Schr\"odinger operators with complex sparse potentials

We establish quantitative upper and lower bounds for Schr\"odinger operators with complex potentials that satisfy some weak form of sparsity. Our first result is a quantitative version of an example, due to S.\ Boegli (Comm. Math. Phys., 2017, 352, 629-639), of a Schr\"odinger operator with eigenvalues accumulating to every point of the essential spectrum. The second result shows that the eigenvalue bounds of Frank (Bull. Lond. Math. Soc., 2011, 43, 745-750 and Trans. Amer. Math. Soc., 2018, 370, 219-240) can be improved for sparse potentials. The third result generalizes a theorem of Klaus (Ann. Inst. H. Poincar\'e Sect. A (N.S.), 1983, 38, 7-13) on the characterization of the essential spectrum to the multidimensional non-selfadjoint case. The fourth result shows that, in one dimension, the purely imaginary (non-sparse) step potential has unexpectedly many eigenvalues, comparable to the number of resonances. Our examples show that several known upper bounds are sharp.


Introduction.
Many examples of Schrödinger operators with "strange" spectral properties involve sparse potentials. In his seminal work [59] Pearson constructed examples of real-valued potentials (on the half-line) leading to singular continuous spectrum. The potentials consists of an infinite sequence of "bumps" of identical profile, and the separation between these bumps increases rapidly. The physical interpretation is that a quantum mechanical particle will ultimately be reflected from a bump. These ideas were further developed in several directions, see e.g. [62] [70], [42], [48], [46], [80], and the references therein. Scattering from sparse potentials in higher dimensions was studied by Molchanov and Vainberg [53], [54]; see also [52], [37], [38], [65], [45]. The discrete spectrum for multidimensional lattice Schrödinger operators was investigated by Rozenblum and Solomyak [63]. They constructed examples of sparse potentials whose number of negative eigenvalues grows like an arbitrary given polynomial power in the large coupling limit. In the recent work [5] Boegli constructed a complex-valued sparse potential with arbitrary small L q norm (q > d) that has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum. Since the proof is based on compactness arguments, there is no quantitative bound on the rate of separation between the bumps, and hence no estimate on the pointwise decay of the potential or the accumulation rate of the eigenvalues is possible.

