A Sharp Lieb-Thirring Inequality for Functional Difference Operators

We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.

Equivalently, dom(U (b)) consists of those functions ψ(x) which admit an analytic continuation to the strip {z = x + iy ∈ C : 0 < y < b} such that ψ(x + iy) ∈ L 2 (R) for all 0 ≤ y < b and there is a limit ψ(x + ib − i0) = lim ε→0 + ψ(x + ib − iε) in the sense of convergence in L 2 (R), which we will denote simply by ψ(x + ib). The domain of the inverse operator U (b) −1 can be characterised similarly.
The operator W 0 (b) is self-adjoint and unitarily equivalent to the multiplication operator 2 cosh(2πbk) in Fourier space. Its spectrum is thus absolutely continuous covering the interval [2, ∞) doubly.
This paper is a contribution to the Special Issue on Mathematics of Integrable Systems: Classical and Quantum in honor of Leon Takhtajan.
The full collection is available at https://www.emis.de/journals/SIGMA/Takhtajan.html Let V ≥ 0, V ∈ L 1 (R) now be a real-valued potential function. The scalar inequality 2 cosh(2πbk) − 2 ≥ (2πbk) 2 implies the operator inequality on dom(W 0 (b)). By Sobolev's inequality, we can conclude that the operator is symmetric and bounded from below on the common domain of W 0 (b) and V . We can thus consider its Friedrichs extension, which we continue to denote by W V (b). This operator acts as Furthermore, by an application of Weyl's theorem (in a version for quadratic forms) and Rellich's lemma together with the fact that the form domain of W 0 (b) is continuously embedded in H 1 (R) (as discussed at the beginning of Section 4) the spectrum of W V (b) consists of essential spectrum [2, ∞) and discrete finite-multiplicity eigenvalues below. Details of this argument in the similar case of a Schrödinger operator can be found in the upcoming book [2, Proposition 4.14]. We will show that the discrete spectrum satisfies a version of Lieb-Thirring inequalities for 1/2-Riesz means. When formulating the main result of the paper it is convenient to parametrise the eigenvalues (repeated with multiplicities) as Note that in the latter case λ j = −2 cosh(|ω j |).
The constant in the inequality (1.2) is sharp in the sense that there is a potential V such that (1.2) becomes equality.
Remark 1.2. Note that Theorem 1.1 does not allow to estimate eigenvalues below −2. In fact, from the proof of this theorem, the case of one eigenvalue below −2 could be included in the inequality (1.2). We expect that the inequality holds true for all eigenvalues below −2. However, the method we use in the proof prevents us from including all eigenvalues due to oscillating properties of the resolvent Lieb-Thirring inequalities were first established for Schrödinger operators in [15]. For a onedimensional Schrödinger operator − d dx 2 −V on L 2 (R) with negative eigenvalues µ 1 ≤ µ 2 ≤ · · · < 0, these bounds state that for any γ ≥ 1/2 there is a constant L γ > 0 such that for all V ≥ 0, V ∈ L γ+1/2 (R). The condition γ ≥ 1/2 is optimal. Inequality (1.1) implies that for all eigenvalues λ j ≤ 2 of W V (b). Under the additional assumption W V (b) ≥ −2, our bound (1.2) presents an improvement of (1.4) for γ = 1/2. This can be seen from the fact that for γ = 1/2 the sharp constant in (1.3) is given by L 1/2 = 1/2 [7] and from the strict inequality for ω j ∈ [0, π). The difference of the terms above vanishes as ω j → π, implying that (1.4) is asymptotically optimal for small coupling. While the necessity of γ ≥ 1/2 in the Lieb-Thirring inequality for Schrödinger operators does not allow us to conclude that (1.4) fails for 0 ≤ γ < 1/2, we will prove the following.
holds for all compactly supported V . This conclusion holds even under the assumption that The study of different properties of functional difference operators W V (b) was considered before. In the case when −V = V 0 = e 2πbx is an exponential function, the operator W V 0 (b) first appeared in the study of the quantum Liouville model on the lattice [1] and plays an important role in the representation theory of the non-compact quantum group SL q (2, R). The spectral analysis of this operator was first studied in [9], see also [17]. In the case when −V = 2 cosh(2πbx) the spectrum of W V (b) is discrete and converges to +∞. Its Weyl asymptotics were obtained in [13]. This result was extended to a class of growing potentials in [14]. More information on spectral properties of functional difference operators can be found in papers [4,5,10,11,16].
The proof method of Theorem 1.1 is similar to the proof of the sharp Lieb-Thirring inequality (1.3) for a one-dimensional Schrödinger operator in the case γ = 1/2 as presented in [6]. It relies on a property of convolutions of the resolvent kernels of the operator under consideration. Such a semigroup property was also recently established for Jacobi operators where it was again used to prove sharp Lieb-Thirring type inequalities [12]. With a different proof (not using the convolution property) the sharp inequalities for the Schrödinger operator and the Jacobi operator were first obtained in [7] and in [8], respectively. Despite formal similarity to the case of Jacobi operators, it is still surprising that the proof method works for functional difference operators W V (b). These operators could be considered as differential operators of infinite order since the symbol cosh(2πbk) can be written as an infinite Taylor series of symbols of even degree w.r.t. the variable k.
Applying the inverse Fourier transform F −1 to (2 cosh(2πbk) + 2 cos(ω)) −1 we find the kernel of the free resolvent . (2.1) is an even and positive kernel for ω ∈ [0, π] and it becomes oscillating if ω ∈ i(0, ∞). This fact is one of the reasons why we are able to study Lieb-Thirring inequalities only for the eigenvalues λ j ∈ [−2, 2]. This interval contains all of the discrete spectrum if the potential V is "small" enough. However, if V generates eigenvalues lying in (−∞, −2), then the oscillating property of the Green's function prevents us from obtaining the desired inequality for all eigenvalues.
Note that the value of G λ on the diagonal x = y takes the form and we can see the relation between the right-hand side of (2.2) and the expression in the left-hand side of (1.2). Due to our parameterisation of the spectral parameter, the convergence λ → 2 − implies ω → π − and thus If λ → −∞, then ω → i∞ and In [17] L. Faddeev and L.A. Takhtajan studied the resolvent in a slightly different form which coincides with (2.1) with σ = i/2b, λ = 2 cosh(2bπκ) and κ = ω−π 2πib . It was pointed out that the free resolvent can be written using the analogues of the Jost solutions that appear in the theory of one-dimensional Schrödinger operators. Namely , where σ σ = −1/4 and where C(f, g) is the so-called Casorati determinant (a difference analogue of the Wronskian) of the solutions of the functional-difference equation For the Jost solutions C(f − , f + )(x, κ) = 2 sinh πiκ σ . The equality (W 0 (b) − λ)G(x − y) = δ(x − y) could be interpreted as an equation of distributions. Since the functions f ± (x, k) are Jost solutions, the distribution defined by (W 0 (b) − λ) × G(x − y) is supported only at x = y, and its singular part coincides with the singular part of the distribution in the neighbourhood of x = y. This singular part is equal to where the authors used the Sokhotski-Plemelj formula. This formula is similar to the respective formula for a Schrödinger operator when the Dirac δ-function appears by differentiating a step function.

