Trace formulae for Schr\"odinger operators with complex-valued potentials on cubic lattices

We consider Schr\"odinger operators with complex decaying potentials on the lattice. Using some classical results from Complex Analysis we obtain some trace formulae and using them estimate globally all zeros of the Fredholm determinant in terms of the potential.


Introduction
Let us consider the Schrödinger operator H acting in ℓ 2 (Z d ), d 3 and given by where ∆ is the discrete Laplacian on Z d given by . Here e 1 = (1, 0, · · · , 0), · · · , e d = (0, · · · , 0, 1) is the standard basis of Z d . The operator V = (V n ) n∈Z d , V n ∈ C, is a complex potential given by We assume that the potential V satisfies the following condition: V ∈ ℓ 2/3 (Z d ). (1.1) Note that the condition (1.1) implies that V can be factorised as with V 1 = |V | 2/3−1 V and V 2 = |V | 1/3 . Here ℓ q (Z d ), q > 0 is the space of sequences f = (f n ) n∈Z d such that f q < ∞, where Note that ℓ q (Z d ), q 1 is the Banach space equipped with the norm · q . It is well-known that the spectrum of the Laplacian ∆ is absolutely continuous and equals It is also well known that if V satisfies (1.1), the essential spectrum of the Schrödinger operator H on ℓ 2 (Z d ) is However, this condition does not exclude appearance of the singular continuous spectrum on the interval [−d, d]. Our main goal is to find new trace formulae for the operator H with complex potentials V and to use these formulae for some estimates of complex eigenvalues in terms of potentials. Note that some of the results obtained in this paper are new even in the case of real-valued potentials due to the presence of the measure σ (see Theorem 2.3) appearing in the canonical factorisation of the respective Fredholm determinants. Non-triviality of such a measure is due to the weak condition (1.1) on the potential V . We believe that it would be interesting to study the relation between properties of V v and σ.
Recently uniform bounds on eigenvalues of Schrödinger operators in R d with complex-valued potentials decaying at infinity attracted attention of many specialists. We refer to [6] for a review of the state of the art of non-selfadjoint Schrödinger operators and for motivations and applications. Bounds on single eigenvalues were proved, for instance, in [1,7,16,11] and bounds on sums of powers of eigenvalues were found in [13,25,8,9,4,15,12]. The latter bounds generalise the Lieb-Thirring bounds [26] to the non-selfadjoint setting. Note that in [15] (Theorem 16) the authors obtained estimates on the sum of the distances between the complex eigenvalues and the continuous spectrum [0, ∞) in terms of L p -norms of the potentials. Note that almost no results are known on the number of eigenvalues of Schrödinger operators with complex potentials. We referee here to a recent paper [14] where the authors discussed this problem in details in odd dimensions.
For the discrete Schrödinger operators most of the results were obtained in the self-adjoint case, see, for example, [36] (for the Z 1 case). Schrödinger operators with decreasing potentials on the lattice Z d have been considered by Boutet de Monvel-Sahbani [5], Isozaki-Korotyaev [20], Kopylova [23], Rosenblum-Solomjak [29], Shaban-Vainberg [34] and see references therein. Ando [2] studied the inverse spectral theory for the discrete Schrödinger operators with finitely supported potentials on the hexagonal lattice. Scattering on periodic metric graphs Z d was considered by Korotyaev-Saburova [24]. Isozaki and Morioka (see Theorem 2.1. in [21]) proved that if the potential V is real and compactly supported, then the point-spectrum of H on the interval (−d, d) is absent. Note that in [10] the author gave an example of embedded eigenvalue at the endpoint {±d}.
In this paper we use classical results from Complex Analysis that lead us to a new class of trace formula for the spectrum of discrete multi-dimensional Scrödinger operators with complex-valued potentials. In particular, we consider a so-called canonical factorisation of analytic functions from Hardy spaced via its inner and outer factors, see Section 6. Such factorisations allied for Fredholm determinants allow us to obtain trace formula that lead to some inequalities on the complex spectrum in terms of the L 2/3 norm of the potential function. Note also that in the case d = 3 we use a delicate uniform inequality for Bessel's functions obtained in Lemma 7.1.