A quantitative version of Boegli's example.
Our first result provides quantitative decay bounds for the example in [5]. Perhaps more importantly, the construction can be used to produce a potential together with an infinite number of eigenvalues (possibly not all of them) satisfying given upper and lower bounds on their accumulation rate. We formulate our result for the most interesting spectral region where ǫ 0 > 0 is small but fixed.
Remark 1. (i) In particular, for any λ ∈ (0, ∞) there exists a sequence (ζ n ) n ⊂ Σ 0 satisfying (2) such that lim n→∞ ζ n = λ. In this way one can find a sequence accumulating to every point of the essential spectrum. This yields a constructive proof of the result of Boegli [5].
(ii) One can remove the logarithm in (2) at the expense of replacing the L q norm of V by the "Davies-Nath norm" (see (28)). (iii) We will give explicit bounds on the polynomial decay β.
(iv) Substituting the trivial lower bound |ζ n | ≥ |Im ζ n | into (2) shows Im ζ n → 0. This is the reason why we say that z n is exponentially close to ζ n .
We believe that the pointwise condition c) is more natural than the L q condition b) for the phenomenon that takes place in Theorem 1. This is because complex analogues of classical phase space bounds, which motivate the consideration on L q norms in the first place, lack many of the features that make them so useful for real potentials (more on that in Subsection 1.5 below). Put simply, the L q norm does not see the separation between the bumps, while the pointwise bound does. We will nevertheless work with L q norms since we allow the bumps to have singularities. In the case where they are bounded the pointwise decay of the whole potential can easily be estimated by comparing the L ∞ norms of the bumps to their spatial separation from the origin. In his fundamental work on non-selfadjoint Schrödinger operators, Pavlov [57], [58] showed that the number of eigenvalues in one dimension is finite if |V (x)| exp(−c|x| 1/2 ), and that this exponential rate is best possible. This means that the potential in Theorem 1 cannot decay too fast. The L q bound imposes no decay whatsoever, but we can at least establish polynomial decay. For recent quantitative improvements of Pavlov's bound we refer to Borichev-Frank-Volberg [6] and Sodin [72].
The proof of the example in [5] is based on "soft" methods like weak convergence, compact embedding and the notion of the limiting essential spectrum. In contrast, our proof uses "hard" estimates for the resolvent and the Birman-Schwinger operator, combined with tools from complex analysis such as Rouché's theorem, Jensen's formula and Cartan type estimates. This allows us to obtain more precise results thatn those in [5]. Rouché's theorem and Jensen's formula are among the most ubiquitous albeit simple tools in non-selfadjoint spectral theory, where such machinery as the variational principle or the spectral theorem is not available. In the present paper Cartan type estimates are crucial to bound a certain Fredholm determinant from below and get upper upper bounds on the norm of the resolvent. This opens the way to proving existence of eigenvalues by means of quasimode construction. We are then in a setting similar to the selfadjoint case where a quasimode of size ǫ guarantees the existence of a spectral point in an ǫ-neighborhood of the quasi-eigenvalue. This follows from the inequality (H V − z) −1 ≤ 1/d(z, σ(H V )), where σ(H V ) is the spectrum. In the non-selfadjoint case the inequality may fail dramatically. This phenomenon gives rise to the notion of pseudospectrum, which we will not discuss here (see e.g. the monograph [19]). The upper bounds obtained by Cartan type estimates generally grow exponentially in 1/d(z, σ(H V )). In order to beat this, we are forced to construct exponentially small quasimodes, a challenging task in all but the simplest models. The strategy is reminiscent of the proof of existence of resonances close to the real axis due to Tang-Zworski [77] and Stefanov [73] (see also the recent book by Dyatlov-Zworski [24]). Our method is perhaps closest to that of Dencker-Sjöstrand-Zworski [22,Section 6] for non-selfajoint dissipative Schrödinger operators. The difference is that we consider decaying potentials and do not assume, as these authors do, that the quasi-eigenvalue is real (see [22,Proposition 6.4]). This means that the amplification of the exponential upper bound through the maximum principle (see [22,Proposition 6.2]) is in general not possible in our case. Another crucial difference is that we need a more quantitative version of the Cartan type estimate (Lemma 33) as well as of the conformal transformations between the spectral region and the model domain (the unit disk). The Riemann mapping theorem is notoriously non-quantitative. Instead, we use Cayley and Schwarz-Christoffel transformations, which have previously been used in other contexts related to non-selfadjoint spectral theory, especially in connection with Lieb-Thirring type inequalities. The combination with Rouché's theorem and the Cartan type bounds is new and leads to results with an inverse problem flavor, as in Theorem 1.

Magnitude bounds.
The second result gives precise bounds on the magnitude of eigenvalues of Schrödinger operators with complex sparse potentials, or more generally, potentials of the form V = N j=1 V j , where the V j have disjoint support and separate rapidly from each other. We will call these "separating" potentials. The Schrödinger operator H V = −∆ + V behaves like an almost orthogonal sum, due to the rapid decoupling between the N "channels". This enables us to improve upon the bounds for general complex potentials due to Frank [27], [28]. For simplicity we state the result here for d ≥ 3. The general case along with further refinements can be found in Subsection 3.4. We refer to Section 2 for a more in-depth explanation of the terminology.
If q > (d + 1)/2, then The bound (3) follows from (27) by a Birman-Schwinger argument. It could also be proved by using the eigenvalue bounds of the author [14], which are inspired by a method of Davies and Nath [20] in one dimension. For N = 1 the estimates (3), (4) coincide with those of Frank [27], [28], respectively. The difference is that here V might decay very slowly or not at all. Nevertheless, on the η 2 energy (spectral) scale the estimate is of the same quality as for N = 1.
We make a short remark about the connection with the Laptev-Safronov conjecture [47], which stipulates that For the range q ∈ [d/2, (d + 1)/2] the conjecture was proven by Frank [27]. The question whether (5) is true for q ∈ ((d + 1)/2, d] is still open. The expectation, based on intuition from counterexamples to Fourier restriction (see e.g. [13], [14] for more explanations) is that the conjecture is false in this range. Incidentally, Theorem 1 clearly implies the necessity of q > d in the conjecture, but this already follows from Bögli's result (without the pointwise bound). In fact, a single bump of the sparse potential used in Boegli's construction (and in Theorem 1) already provides a counterexample. Since there seems to be some confusion about this issue we use the results of Section 7 to show necessity of the condition q > d. Indeed, Lemma 22 implies that for small ǫ > 0 there is a potential V (ǫ) and an eigenvalue z(ǫ) such that |z(ǫ)| q− d 2 / V (ǫ) q q ǫ q−d log q (1/ǫ). The example (a complex step potential) is simple but, quite amazingly, generic enough to show optimality of several estimates in the literature (see [14]). In one dimension, the step potential can be tuned to essentially saturate any of the known magnitude bounds. For example, the last inequality also shows that the Davies-Nath bound [20] is sharp and, since Im z(ǫ) ≍ ǫ, that Frank's bound [28,Theorem 1.1] is sharp up to a logarithm. In Subsection 1.5 we will show how the complex step potential also implies sharpness of another bound in [28].

