OLEG SAFRONOV 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 = ∑


Introduction and main results
We study the behavior of eigenvalues of the operator H = − + V acting on a Hilbert space L 2 ‫ޒ(‬ d ), where d ≥ 3. The potential V is assumed to be a complex-valued function of the form where the ω n are independent random variables taking values in the interval [−1, 1] and χ is the characteristic function of the unit cube [0, 1) d .
The probability space in our theorems is the set of all infinite sequences ω = {ω n } n∈‫ޚ‬ d . The probability measure is defined on as the infinite product of corresponding measures on intervals [−1, 1].
Since ω n can be viewed as a function on whose value is equal to the n-th coordinate of ω, its expectation ‫[ޅ‬ω n ] can be viewed as an integral over . We impose the condition ‫[ޅ‬ω 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 that V is a nonnegative function such that |V | ≤ V.

1.2)
It is assumed that Im λ j ≥ 0. The constant C in (1.1) depends only on d, ν and q.
Theorem 1.1 is a particular case of the following statement, which has rather complicated looking conditions imposed on the parameters.
Assume also that V ∈ L p ‫ޒ(‬ d ). Then the eigenvalues λ j of the operator − + V satisfy It is assumed that Im λ j ≥ 0. If = 1 2 (d + 1), then (1.3) holds with p = . The constant C in (1.3) depends only on d, p, and q.
It is known that, if v n ∈ ‫,ޒ‬ the eigenvalues λ j obey the Lieb-Thirring estimate (see [Helffer and Robert 1990;Laptev and Weidl 2000;Lieb and Thirring 1976 (1.4) Theorem 1.1 allows one to consider real potentials V for which the right-hand side of (1.4) is infinite, while the left-hand side is finite almost surely. Indeed, let 1 < 2γ = q < d/(d − 1). Then the parameter p in (1.2) satisfies the inequality p > 1 2 d + γ .
(1.5) Similar results for real random potentials V = V were obtained by the author and Vainberg in [Safronov and Vainberg 2008]. However, there is a big difference between Theorem 1.1 and the results of that earlier work, since the only point of accumulation of eigenvalues of the operator H considered there 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 ‫ޒ‬ + are constructed in [Bögli 2017]. Because of the difference between the cases of real and complex potentials, it would be more appropriate to ask what new information Theorem 1.1 provides compared to [Frank 2018;Frank and Sabin 2017], rather than realize that this theorem does not follow from the Lieb-Thirring estimate even in the selfadjoint case.
The related result of [Frank and Sabin 2017] says that there is a constant C that depends on d, p and γ such that , (1.6) under conditions on γ and p implying that p < γ + 1 2 d. One can now refer to (1.5) to conclude that our results do give new information about the distribution of eigenvalues in the complex plane.
The same conclusion could be made by an analysis of the results of [Frank 2018], where the eigenvalues in the disk are considered separately from the rest of the eigenvalues; here p > 1 2 d. R. Frank [2018] proves that under some restrictions on p, ( 1.7) for γ equal to either p or 2 p − d + ε. The constants C > 0 and σ > 0 depend only on d and p in the first case but also on ε > 0 in the second. In its turn, ε > 0 belongs to the interval whose size depends on p. The observation we make is that p < γ + 1 2 d in (1.7). On the other hand, in deterministic results, p simply can not be larger than γ + 1 2 d. Theorem 1.1 gives information about the eigenvalues of H situated in a finite disk about the origin. The behavior of the eigenvalues outside of this disk is described below. Theorem 1.3. Let d ≥ 3, let R > 0 and let 1 < ν < q < 2. Then the eigenvalues λ j of the operator .
It is assumed that Im λ j ≥ 0. The constant C in (1.1) depends only on d, ν and q.
According to Theorem 1.3, the condition V ∈ L p implies that, for any R > 0, almost surely. Eigenvalues of H outside a finite disk about the point z = 0 were also studied in [Frank 2018]. However, in the theorems of that work the radius R of the disk depends on V. Moreover, when d ≥ 3, these theorems guarantee convergence of |λ j |≥R 2 |Im λ j | α |λ j | −β for some α > 1 and β > 0 rather than convergence of the series (1.8).
Theorem 1.3 immediately implies the following assertion.
It is assumed that Im λ j ≥ 0. The constant C in (1.1) depends only on d, ν and q.
We also mention the article [Frank 2018] because Theorem 1.1 of that paper deals with the question about the shape of the domain containing all eigenvalues of H. In particular, it implies that the imaginary part of an eigenvalue tends to zero as the real part tends to infinity (in a quantitative way) once V ∈ L p with p > 1 2 (d + 1). Despite a vague visual resemblance of Corollary 1.4 to such a theorem, it does not give new information about the region containing all eigenvalues of H.
The next statement is an improvement of Theorem 1.1 for 3 ≤ d ≤ 5 and R 0 ≤ 1.
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 1.5 gives new information about eigenvalues 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. On the other hand, since it allows us to consider ratios σ/r smaller than ratios q/ p allowed by Theorem 1.1, Theorem 1.5 is an improvement of Theorem 1.1 for dimensions 3 ≤ d ≤ 5 and the values of the parameter R 0 < 1.
One of the difficulties we encountered in this paper is that our statements can not be derived by taking expectations in the inequalities obtained by Borichev, Golinskii and Kupin [Borichev et al. 2009]. The reason is that operators of the Birman-Schwinger type we are dealing with might have different properties for different ω. This difficulty was overcome through an application of the Joukowski transform to a half-plane with a removed semidisk and consecutive integration with respect to the radius.
Eigenvalue bounds for Schrödinger operators with complex potentials have been studied for a long time. First of all, one should mention the related work of B. Pavlov, who found sharp conditions on V guaranteeing that H has only finitely many eigenvalues in ‫ރ‬ \ ‫ޒ‬ + . In particular, this is true for the one dimensional operator on the half-line ‫ޒ‬ + (see [Pavlov 1966]) if for some constants C and c > 0.
In 2001 While we do not intend to describe all results related to the theory of operators with complex-valued potentials, we would like to mention the articles [Briet et al. 2021;Cuenin 2017;Cuenin et al. 2014;Demuth et al. 2009;Demuth and Katriel 2008;Hansmann 2011;2017;Korotyaev 2020;Korotyaev and Laptev 2018;Korotyaev and Safronov 2020;Laptev and Safronov 2009;Pavlov 1967] in addition to those already mentioned, all of which could be viewed as valuable contributions in this area.