Some notations and statements of main results
We denote by D r (z 0 ) ⊂ C the disc with radius r > 0 and center z 0 ∈ C D r (z 0 ) = {z ∈ C : |z − z 0 | < r}, and abbreviate D r = D r (0) and D = D 1 . Let also T = ∂D. It is convenient to introduce a new spectral variable z ∈ D by The function λ(z) has the following properties: • λ(z) maps z = 0 to λ = ∞.
• The inverse mapping z(·) : Λ → D is given by Next we introduce the Hardy space H p = H p (D). Let F be analytic in D. For 0 < p ∞ we say F belongs the Hardy space H p if F satisfies F Hp < ∞, where F Hp is given by Let B denote the class of bounded operators and B 1 and B 2 be the trace and the Hilbert-Schmidt class equipped with the norm · B 1 and · B 2 respectively.
Theorem 2.1. Let a potential V satisfy (1.1). Then the determinant D(z) = det(I + V R 0 (λ(z)), z ∈ D, is analytic in D. It has N ∞ zeros {z j } N j=1 , such that Moreover, it satisfies D H∞ e C V 2/3 , (2.5) where the constant C depends only on d.
Furthermore, the function log D(z) whose branch is defined by log D(0) = 0, is analytic in the disk D r 0 with the radius r 0 > 0 defined by (2.4) and it has the Taylor series as |z| < r 0 :
Define the Blaschke product B(z), z ∈ D by ii) The determinant D has the factorization in the disc D: where D B is analytic in the unit disc D and has not zeros in D.
iii) The Blaschke product B has the Taylor series at z = 0: where B n satisfy The next statement describes the canonical representation of the determinant D(z). Theorem 2.3. Let a potential V satisfy (1.1). Then i) There exists a singular measure σ 0 on [−π, π], such that the determinant D has a canonical factorization for all |z| < 1 given by where log |D(e it )| ∈ L 1 (−π, π).
ii) The measure σ satisfies
3) The closure of the set {z j } ∪ supp σ is called the spectrum of the inner function D in .

Remark.
Note that some of the results stated in Theorems 2.4 and 2.5 are new even for real-valued potentials, see Section 5.

3.1.
Properties of the Laplacian. One may diagonalize the discrete Laplacian, using the (unitary) Fourier transform Φ : Here (·, ·) is the scalar product in R d . In the so-called momentum representation of the operator H, we have: and the potential V becomes a convolution operator

Trace class operators.
Here for the sake of completeness we give some standard facts from Operator Theory in Hibert Spaces.. Let H be a Hilbert space endowed with inner product ( , ) and norm · . Let B 1 be the set of all trace class operators on H equipped with the trace norm · B 1 . Let us recall some well-known facts.
where the right-hand side is absolutely convergent for |λ| > r 1 , r 1 > 0 being a sufficiently large constant. In particular,
Theorem 3.2. Let V satisfy (1.1). Then the operator-valued function Y (λ(z)) : D → B 1 is analytic in the unit disc D and satisfies Moreover, the function D(z), z ∈ D belongs to H ∞ and Proof. The operator V 2 belongs to B 2 and due to Theorem 7.3 (see Appendix 2) the operatorfunction X(·) : Λ → B 2 satisfies the inequality Thus the operator-valued function Y (λ(z)) = V 2 X(λ(z)) : D → B 1 is of trace class. Moreover, the function D(z), z ∈ D, belongs to H ∞ and due to (3. and where a = 2 d and the coefficients d j are given by (3.2).

Proof of the main results
We are ready to prove main results. (1 − |z j |) < ∞ and the Blaschke product B(z) given by converges absolutely for {|z| < 1}. We have D(z) = B(z)D B (z), where D B is analytic in the unit disc D and has no zeros in D. Thus we have proved ii). i) Lemma 6.1 gives (2.11) and (2.12).
iii) For small a sufficiently small z and for t = z j ∈ D for some j we have the following identity: Besides, where the function b is analytic in the disk {|z| < r 0 2 } and B n satisfy Proof of Theorem 2.3. i) Theorem 2.1 implies D ∈ H ∞ . Therefore the canonical representation (2.14) follows from Lemma 6.3.
In order to determine the next two estimates we use the trace formula (2.16). Let Im V 0. Then Im λ j 0 and the estimates (2.5) and (2.21) and the second trace formula (2.20) which yields (2.22). Similar arguments give (2.23).