1.4.
A generalization of Klaus' theorem. The following is a generalization of a result due to Klaus [43] on the characterization of the essential spectrum. The generalization is twofold: First, we admit complex potentials and second, we prove it for any dimension (whereas Klaus only proved the one-dimensional case). In this introduction we again focus on the case d ≥ 3, but the statement is valid in d = 1, 2 for q in the range (15). [43] for complex potentials). Assume that d ≥ 3 and d/2 ≤ q ≤ (d + 1)/2. If V is a separating potential and sup j∈ where S is the set of all z ∈ C \ [0, ∞) such that there exist infinite sequences (i n ) n , (z n ) n with z n ∈ σ(H Vi n ), i n → ∞ and z n → z as n → ∞.
An alternative proof (also in one dimension) of Klaus' theorem can be found in [16]. The role of Theorem 3 in this paper will be an auxiliary one, and we will only use it to argue that the essential spectrum is invariant under the perturbations we consider in Theorem 1. Although our proof follows the general strategy of [43] it is still worth emphasizing that some parts of it require somewhat novel techniques.  [21] for one-dimensional Schrödinger operators with complex potentials by establishing a lower bound on a certain Riesz means of eigenvalues. More precisely, consider H αV , where V = i1 [−1,1] is a purely imaginary step potential and α is a large semiclassical parameter. Boegli anď Stampach prove that, for any p ≥ 1, The interesting feature of this bound is that it shows a logarithmic violation of Weyl's law. To recall Weyl's law, consider a self-adjoint operator, with a smooth real-valued potential. Note that if we set h = 1/ √ α, then α −1 H αV takes the form of a semiclassical Schrödinger operator, −h 2 ∂ 2 x + V (x). Semiclassical asymptotics (Weyl's law) yield, for a suitable class of functions f , as α → ∞. In particular, if f is homogeneous of degree γ, then For f (λ) := λ γ − and γ ≥ 1/2 (since we are considering d = 1) the Lieb-Thirring inequality captures the semiclassical behavior (8), but is valid for any α > 0, not only asymptotically. Returning to the complex potential V = i1 [−1,1] and noticing that f (z) := (Im z) p |z| 1/2 is homogeneous of degree γ = p − 1/2, we observe that (6) implies that the formal analogue of (8) cannot hold, i.e. that lim inf hence violating Weyl's law (8). The comparison with Weyl's law is formal because f (H α ) does not make sense in general for a non-normal operator. However, (6) also shows that the complex analogue of the Lieb-Thirring inequality (9), cannot be true, thus disproving the conjectured bound in [21]. For a non-selfadjoint (pseudo)-differential operator with analytic symbol and in one dimension the eigenvalues typically lie on a complex curve, hence violating Weyl's law (in terms of complex phase space). Small random perturbations typically restore the Weyl law (in real phase space). The literature on the subject is vast and we merely refer the interested reader to the book of Sjöstrand [71] for an overview of recent developments. In contrast, a classical result of Markus and Macaev [51] implies that Weyl's law holds for the real part of the eigenvalues, provided the non-selfadjoint perturbation is small.
A slightly different view on the same phenomenon (violation of Weyl's law) is connected with the notion of "locality". In the semiclassical limit, in the selfadjoint case, each state occupies a volume of (2π This means that in the semiclassical limit each bump is responsible for an equal number of eigenvalues. In particular, two distinct realizations of V as a sum of bumps have the same number of eigenvalues, as long as the bumps are disjoint. This feature of locality is also captured by the Lieb-Thirring inequalities since the bound involves the integral linearly. Our third result shows that this kind of locality can be violated in the nonselfadjoint case. We adopt the same notation N (H; Σ) for the number of eigenvalues of H in Σ as in the self-adjoint case, but emphasize that these are counted according to their algebraic multiplicity. We consider one sparse one non-sparse (or non-separating) realization of V and denote these by V s and V n , respectively. For simplicity, we will consider the same potential as in [4], i.e. the purely imaginary step potential W = i|W 0 |1 [−R0,R0] of size |W 0 | and width R 0 . For simplicity we fix these scales to be of order one. Then V n is the single well of width R = N R 0 , while V s is a sum of N disjoint wells W (x − x j ) of width R 0 . We will fix the coupling strength α and consider the limit N → ∞. For the non-sparse operator this is in fact still a semiclassical limit, as can easily be seen by rescaling.
where C is a large constant. Then we have Moreover, there exists a sequence (x j ) j such that The estimates in [4] would be sufficient to prove a lower bound of size N 2−ǫ in (10). Their argument proceeds by approximating the characteristic eigenvalue equation and finding the roots of this equation in an asymptotic regime. They did not prove that the original equation has nearby roots. This can be done e.g. by a contraction mapping argument [75]. We will give a proof using Rouché's theorem. Note that, by power counting (dimensional analysis), the constants in (10), (11) only depend on the dimensionless quantity |W 0 | 1/2 R 0 .
