EIGENVALUE BOUNDS FOR SCHRÖDINGER OPERATORS WITH RANDOM COMPLEX POTENTIALS

We consider the Schrödinger operator perturbed by a random complex-valued potential. We obtain an estimate on the rate of accumulation of the eigenvalues of this operator to the positive half-line. 1. Main result In this paper, we study the behavior of eigenvalues of the operator H = −∆ + V. The potential V is assumed to be a complex-valued function of the form V = ∑ n∈Zd ωnvnχ(x− n), vn ∈ C, x ∈ R, where ωn are independent random variables taking values in the interval [0, 1] and χ is the characteristic function of the unit cube [0, 1). We impose the condition E[ωn] = 0 on ωn guaranteeing oscillations of V . The coefficients vn do not have to be real. To formulate the main result, we set Ṽ = ∑


Main result
In this paper, we study the behavior of eigenvalues of the operator H = −∆ + V. The potential V is assumed to be a complex-valued function of the form where ω n are independent random variables taking values in the interval [0, 1] and χ is the characteristic function of the unit cube [0, 1) d . We impose the condition E[ω n ] = 0 on ω n guaranteeing oscillations of V . The coefficients v n do not have to be real.
To formulate the main result, we set Note thatṼ is a non-negative function such that |V | Ṽ .
It is assumed that Im λ j 0. The constant C in (1.1) depends only on d, ν and q.
It is known that, if v n ∈ R, then the eigenvalues λ j obey the Lieb-Thirring estimate (1.2) Theorem 1.1 alows one to consider real potentials V for which the right hand side of (1.2) is infinite for while the left hand side is finite almost surely. Similar results for real random potentils V =V were obtained earlier in [5]. However, there is a big difference between Theorem 1.1 and the results of [5], since the only point of accumulation of eigenvalues of the operator H in the case considered by the authors of [5] is the point λ = 0. When one studies complex-valued potentials, the fact that the eigenvalues λ j might accumulate to points other than λ = 0 should not be excluded. Examples of decaying complex potentials V such that eigenvalues of H = −∆ + V accumulate to points of the positive real line R + are constructed in [1]. Because of the difference between the cases of real and complex potentials, it would be more appropriate to ask what new information does Theorem 1.1 provide compared to [4] and [3], rather than realize that this theorem does not follow from the Lieb-Thirring estimate. The next statement is an improvement of Theorem 1.1 for 3 d 5.
Theorem 1.2. Let 3 d 5 and let 0 < R 0 1. Assume that τ 1 satisfies with η and ν such that 1 < ν < η < 2. If d = 3, then we assume additionally that 8ν +9η < 26. Let p, q and r be the numbers defined by where θ is the solution of the equation Then the eigenvalues λ j of the operator −∆ + V satisfy Besides its dependence on d, the constant C τ 1 ,σ in this inequality depends on a choice of the parameters τ 1 and σ.
Theorem 1.2 gives new information about eigenvaues of H. Even in the case V =V , this theorem does not follow from the Lieb-Thirring estimates. It turns into Theorem 1.1 for dimensions 3 d 5 once we set τ 1 = 0. However, Theorem 1.2 allowes one to consider ratios σ/r which are smaller than the ratios q/p allowed in Theorem 1.1. Thus, Theorem 1.2 is an improvement of Theorem 1.1 for dimensions 3 d 5.
One of the difficulties we encountered in this paper is that Theorem 1.1 and Theorem 1.2 can not be obtained by taking expectations in the inequalities obtained by Borichev, Golinski, and Kupin [2]. This difficulty was overcome through an application of the Joukowsky transform to a halh-plane with a removed semi-disk and consecutive integration with respect to the radius.

Preliminaries
Everywhere below, S p denotes the class of compact operators K obeying Note that if K ∈ S p for some p > 1, then K ∈ S q for q > p and K q K p . Let z j be the eigenvalues of a compact operator K ∈ S n where n ∈ N \ {0}. We define the n-th determinant of I + K by There exists a constant C n > 0 depending only on n such that det n (I + X) e Cn X n n , ∀X ∈ S n .
Moreover, the following statement holds: Proposition 2.1. Let n 2. Then for any n − 1 p n, there exists a constant C p,n > 0 depending only on p and n such that Let Let us also set D n (k) = det n (I − X(k)), n = [p] + 1, for a compactly supported V .
The algebraic multiplicity of the eigenvalue λ does not exceed the multiplicity of the root of the function D n (·).
Proof. The statement about the multiplicity follows from the fact that eigenvalues of H can be turned into simple eigenvalues by a small perturbation (which does not have to be a function). The rest is implied by the Birman-Schwinger principle, saying that a point λ is an eigenvlue of H if and only if −1 is an eigenvalue of |V | 1/2 (−∆ − λ) −1 V |V | −1/2 . Therefore, 1 is an eigenvalue of X(k).

