special diagonal form is crucial here.

1D PERIODIC CASE
We consider here the magnetic Lieb-Thirring inequality in the 1D periodic case.We assume that the period equals Of course, one can use scaling and consider only the case ε = 1, but we prefer to consider the general case in order to trace down the corresponding constants in the most explicit way. where Here α is the magnetic flux and the constants K 1 (α) and K 2 (α) are defined in (1.4), (1.5).
Proof.We first point out that estimate (1.4) was obtained in [8, (6.8)],where k(α) is the constant in the 1D magnetic interpolation inequality 3) The sharp constant k(α) (shown in Figure 2) was found in [8, (3.5)] and is given in (1.4).For the sake of completeness we briefly recall the proof of (2.3).We further assume for the moment that the magnetic potential is constant a(x) ≡ a = const.We use the Fourier series We consider the self-adjoint operator and its Green's function G λ (x, ξ) which is found in terms of the Fourier series On the diagonal we obtain Using a general result (see Theorem 2.2 in [14] with θ = 1/2) we find that the sharp constant in (2.3) is as follows where An elementary analysis of the dependence of the behaviour of the function F (ϕ) on the parameter α = a/ε (see [8] for the details) gives the expression for k(α) in (1.4).
We now consider the case of a non-constant magnetic potential a(x).It this case instead of the complex exponentials we consider the orthonormal system of functions that are periodic with period 2π/ε in view of (2.2) and satisfy Therefore the Green's function of the operator A(λ) is giving the same expression for G λ (ξ, ξ) as in (2.4) and hence the same expression for k(α) as in the case a(x) = const.
It now remains to prove (1.5): Let a(x) = const.We use the Fourier series with respect to system (2.5): Then we obtain that where where For any δ > 0 we have In view of orthonormality, Bessel's inequality, (2.6) and the fact that (2.9) Next, following [6,7] (see Remark 2.1) we set This gives where and where we singled out the factor √ E, set b := µE 3/2 /ε 3 , and recalled the definition of µ.
Substituting this into (2.8) and optimizing with respect to δ we obtain which gives that The proof is complete.
Remark 2.1.The series over k ∈ Z in (2.9), which we obviously want to minimize under the condition [6]: (2.12) This explains the choice of f (t) in (2.10).

1D PERIODIC CASE FOR MATRICES
Let {ψ n } N n=1 be an orthonormal family of vector-functions We consider the M × M matrix U(x) Theorem 3.1.The following inequality holds where K(α) is defined in Theorem 1.1.
As before, let f be a scalar function with where Let e ∈ C M be a constant vector.Then where •, • denotes the scalar product in C M .For the first term we have where the scalar function χ E (x ′ , x) is as in (2.7).Now, again by orthonormality, Bessel's inequality and (2.11) we obtain For the second term we simply write Combining the above we obtain If we denote by λ j (x) and λ E j (x), j = 1, . . ., M the eigenvalues of the (Hermitian) matrices U(x) and U E (x), respectively, then the variational principle implies that Optimizing with respect to δ we find that Integration with respect to E gives that and integration with respect to x gives (3.1) with (1.5).
We finally point out that matrix inequality (3.1) with estimate of the constant (1.4) was previously proved in [8,Theorem 6.2].The proof given there holds formally for the case of a constant magnetic potential.However, if a(x) = const we only have to use the orthonormal family (2.5) as we have done in the proof of the scalar Lieb-Thirring inequality in Theorem 2.1.The proof is complete.
It is well known [2,7] that inequalities for orthonormal systems are equivalent to the estimates for the negative trace of the corresponding Schrödinger operator.In our case we consider the magnetic Schrödinger operator Then the spectrum of operator (3.2) is discrete and the negative eigenvalues −λ n ≤ 0 satisfy the estimate where Proof.Let {ψ n } N n=1 be the orthonormal vector valued eigenfunctions corresponding to {−λ n } N n=1 : Taking the scalar product with ψ n , using inequality (3.1), Hölder's inequality for traces and setting X = Calculating the maximum with respect to X we obtain (3.3).
The higher-order Riesz means of the eigenvalues for magnetic Schrödinger operators with matrix-valued potentials are obtained by the Aizenmann-Lieb argument [1,7].

MAGNETIC SCHR ÖDINGER OPERATOR ON THE TORUS
Proof of Theorem 1.1.We use the lifting argument with respect to dimensions developed in [12].More precisely, we apply estimate (3.5) d−1 times with respect to variables x 1 , . . ., x d−1 (in the matrix case), so that γ is increased by 1/2 at each step, and, finally, we use (3.5) (in the scalar case) with respect to x d .Using the variational principle and denoting the negative parts of the operators by Remark 4.2.The method of Theorem 2.1 (namely, its second part) is difficult to apply in the case orthonormal system on the torus T d with d > 1, because the corresponding series (2.11) is now over the lattice Z d and depends on d parameters.However, the Lieb-Thirring inequality for an orthonormal system {ψ j } N j=1 ∈ H 1 (T d ) follows from Theorem 1.1 γ=1 by duality.For example, for d = 2 it holds

SOME COMPUTATIONS
We now present some computational results.We denote by F (b, α) the key function in (2.The unique point of maximum b * (α) has the following asymptotic behaviour as α → 0 + .For a small α the main contribution in the sum in (5.1) comes from the term with k = 0, that is, from The minimum with respect to α is attained at α * = 0.273 giving K 2 (α * ) = K 2 (1 − α * ) = 0.811.