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 ∈ ‫ގ‬ \ {0}. We define the n-th determinant of I + K as There exists a constant C n > 0 depending only on n such that Moreover, we have the following statement; see Proposition 2.1 of [Korotyaev and Safronov 2020].
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 The way the eigenvalue bounds are obtained in [Korotyaev and Safronov 2020] uses applications of the following abstract result.
Theorem 2.2. Let H 0 be a selfadjoint operator on a Hilbert space H. Let W 1 and W 2 be two bounded operators on H, and let V = W 2 W 1 . Assume that the function is analytic in the upper half-plane ‫ރ‬ + = {z ∈ ‫ރ‬ : Im z > 0} and continuous up to the real line ‫.ޒ‬ Assume also that where C p depends only on the parameter p.
Proof. The proof of this statement relies on Jensen's inequality for zeros of an analytic function, which is (also) justified in Proposition 3.11 of [Korotyaev and Safronov 2020]. □ Proposition 2.3. Let a(z) be an analytic function on ‫ރ‬ + satisfying the condition Assume that for some γ > 0, Then zeros of a(z) situated above the line Im z = γ satisfy the inequality The statement also holds for γ = 0, if a(z) is continuous up to the real line ‫.ޒ‬ The bound (2.3) follows from (2.1) and the estimate (2.4) with γ = 0 once we set According to the Birman-Schwinger principle, z is an eigenvalue of H 0 + V if and only if a(z) = 0 (multiplicities coincide). This completes the proof of Theorem 2.2. □ One of the tools used in the present paper is an interpolation. Interpolation has been also used to prove Theorem 1.2 of [Korotyaev and Safronov 2020], which can be generalized and formulated as follows.
Theorem 2.4. Let ( , µ) be a space with an σ -finite measure µ such that L 2 ( , µ) is separable. Let H 0 be a selfadjoint operator on the Hilbert space L 2 ( , µ). Assume that the integral kernel of the operator e −it H 0 satisfies the estimate for p > such that p ≥ 1. Assume also that (2.2) holds for all W 1 and W 2 that belong to a class of functions dense in L 2 p ( , µ). Then eigenvalues of the The proof of this result is a counterpart of the proof of Theorem 1.2 from [Korotyaev and Safronov 2020], with the only differences being that the value of the parameter in Theorem 1.2 of that work is 3 2 and = ‫ޒ‬ 3 . However, one can consider different as well as spaces which are different from ‫ޒ‬ d . Especially interesting are spaces of fractional dimensions for which 2 is not an integer.
Another object that we will work with is the operator If V is a bounded compactly supported function, then X (k) is a trace class operator for d ≤ 3, and X (k) ∈ S p for p > 1 4 d and d ≥ 4. In this case, we set The algebraic multiplicity of the eigenvalue λ does not exceed the multiplicity of the root of the function D n ( · ).
Proof. According to the Birman-Schwinger principle, a point λ is an eigenvalue of H if and only if −1 is an The statement about the multiplicity follows from the fact that an isolated eigenvalue of H whose multiplicity m is larger than 1 can be turned into m simple eigenvalues by an arbitrarily small perturbation of finite rank (which does not have to be a function). For any ε > 0 there is a finite rank operator K ε such that ∥K ε ∥ < ε and that all eigenvalues of − + K ε + V near λ are simple. Define now the function λ for sufficiently small ε > 0. In this neighborhood of the point k 0 , we have d ε (k) → D n (k) uniformly, as ε → 0. Since the function d ε (k) has at least m zeros near k 0 , the multiplicity of the zero of the function D n (k) at k = k 0 can not be smaller than m by the argument principle. □