Some auxiliary results
We first collect some results from [6] verbatim. Let A be a compact operator on a Hilbert space G and let us denote where λ j (A * A) are the eigenvalues of A * A in decreasing order. Then by Ky Fan's inequality (see for example [3, Lemma 4.2]) the functionals · n , n = 1, 2, . . . , are norms and thus for any unitary operator Y in G we have Proof . This is a simple consequence of the triangle inequality Let λ j = −2 cos ω j ≤ 2 be the eigenvalues of W 0 (b) − V with V ≥ 0. In order to slightly simplify the notations it is convenient to write λ j = −2 cos θ j with θ j ∈ −∞, π 2 and ω 2 j = θ j . Let us denote by K λ the Birman-Schwinger operator Let µ j (K λ ) be the eigenvalues (in decreasing order) of the Birman-Schwinger operator K λ defined in (3.1). Then due to the Birman-Schwinger principle we have Let us define the operator This is the interesting convolution/semigroup property mentioned in the introduction. In the special case −∞ < θ < 0 = θ analogous computations lead to the same result with g 0,θ (k) = χ [−1,1] 2πbk/ |θ| πb/ |θ|.  For (θ, θ ) such that −∞ < θ ≤ θ and 0 ≤ θ < π 2 we have L θ ≺ L θ .
Proof . Let Y (k) : L 2 (R) → L 2 (R) be the unitary multiplication operator (Y (k)ψ)(x) = e −2πikx ψ(x) and let T be the projection onto V 1/2 , i.e., Using Y (k + k ) = Y (k )Y (k ) and Lemma 3.2 we obtain where we have used that g θ ,θ dk is a probability measure.