In the selfadjoint case, i.e. when W is replaced by |W 0 |1 [−R0,R0] , the number of negative eigenvalues of V n is of order N , in agreement with semiclassics. Also note that, since in one dimension each H Vj has at least one negative eigenvalue, we have N (H Vs ; R − ) ≍ N in this case. Hence the left hand sides of (10) and (11) are equal in magnitude, which may be seen as a manifestation of locality. However, this locality is violated if one takes into account not only eigenvalues but also resonances. Zworski [81] proved that, for a compactly supported, bounded, complex potential, the number n(r) of resonances λ 2 j in a disk |λ j | ≤ r asymptotically satisfies as r → ∞. Moreover, for any ǫ 0 > 0, the number of resonances in |λ j | ≤ r but outside the sector Σ 0 (see (1)) is o(r). This and the fact that eigenvalue bounds outside Σ 0 are "trivial" (in the sense that they can be proved by the same standard estimates as for real potentials, with the provisio that the constants blow up as the implicit constant in (1) becomes small) motivates us to often restrict attention to the spectral set Σ 0 . The result (12) was obtained earlier by Regge [61] in some special cases. Different proofs were given by Froese [31] and Simon [68]. Froese's proof also works for complex potentials. Note that eigenvalues are included in the definition of resonances. Formula (12) can be seen as a Weyl law for resonances but is nonlocal as it includes the convex hull of the support of the potential. An obvious corollary of Zworski's formula is that the right hand side of (12) is an upper bound for the number of eigenvalues λ 2 j in the disk |λ j | ≤ r (resonances in the upper half plane). For the potential V n , taking r ≍ N/ log N and observing that ch supp(V n ) ≍ N , we find that the leading term in (12) is of order N 2 / log N , in agreement with the lower bound (10). Note, however, that the asymptotics are not uniform, i.e. the error may depend on N (recall that V n depends on N ).
Korotyaev [44,Theorem 1.1] proved an upper bound which yields for r ≍ N/ log N . Hence, (10) shows that, at the scale considered here, a substantial fraction of resonances are actual eigenvalues. In the special case of a step potential Stepanenko [76] proved that the total number of eigenvalues is bounded by N 2 / log N . The lower bound (10) shows that this is sharp up to constants; this was observed independently by Stepanenko [75]. In dimension d ≥ 3 it may of course happen that all the H Vj , and thus also H Vs , have no eigenvalues at all if |W 0 | 1/2 R 0 is small (by the CLR bound), while H Vn has of the order N d eigenvalues. This is not the kind of phenomenon that takes place in Theorem 4. Indeed, there we allow |W 0 | 1/2 R 0 to be of unit size. We also note that in higher dimensions the results on the asymptotics of the resonance counting function are weaker, and there are example of complex potentials with no resonances at all (see Christiansen [8]). On the other hand, Christiansen [7] and Christiansen-Hislop [9], [10] proved that n(r) has maximal growth rate r d for "most" potentials in certain families.
A final comment regarding the implications of Theorem 4, also related to locality (or the lack thereof), concerns a general observation on Lieb-Thirring type inequalities for complex potentials. It is a fact that all known upper bounds for eigenvalue sums have a superlinear dependence on N . For example, in d = 1, [28, with (a, b, c) = (1, 1, 2). The lower bound (10) shows that the power c cannot be decreased, while preserving the homogeneity condition −2a = (−2b + 1)c. Indeed, so that the ratio of the first to the second is bounded below by , which tends to infinity as N → ∞, for every c < 2. On the other hand, for sufficient rapid separation of the bumps in V s , we can prove Frank's bound (13) with a linear dependence on N , at least locally in the spectrum.
Our methods could be used to prove similar generalizations of the higherdimensional results of [28], but we will not pursue this.
Notation. We write σ(H), ρ(H) for the spectrum, respectively the resolvent set of a closed linear operator H. The free and the perturbed resolvent operators are denoted by R 0 (z) = (−∆ − z) −1 and R V (z) = (H V − z) −1 , respectively. We denote by S p the Schatten spaces of order p over the Hilbert space L 2 (R d ) and by · p the corresponding Schatten norms. We also write · = · ∞ for the operator norm. To distinguish it from L p norms of functions we denote the latter by · L p . We will use the notation V 1/2 = V /|V | 1/2 and x := 2 + |x|. The statement a b means that |a| ≤ C|b| for some absolute constant C. We write a ≍ b if a b a. If the estimate depends on a list of parameters τ , we indicate this by writing a τ b. The dependence on the dimension d and the Lebesgue exponents p, q is usually suppressed. We write a ≪ b if |a| ≤ c|b| with a small absolute constant c, independent of any parameters. By an absolute constant we always understand a dimensionless constant C = C(d, p, q). Here we choose units of length l such that position, momentum and energy have dimensions l, l −1 and l −2 , respectively. We chose the branch of the square root Acknowledgements. The author gratefully acknowledges correspondence with Sabine Boegli and comments of Rupert Frank, who pointed out the failure of Weyl's law and the connection with nonlocality. Many thanks also go to Stéphane Nonnenmacher for useful discussions on resonances and to Alexei Stepanenko for explaining his recent preprint. Special thanks go to Tanya Christiansen for many helpful remarks on a preliminary version of the introduction.