Operators of the Birman-Schwinger type
Let a, b and V be functions on R d . Define where F is the unitary Fourier transform operator. For any complex number z, we understand V z as the sum V z (x) := n ω n |v n | z e i arg vn χ(x − n).
Also, we will use the following notatioñ V = n |v n |χ(x − n).
Proof. We are going to prove (3.3) for one point ζ 0 such that Re ζ 0 = 2/p. For that purpose, we define the operator K(ω) = |A ζ 0 | p/2 . Then, obviously, Sp . Let Ω = Ω(ω) be the partially isometric operator appearing in the polar decomposition of A ζ 0 We introduce the analytic function which will be treated by the three lines lemma. Since ||A ζ || 1 for Re ζ = 0, we obtain that On the other hand, by Hölder's inequality. Using the three lines lemma, we obtain from (3.4) and (3.5) that Note now that f (ζ 0 ) = β. Consequently, Corollary 3.2. Let T be a random operator of the form Let a ∈ L p , b ∈ L p , v n ∈ p and p 2. Then Proof. Observe that the functions |a| p/2 , |b| p/2 and V p/2 belong to L 2 . Therefore, according to the proposition, the S p -norm of the operator K = |a| pζ/2 F V pζ/2 F * |b| pζ/2 obeys the inequality  The following result is a very well known bound obtained by E. Seiler and B. Simon [6]. Moreover, the reader can easily prove it using standard interpolation. Proposition 3.3. Let a and W be two functions on R d . Let T be the operator where F is the operator of Fourier transform. Then Corollary 3.4. Let q p 2. Let T be a random operator of the form Let a ∈ L p , b ∈ L q and v n ∈ p . Then Proof. According to Proposition 3.3, On the other hand, according to Corollary 3.2, It remains to interpolate between the two cases.
For that purpose, we introduce the function where K = |a|F V F * |b| and Ω is the partially isometric operator appearing in the polar decomposition |a|F V F * |b| = ΩK. For our convinience, we denote β := E Tr K p ) q/p If Re ζ = 0, then by Hölder's inequality, Observe also that f (p/q) = β.
Thus, by the three lines lemma, we obtain that The proof is completed. 2

Small values of Re ζ
The arguments of this paper will often rely on the properties of the operator (−∆ − z) −ζ for different values of ζ. In this section, we discuss the case 0 Re ζ < 1.
Let R > 0. Let χ 0,k be the characteristic function of the ball and let χ 1,k = 1 − χ 0,k be the characteristic function of its complement We introduce the operators P n,k = F χ n,k F * , which are the spectral projections of −∆ corresponding to the intervals [0, 4|k| 2 ] and (4|k| 2 , ∞). For any complex number z, we understand V z as the sum In the next two propositions, we discuss the properties of the operators for Re ζ = γ/2 and 0 < γ < 3/2. We will study the spectral properties of the operator However, the terms in this representation will be studied separately. A this point, we do not discuss X 1,1 (ζ) at all. For the sake of convenience, we set Proof. This statement follows from Corollary 3.2 and Proposition 3.3. If r = q/2 = 2p, then 1/r + 2/q = 1/p. Moreover, since It remains to realize that while a similar argument shows that Proof. Since we obtain the estimate It remains to realize that Let us now talk about the operator The following estimate plays a very important role in our rguments.
In particular, we can set p = 1 and prove the following statement. Then

Large values of Re ζ
The arguments of this paper will often rely on the properties of the operator (−∆ − z) −ζ for different values of ζ. In this section, we discuss the case (d − 1)/2 Re ζ (d + 1)/2.
Let R > 0. Let χ 0,k be the characteristic function of the ball and let χ 1,k = 1 − χ 0,k be the characteristic function of its complement We introduce the operators P n,k = F χ n,k F * , which are the spectral projections of −∆ corresponding to the intervals [0, 4|k| 2 ] and (4|k| 2 , ∞). For any complex number z, we understand V z as the sum V z (x) := n ω n |v n | z e i arg vn χ n (x).
The following proposition plays an important role in our arguments. The proof of this proposition, as well as related references, can be found in [4].
for z / ∈ R + and j = 0, 1. The positive constants β and α in this inequality depend only on d and Re ζ.