Proof of inequality (1.2)
We now enumerate the eigenvalues of the operator W V (b) belonging to the interval [−2, 2) such that −2 ≤ λ 1 ≤ λ 2 ≤ λ 3 ≤ · · · repeated with multiplicity. By using the monotonicity established in Lemma 3.3 we have a sequence of inequalities Note that we do not use any assumptions on the multiplicities of the eigenvalues, other than their finiteness. Furthermore, by Lemma 3.3 the same results also hold true if a single eigenvalue is below −2. Continuing the above process and noting that the trace of L θ is R V dx for all θ, we finally obtain The proof is complete.

Sharpness of inequality (1.2)
Similarly to the case of Schrödinger operators, we aim to prove that the Lieb-Thirring inequality becomes an equality for Dirac-delta potentials. To this end let c > 0 and consider the potential V c (x) = cδ(x). To properly define W Vc (b), we first note that the quadratic form ψ, (W 0 (b) − 2)ψ can be written as This can be seen by introducing the self-adjoint operator D(b) = U (b/2)−U (b/2) −1 = 2 sinh bP 2 and checking that D(b) 2 = W 0 (b) − 2 either directly or by means of the identity cosh(2πbk) − 1 = 2 sinh(πbk) 2 . The form domain of W 0 (b) is thus dom(D(b)) = dom(W 0 (b/2)) ⊂ H 1 (R) and on this domain Sobolev's inequality yields that for any choice of ε > 0. The KLMN theorem thus allows us to define W 0 (b) − V c . As a rank one perturbation of the operator W 0 (b) the potential V c generates no more than one eigenvalue below the continuous spectrum [2, ∞).
by means of the formal identity F(δψ c ) = ψ c (0). Writing again λ = −2 cos ω we obtain and therefore Of course we could have seen this immediately by using the equation for the Green's function . Since sin √ θ √ θ is a monotone decreasing function of θ = ω 2 ∈ −∞, π 2 that takes all values in [0, ∞), for any c > 0 there is a unique solution ω c to (4.4) and vice versa. If c/(2πb) < 1 then ω c ∈ (0, π) and otherwise ω c ∈ i[0, ∞). Since V c dx = c, the identity (4.4) can be rewritten as showing that the Lieb-Thirring inequality is satisfied for potentials −cδ with a single eigenvalue that can be placed anywhere in (−∞, 2) by choosing c > 0 suitably.
is positive assuming that the coupling constant c is small enough satisfying the inequality c/(2πb) ≤ 1 and thus ω c ∈ [0, π). Note that if c/(2πb) = 1 then ω c = 0 and However, if the coupling constant c > 2πb then ω c ∈ i(0, ∞) and hence x is an oscillating function and in particular has an infinite number of zeros. This contradicts a possible conjecture that the eigenfunction for the lowest eigenvalue is strictly positive.
By the min-max principle this proves the first part of Theorem 1.3.