Large values of Re ζ without projections
The following proposition gives an important estimate for the integral kernel of (− − z) −ζ .
Proposition 3.1. Let d ≥ 2, and let 1 for z / ∈ ‫ޒ‬ + . The positive constants β and α in this inequality depend only on d and Re ζ .
The proof of this proposition, as well as related references, can be found in [Frank and Sabin 2017].
Everywhere below, we use the notation for z / ∈ ‫ޒ‬ + . The positive constants β and α in this inequality depend only on d and Re ζ . If Re ζ = 1 2 (d +1) and d ≥ 2, then (3.2) holds with r = 2.
Proof. It follows from (3.1) that A simple application of Hölder's inequality leads to (3.2). □ We need to turn (3.2) into a similar estimate for the S 4 -norm of the operator corresponding to smaller values of Re ζ . For that purpose, we employ the inequality for Re ζ = 0. By interpolation we obtain the following proposition from (3.2) and (3.3).
Let us now consider the operator where W is a fixed function independent of ω. The proof of the following proposition is based on the fact that ‫[ޅ‬ω n ] = 0.

An estimate for the square of the Birman-Schwinger operator
According to the observations that we made, if W = V , then X(ζ ) is a function that obeys (3.7) for some rather large values of Re ζ , and it also obeys To obtain our first result about eigenvalues, we can interpolate between these two cases. Let where W is a fixed function independent of ω. What follows is the result of the interpolation (which does not work for d = 2). (4.1) If = 1 2 (d + 1) and d ≥ 3, then (4.2) holds with p = . Proof. Note that X (k) = X(1). The logic of interpolation says that (4.2) holds for p defined as p = 2/θ, for θ such that 1 = θ τ, where 1 2 ≤ τ < min{ , d /(4 − 2)}. Of course, this interpolation works only if τ > 1, which is impossible for d = 2. Observe that, with this notation, p = 2τ . Let be the polar decomposition of the operator X (k). Consider the function Consequently, by the three lines lemma, Put differently, The latter inequality implies (4.2) because 2τ = p. □ Now we can formulate and prove the following result.
Theorem 4.2. Let d ≥ 3, and let 1 < ν < q < 2. Assume that W = |V | 1/2 . Then for p defined by . (4.4) Proof. Observe that the assumption ν < q < 2 leads to the inequalities . (4.5) We will show that the conditions of Proposition 4.1 are fulfilled for the parameter defined by The latter relation simply means that Consequently, (4.3) follows from (4.2). The second inequality in (4.5) implies while the first inequality in (4.5) combined with the condition ν < 2 implies One can also see that the first inequality in (4.7) is equivalent to the estimate Finally, note that when d ≥ 3, the condition p < 2 follows from the fact that ν + q > 2. □ 5. 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 ∈ ‫ރ‬ : Im k > 0, |k| > R} onto the upper half-plane {z ∈ ‫ރ‬ : Im z > 0}. Rather standard arguments lead to the estimate where the z j are the zeros of the function d(z) situated in the upper half-plane ‫ރ‬ + . In fact, (5.1) could be established in the same way as Jensen's inequality for zeros of an analytic function on a unit disk. In (5.1) we assume that V is compactly supported. The relation (5.1) leads to the estimate Taking the expectation we obtain Due to Theorem 4.2, the latter inequality leads to Now, suppose that we consider only the eigenvalues λ j = k 2 j that satisfy the inequality Multiplying (5.3) by R q−1 and integrating with respect to R from 0 to R 0 , we obtain This implies Theorem 1.1. □ Theorem 1.2 can be proved in the same way. The only difference is that one needs to use Proposition 4.1 instead of Theorem 4.2.