Definitions and preliminaries
2.1. Separating and sparse potentials. We consider sparse potentials of the form where j ∈ [N ] = {1, 2, . . . , N }, N ∈ N ∪ {∞}, and Ω j ⊂ R d are mutually disjoint (not necessarily bounded) sets. We then set We assume that V j ∈ ℓ p L q , where the norms are defined by Here q will be in the range In particular, we have The constant sep(L, η) only depends on L and not on V itself. We will sometimes abuse terminology and call the sequence L separating. Note that since ηL j is dimensionless, η has the dimension of inverse length. We shall always assume that the sequence L is increasing.
Definition 2. We say that V = V (L, Ω) is sparse if it is separating, the supports Ω n are bounded, and lim n→∞ diam(Ω n )/L n = 0.
Most of our results hold for separating potentials. The strong separation condition is convenient and facilitates some of the proofs. Typical examples of strongly separating sequences are (η ≪ η 0 ): See Subsection 8.2 for a proof. The explicit example used to prove Theorem 1 turns out to be sparse. Note that, by the disjoint supports, the definition (14) is

2.2.
Comparison with a direct sum. In Section 5 we will compare the two operators Note that the point spectrum (eigenvalues) of H diag , is independent of the sequence L. We will consider σ p (H Wj ) as part of the data and seek to prove lower bounds on . In fact, we will consider a subset of the point spectrum, the discrete spectrum. We will consider the V j as given only up to translations, i.e. we stipulate that where W j ∈ ℓ p L q contains the origin in its support. Using the triangle inequality, it is easy to see that |x i − x j | ≥ L j for i = j, and therefore Straightforward arguments also show that In particular, for sparse potentials, Note that diam(Ω j ) = diam(supp W j ) is part of the data. We can thus obtain a lower bound for |x i − x j | in terms of L j and vice versa. For this reason we restrict attention to L j here.