Schrödinger operators with real potentials
Consider Schrödinger operators H = −∆ + V , where the potential V is real and satisfies condition (1.1) . The spectrum of H has the form Note that each eigenvalues of H has a finite multiplicity.
The corresponding point from z j ∈ D is real and satisfy Moreover, we have the identity for all λ ∈ Λ and z ∈ D. If λ is the eigenvalue of H, then we have the identity Proof. The eigenvalue of H, then we have the identity Remark. 1) We consider the case (5.2). If λ > d, then we have z ∈ (0, 1) and then This yields Corollary 5.2. Let a potential V be real and satisfy (1.1). Then the following estimates hold true:

6.
Appendix, Hardy spaces 6.1. Analytic functions. We recall the basic facts about the Blaschke product (see pages 53-55 in [17]) of zeros {z n }. The subharmonic function v(z) on Ω has a harmonic majorant if there is a harmonic function U(z) such that v(z) U(z) throughout Ω. We need the following well-known results, see e.g. Sect. 2 from [17].
Lemma 6.1. Let {z j } be a sequence of points in D \ {0} such that (1 − |z j |) < ∞ and let m 0 be an integer . Then the Blaschke product converges in D. Moreover, the function B is in H ∞ and zeros of B are precisely the points z j , according to the multiplicity. Moreover, Let us recall a well-known result concerning analytic functions in the unit disc, e.g., see Koosis page 67 in [22]. Theorem 6.2. Let f be analytic in the unit disc D and let z j = 0, j = 1, 2, ..., N ∞ be its zeros labeled by 0 < |z 1 | ... |z j | |z j+1 |z j+2 | .... Suppose that f satisfies the condition sup r∈(0,1) 2π 0 log |f (re iϑ )|dϑ < ∞.
The Blaschke product B(z) given by where m is the multiplicity of B at zero, converges absolutely for {|z| < 1}. Besides, f B (z) = f (z)/B(z) is analytic in the unit disc D and has no zeros in D.
We define the functions (after Beurling) in the disc by the outer f actor of f, for |z| < 1. Note that we have |f in (z)| 1, since dσ 0. Thus f B (z) = f (z) B(z) has no zeros in the disc D and satisfies where the measure µ equals For a function f continuous on the disc D we define the set of zeros of f lying on the boundary ∂D by S 0 (f ) = {z ∈ S : f (z) = 0}. It is well known that the support of the singular measure σ = σ f satisfies see for example, Hoffman [19], p. 70.
In the next statement we present trace formulae for a function f ∈ H p , p > 0.
Lemma 6.4. Let f ∈ H p , p > 0 and f (0) = 1 and let B be its Blaschke product. Let the functions log f and F = log f B have the Taylor series in some small disc D r , r > 0 given by

Appendix, estimates involving Bessel's functions
In order complete the proof of Theorem 3.2 we need some uniform estimates for the Bessel functions J m , m ∈ Z with respect to m for which we their integral representation Note that for all (t, m) ∈ R × Z: Our estimates are based on the following three asymptotic formulae, see [35], Ch IV, § 2. Let Then for a fixed ε, 0 < ε < 1 we have Therefore if ξ > 1 + ε then this formula implies the uniform with respect to m exponential decay of the Bessel function in t.
In this case the Bessel function oscillates as t → ∞ and obviously the latter formula implies the uniform with respect to m estimate 3. We now consider the third case 1 − ε ξ 1 + ε which is more difficult.
Proof. If 1 − ε ξ 1 + ε, then (see [35], Ch IV, § 2) where v is the Airy function and Besides the functions c 0 (ξ) and d 0 (ξ) are bounded with respect to ξ and, for example, , (see [28] formulae ( In what follows all the constants depend on ε, 0 < ε < 1, but not on m and t. Due (7.7) there are constants c and C such that Applying estimates for the Airy functions in (7.6) and using (7.8), (7.9) we find that if |t| 1 |J m (t)| C 1 (1 + |y|) The proof is complete.
Let us now consider the operator e it∆ , t ∈ R. It is unitary on L 2 (S d ) and its kernel is given by Then where |n| = |n 1 | + .... + |n d |. Moreover, the following estimates are satisfied: for all (t, n) ∈ R × Z d and some constants C 1 = C 1 (d) and C 2 = C 2 (d).
which yields (7.11) for d = 1. Due to the separation of variables we also obtain (7.11) for any d 1.
ii) Consider the case C − , the proof for C + is similar. We have the standard representation of the free resolvent R 0 (λ) in the lower half-plane C − given by and summing results for R 01 and R 02 we obtain (7.16).