Operators of the Birman-Schwinger type
Let a, b and V be functions on ‫ޒ‬ 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 v n χ (x − n).
Note that the operator A ζ can be viewed as a sum over the lattice ‫ޚ‬ d : where A ζ,n = ω n |a| ζ F|v n | ζ e i arg v n χ ( · − n)F * |b| ζ .
We will show that while A ζ might not be bounded at some points ω, it is still a compact operator almost surely if a, b and V are in L 2 . We remind the reader that V was defined as the function Remark. Operators of the form a F W F * b do not have to be bounded for all a, b and W from L 2 . Indeed, let W (x) = (|x| + 1) −s , with 1 2 d < s < 2 3 d, and let If a F W F * b was bounded, the operator T = a F √ W would be bounded as well. The latter is not true, simply because T ψ / ∈ L 2 for ψ = W (the singularity of T ψ at zero is |ξ | 3s/4−d ).
Proposition 6.1. Let a ∈ L 2 , b ∈ L 2 and V ∈ L 2 . Let also p ≥ 2. Then the sum (6.1) with Re ζ = 2/ p converges almost surely in S p . Moreover, Proof. We are going to prove (6.2) 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, Let = (ω) be the partially isometric operator appearing in the polar decomposition We introduce the analytic function which will be treated by the three lines lemma. Since ∥A ζ ∥ ≤ 1 for Re ζ = 0, and ∥|K | i Im ζ 0 * ∥ ≤ 1, we obtain that On the other hand, by an analogue of Hölder's inequality valid for Schatten classes. Indeed, for Re ζ = 1, Using the three lines lemma, we obtain from (6.3) and (6.4) that Note now that f (ζ 0 ) = β. Consequently, Corollary 6.2. Let T be a random operator of the form Let a ∈ L p , b ∈ L p , v n ∈ ℓ 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 obeys the inequality The following result is a very well-known bound obtained by E. Seiler and B. Simon [Seiler and Simon 1975]. Moreover, the reader can easily prove it using standard interpolation. Proposition 6.3. Let a and W be two functions from L p ‫ޒ(‬ d ) with p ≥ 2. Let T be the operator where F is the operator of the Fourier transform. Then Corollary 6.4. Let q ≥ p ≥ 2. Let T be a random operator of the form Let a ∈ L p , b ∈ L q and v n ∈ ℓ p . Then Proof. According to Proposition 6.3, On the other hand, according to Corollary 6.2, It remains to interpolate between the two cases. For that purpose, we introduce the function where K = ||a|F V F * |b|| and is the partially isometric operator appearing in the polar decomposition |a|F V F * |b| = K . For convenience, we write β := ‫([ޅ‬Tr K p ) q/ p ].
Thus by the three lines lemma,