2.3.
Truncations. For technical reaons, it will turn out to be convenient to consider finite truncations. For n ∈ [N ] we define 2.4. Abstract Birman-Schwinger principle. We mostly disregard operator theoretic discussions here and refer e.g. to [28] for the (standard) definition of H Vj as m-sectorial operators. The rigorous definition of H V is a bit more subtle since V need not be decaying. However, a classical construction of Kato [40] produces a closed extension H V of −∆ + V via a Birman-Schwinger type argument. This approach works as soon as one can find point z 0 in the resolvent set of H 0 = −∆ at which the Birman-Schwinger operator has norm less than one. Such bounds are provided by Lemma 6, but to avoid technicalities it is useful to think of the potential as being bounded by a large cutoff (and all of the bounds will be independent of that cutoff). By iterating the second resolvent identity, it is then easy to see that For more background on the abstract Birman-Schwinger principle in a nonselfadjoint setting we refer to [32], [28], [3], [36].
2.5. The essential and discrete spectrum. We briefly recall some facts about the essential and discrete spectrum of a closed operator H. There are several inequivalent definitions of essential spectrum for non-selfadjoint operators (but these all coincide for Schrödinger operators with decaying potentials [28,Appendix B]). We use the following standard definition.
The discrete spectrum is defined as In the situations we consider here (20) will always be true for H = H V and H diag . In fact, Corollary 11 tells us that σ e (H) = [0, ∞), just as for decaying potentials.

Universal bounds for separating potentials
In this section we consider H V as a perturbation of −∆. We will thus only make assumptions about V and not about H diag .
3.1. Birman-Schwinger analysis. Since the V j have mutually disjoint supports, we can write the Birman-Schwinger operator (18) as The first term is unitarily equivalent to the orthogonal sum (Ω i ). By abuse of notation we will always identify these two Hilbert spaces and the corresponding operators. The off-diagonal contribution is In the following we will use the notation where We use the abbreviation and set α(q) := 2 max(q, q d ).
(iii) The diagonal part satisfies (iv) The off-diagonal part satisfies (v) The full Birman-Schwinger operator satisfies Proof. It follows from (iii) and (iv). Moreover, (iv) follows from (ii) and the triangle inequality, using the estimate d(  [30,Theorem 12] in the case q ≤ q d and in [28, Proposition 2.1] for q ≥ q d . The only difference is that we include the (second) exponential in the pointwise bound for a ∈ [(d − 1)/2, (d + 1)/2] and d ≥ 2, see e.g. [49, (2.5)]. For d = 1 one can use the explicit formula for the resolvent kernel.
Remark 2. Using the results of [14] (or [20] in one dimension) in the proof of Lemma 6 we could replace the bounds (23), (24) by the following. For q ≤ q d and i, j ∈ [N ], where F V,q (s) is the "Davies-Nath norm" This implies the bounds F Vi,q .