An estimate for the square of the Birman-Schwinger operator
To obtain our first result about eigenvalues, we can interpolate between the case V ∈ L ∞ for Re ζ = 0 and the case V ∈ L 2 for Re ζ described in Corollary 5.5. Let If κ = (d + 1)/2 and d 3, then (6.25) holds with p = κ.
Let d 3. Fix the value of ν so that 1 < ν < 2 and choose q so that ν < q < 2. Let us now define p by .
Observe that the assumption ν < q < 2 leads to the inequalities We also introduce the parameter κ setting The latter relation simply means that The second inequality in (6.22) implies while the first inequality in (6.22) combined with the condition ν < 2 implies that One can also see that the first inequality in (6.24) is equivalent to the estimate Finally, note that in d 3, the condition p < 2κ follows from the fact that ν + q > 2.
7. Proof of Theorem 1.1 We will work with the function where z is related to k via the Joukowski mapping which maps the set {k ∈ C : Im k > 0, |k| > R} onto the upper half-plane {z ∈ C : Im z > 0}. Rather standard arguments lead to the estimate where z j are zeros of the function d(z) situated in the upper half-plane C + . In fact, (7.26) could be established in the same way as Jensen's inequality for zeros of an analytic function on a unit disc. In (7.26) we assume that V is compactly supported. The relation (7.26) leads to the estimate Taking the expectation we obtain Due to Theorem 6.2, the latter inequality leads to Now, suppose that we consider only the eigenvalues λ j = k 2 j that satisfy the inequality |k j | R 0 .
Multiplying (9.42) by R q−1 and integrating with respect to R from 0 to R 0 , we obtain This implies Theorem 1.1
By small values of Re ζ we mean the values that are considered in Corollary 4.3, which states that, for any p 1 and d/(8p) < Re ζ < 3/(4p), In this corollary we had to assume that 2 d 5 and |k| R where 0 < R 1. One should not forget also that our assumptions about γ = 2 Re z imply that Re z < 3/4. The next result follows once we change 4 Re z to d/(2p) in the right hand side of (8.29).
for |k| R.
For the sake of simplicity, we choose In this case, because of the assumption p 1 that we made, we have to assume that 0 < Re ζ d 7 .
By the large values of Re ζ we mean the values appearing in Corollary 5.7. We will use only a simpler version of this result. Let d 3. Fix the value of ν so that 1 < ν < 2 and choose η so that ν < η < 2. Let us now consider Re ζ given by .
Observe that the assumption ν < η < 2 leads to the inequalities We also introduce the parameter κ setting The latter relation simply means that The second inequality in (8.31) implies while the first inequality in (8.31) combined with the condition ν < 2 implies that κ < d + 1 2 .
One can also see that the first inequality in (8.33) is equivalent to the estimate Finally, note that in d 3, the condition Re ζ < κ follows from the fact that ν + η > 2.
Consequently, Corollary 5.7 implies the following statement: Assume that V ∈ L 2 (R d ) and α 0 > 2α. Then (8.34) We will interpolate between Corollary 8.2 and Theorem 8.3. Let us choose now τ 1 so that Consider now Y γ (ζ) for After that we find θ ∈ (0, 1) satisfying the equation and define q and r so that Since p = d 7τ 1 , we obtain by interpolation that To guarantee integrability of Sq at infinity , we need the parameters to satisfy the condition which is equivalent to the inequality implying that The latter can be written in the form Put differently, The condition that θ is large can be converted into an inequality showing that τ 1 is small. The relation (8.35) is satisfied, if Since η > ν, that condition is obviously fulfilled, if In this case, we also have if 8ν + 9η < 26, and d = 3, Let us try to formulate the result.
Theorem 8.4. Let 3 d 5. Assume that τ 1 satisfies (8.36) with η and ν such that 1 < ν < η < 2. If d = 3, then we assume additionally that 8ν + 9η < 26. Let p, q and r be the numbers defined by where θ is the solution of the equation In the next statement, we estimate the remainder X 1,1 (ζ) for ζ = 1.
Theorem 8.5. Let p > 3d/4 2 and let ζ = 1. Then Proof. In this theorem, we deal with the operator On the one hand, we see that which implies the inequality On the other hand, Consequently, The next statement follows by Hölder's inequality.
Theorem 8.7. Let 3 d 5. Assume that τ 1 satisfies (8.36) with η and ν such that 1 < ν < η < 2. If d = 3, then we assume additionally that 8ν + 9η < 26. Let p, q and r be the numbers defined by where θ is the solution of the equation where z j are zeros of the function d(z) situated in the upper half-plane C + . In fact, (9.41) could be established in the same way as Jensen's inequality for zeros of an analytic function on a unit disc. In (9.41) we assume that V is compactly supported. The relation (7.26) leads to the estimate Taking the expectation we obtain E[ X(R · e iθ ) q q ] sin θdθ .
Due to Theorem 6.2, the latter inequality leads to j E Im k j (|k j | 2 − R 2 ) + |k j | 2 R C|R| −θqν/2 Ṽ 2q r . (9.42) Now, suppose that we consider only the eigenvalues λ j = k 2 j that satisfy the inequality |k j | R 0 .