Large values of Re ζ
Let 0 < R ≤ 1. 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 = Fχ n,k F * , which are the spectral projections of − corresponding to the intervals [0, 4|k| 2 /R 2 ] and (4|k| 2 /R 2 , ∞). Besides depending on the properties of (− − z) −ζ , the arguments of this paper also rely on the properties of the operators P n,k (− − z) −ζ for different values of ζ . In this section, we discuss relatively large values of Re ζ . The following proposition gives an important estimate for the integral kernel of P n,k (− − z) −ζ .
Proof. Indeed, let Re ζ 0 = τ , and let A = |A| be the polar decomposition of the operator Consider the function Consequently, by the three lines lemma, Put differently, The latter inequality implies (7.8), and the proof is completed. □ In particular, once we set r /τ = 4, we obtain the following.

Small values of Re ζ
The notations we use in this section are the same as in the previous one. In particular, the projections P n,k are the same as before. As was mentioned, the arguments of this paper rely on the properties of the operators P n,k (− − z) −ζ for different values of ζ . In this section, we discuss the case 0 ≤ Re ζ < 1.
In the next two propositions, we discuss the properties of the random operators for Re ζ = 1 2 γ and 0 < γ < 3 2 . Here W is a fixed function which does not depend on ω. The value of the parameter α 0 should be sufficiently large as in Corollary 7.6.
Later, we will also 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.

Another interpolation between small and large values of Re ζ
Let us recall two theorems that hold for the operator with W = V 1/2 . By small values of Re ζ we mean the values that are considered in Corollary 8.3, which states that, for any p ≥ 1 and d/(8 p) < Re ζ < 3/(4 p), In this corollary, we had to assume that 2 ≤ d ≤ 5 and |k| ≥ R, where 0 < R ≤ 1. One should also not forget that our assumptions about γ = 2 Re ζ imply that Re ζ < 3 4 . In the next result, we only replace 4 Re ζ by d/(2 p) in the right-hand side of (9.1).
For the sake of simplicity, we choose p = d 7 Re ζ .
In this case, because of the assumption p ≥ 1 that we made, we have to assume that 0 < Re ζ ≤ 1 7 d. Note that 1 7 d < 3 4 . Thus, we can formulate the following assertion.
By the large values of Re ζ we mean the values appearing in Corollary 7.6. We will use only a simpler version of this result. .
The latter relation simply means that (9.5) Thus (9.3) coincides with (7.11). Let us check that all conditions of Corollary 7.6 are fulfilled. The second inequality in (9.4) implies while the first inequality in (9.4) combined with the condition ν < 2 implies that One can also see that the first inequality in (9.6) is equivalent to the estimate Finally, note that when d ≥ 3, the condition Re ζ < follows from the fact that ν + η > 2. Consequently, Corollary 7.6 implies Theorem 9.3. □ We interpolate between Corollary 9.2 and Theorem 9.3.
Theorem 9.4. Let 3 ≤ d ≤ 5. 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 for |k| ≥ R and 0 < R ≤ 1.
Since we have some information about the values of this function on the boundary of the strip, we obtain (9.10) by interpolation between Corollary 9.2 and Theorem 9.3. □ Remark. We need to explain why the parameters were selected as described in Theorem 9.4. The work with perturbation determinants requires convergence of integrals of the form (1 − θ )τ 1 d , implying that τ 1 (1 − θ ) < θ (ν − 1)(d + 1) 14d .
The latter can be written differently as 14d .
In other words, . (9.11) The condition that θ is large can be converted into an inequality showing that τ 1 is small. The relation (9.11) is satisfied if 7d .
On the one hand, we see that which implies the inequality On the other hand, Consequently, The next statement follows by Hölder's inequality.
10. Proof of Theorem 1.5 Again, we work with the function d(z) = det n (I − X (k)), n = [q] + 1, where z is related to k via the Joukowski mapping Standard arguments allow us to rewrite (5.2) with p replaced by q as where the k j are defined as square roots of eigenvalues of H. Due to Theorem 9.7, the latter inequality yields ‫ޅ‬ j Im k j (|k j | 2 − R 2 ) + |k j | 2 R ≤ C|R| −θqν/2 ∥ V ∥ 2q r . (10.1) Now, suppose that we consider only the eigenvalues λ j = k 2 j that satisfy the inequality |k j | ≤ R 0 .