Norm resolvent convergence.
Lemma 7. Under the assumptions of Lemma 6 we have The claim readily follows from (27), the identity and a T T * argument. Proof. We first note that all the bounds of Lemma 6 also hold for H V (n) , uniformly in n. Since V (n) converges to V in ℓ p L q , it follows that the Birman-Schwinger operator associated to H V (n) converges to BS V (z). Moreover, by Lemma 7, the operators |V (n) | 1/2 R 0 (z) converge in S 2p -norm. Note that the square root of V is trivial to compute, owing to the disjointness of supports of the V j . We choose z = it, t ≫ 1. For such z the norm of the Birman-Schwinger operator is < 1, hence a Neumann series argument and the resolvent identity (19) yield the claim. Proof. Only the inclusion ⊂ is nontrivial. The resolvent set of a direct sum of bounded operators A = i A i is known to be ). This clearly implies λ ∈ i ρ(A i ). It remains to prove that sup i (A i − λ) −1 < ∞.
By assumption, there exists δ > 0 such that d(λ, σ(A i )) ≥ δ. By [2, Theorem 4.1], With Proposition 10 in hand is immediate that [43, Lemma 2.4] (K i = A i in our notation) holds in the generality needed here. The Schatten bound (26) provides a substitute for the compactness arguments [43, Proposition 2.1-2.2]. For the remainder of the proof one can follow the arguments in [43] verbatim.
Corollary 11. If V ∈ ℓ p L q with q in the range (15) and p < ∞, then we have S = ∅, i.e. σ e (H V ) = [0, ∞). The same holds for H diag .
Proof. By (27), we have that lim t→∞ BS V (it) = 0, which means that the inverse exists as a bounded operator for z = it and t ≫ 1. Corollary 7 implies that R 0 (z)V 1/2 is compact, whence the resolvent difference (19) (27), Remark 2 and the Birman-Schwinger principle.
Theorem 12. Let q be in the range (15). If V is separating, then every eigenvalue as well as Remark 3. (i) In the case of a single "bump" (N = 1) the bound (29) was proved by Frank in [29] for q ≤ q d and in [28] for q > q d . In the latter case it was observed that the inequality implies Im z → 0 as Re z → +∞ for eigenvalues z of H V . More precisely, if we fix the norm of the potential, then In the case N = 1 (29) implies that the above holds with an additional factor s(L, z) 2 on the right. If L grows at least polynomially, η 0 L k k α , then we obtain see Example a) after Definition 2. Hence, for sufficiently large α (depending on q and d) the exponent of |Im z| remains positive, while that of Re z remains negative, and we still get the conclusion that Im z → 0 as Re z → +∞. (ii) The N = 1 case of (30) was proved in one dimension by Davies and Nath [20] and in higher dimensions by the author [14]. The inequality is similar to (29) for q > q d . Both are relevant for "long-range" potentials. In the case of the step potential (30) is sharp, while (29) (both for N = 1) loses a logarithm (see (56) and (57)).
The assumption V ℓ p L q 1 is for convenience only and can easily be removed by power counting arguments (Since the estimates are scale-invariant).

Upper bounds.
We collect some useful estimates that we will repeatedly use (these follows from [69, Theorem 9.2]): where f = f diag or f V and BS(z) = BS diag (z) or BS V (z). Formulas (31), (32), together with the bounds of Lemma 6 motivates the following definitions, where we suppressed the dependence on V . We recall that ω q (z) and s(L, z) were defined in (21) and (22). The constant O p (1) is allowed to vary from one occurence to another. Thus, for example, the inequality ϕ p (t)ψ p (t) ϕ p (t) holds, but ψ p (t) 1 need not be true.

4.2.
Lower bounds away from zeros. The following lemma can be considered as one of the key technical results. To state it we introduce the notation In the following H denotes either H diag or H V . We then write δ diag (z) := δ H diag (z) and δ V (z) := δ HV (z).
Proof. In the following, f denotes either f diag or f V . We are going to apply Lemma 34 with parameters (in the notation of that lemma) and with U j , j = 1, 2, 3, the wedges defined in (85). Here c ≪ 1 is a small absolute constant and ǫ = ǫ(z) is a small parameter that will be chosen momentarily. Note that | arg(z)| ≪ 1 since z ∈ Σ 0 and thus sin(2ϕ j ) ≍ ϕ j ≍ tan(2ϕ j ), which will be used repeatedly in the proof. We will now verify the assumptions of Lemma 34. The first condition in (86) is satisfied by definition. Using that, for j = 1, 2, d(∂U j , ∂U 3 ) = sin(2ϕ j − 2ϕ 3 )r 2 j ≍ (cǫ) j |Im z|, we find that the second and third condition in (86) become respectively, which means that the third condition is trivially satisfied. Since c 2 (|z|/ z ) 1/2 ≪ 1 and 2π θ3−ϕ3 ≤ 3, the second condition is satisfied if we choose e.g. ǫ = c 10 (|z|/ z ) 5/2 , which we do. We will next show that, for a suitable choice of where ν = 0 if f = f diag and ν = 1 if f = f V . Since in the present case we see that (87) holds. Lemma 34 thus implies that (36), (37) hold. To prove (38) we first observe that, by the maximum principle, |f | attains its maximum on the boundary of the wedge U 3 . We estimate this on the boundary component corresponding to the ray wρ 2 e 2iϕ3 , ρ > r 3 , the estimates for the other two boundary components being similar. By (31) and the bounds of Lemma 6, this maximum is bounded by the right hand side of (38), where we used that sup ρ>r3 ω q (ρ 2 e 2iϕ3 ) c ǫ −2( q d q −1)− ω q (z) and that L is strongly separating. This proves (38) for the first term on the left. The other part follows by selecting, for instance, z 2 = i 4 R 2 2 and using the estimates (similar to (32), see [69,Theorem 9 where we once again used Lemma 6 and ω q (z 2 ) ω q (z). Note that in the case of f V we can absorb the factor s(L, z 2 ) 2 in the definition of Φ p,q (L, z 2 ) into the O(1) term since Im √ z 2 1 and L is strongly separating. Since ω q (z 2 ) ≪ 1 for R 2 ≫ 1 (which is true whenever c ≪ 1), it follows that |f (z 2 )| ≥ 1/2 for c sufficiently small. This finishes the proof of (38).

4.3.
Upper bound on the resolvent away from eigenvalues. As a consequence of Lemma 13 we also obtain an upper bound for the resolvent of H V away from the spectrum. The idea is to use the following infinite-dimensional analogue of Caramer's rule (see [67, (7.10)]), Proposition 14. Suppose Assumption 1 holds. Then for all z ∈ Σ 0 , where M q (z), M q (L, z) are given by (34).
Proof. In view of the trivial bound the claim follows from Lemma 15, the resolvent identity (19), Corollary 7 and the fact that |Im z| −1 M p,q (z).

Ghershgorin type upper bounds.
We record the following Ghershgorin type bound. We temporarily restore the norm of the potential and define ω q,i (z) := ω q (z) V i L q , and M p,q,i (z) is defined by (34) with ω q (z) replaced by ω q,i (z).

Proposition 16. Under Assumption 1 the discrete spectrum of σ(H
Proof. Assume first that N < ∞, and consider the Hilbert spaces H n = L 2 (Ω n ), H = n∈[N ] H n , with operators A ij = δ ij I H + BS ij (z) and A = (A ij ) N i,j=1 . Applying the Gershgorin theorem for bounded block operator matrices due to Salas [64] (see also [78,Theorem 1.13 Note that here we are using the convention that (A ii − λ) −1 = ∞ if λ ∈ σ(A ii ). By the Birman-Schwinger principle, this implies that Again, we include the spectrum of A ii in the set on the right. By Lemma 15, we have ).
On the other hand, by (24) and the strong separation property, The claim for N < ∞ follows. Similarly, it follows for the truncated operators H (n) . Since the set (42) is independent of n the claim for the case N = ∞ then follows from the norm resolvent convergence of the truncated operators (Proposition 8). Here we consider a finite system of eigenvalues of H diag in some compact subset Σ ⋐ C \ [0, ∞). By Corollary 11, Assumption 1 implies that each point in Σ is either in the resolvent set or a discrete eigenvalue of H diag . By compactness, Σ ∩ σ(H diag ) is a finite set. We then have

Lower bounds
For δ ∈ (0, 1 3 min(1, δ 0 (Σ))) we set In general it is hard to determine δ 0 (Σ), but we still have Γ δ ⊂ ρ(H diag ) for generic δ. This is all that is needed for a lower bound on the number of eigenvalues in U δ . The norm resolvent convergence (Proposition 9) implies the following proposition. In Subsection 5.4 we will give an alternative proof using the argument principle. Let us first state our assumptions for the remainder of this section.

Argument principle.
The argument in the previous subsection involved compactness and continuity and is obviously non-quantitative. The issue is of course the need for a quantitative estimate of the resolvent on the curve Γ δ . We will prove such estimates in Proposition 14. Here we argue in a slightly different (albeit closely related) manner. We will use the regularized Fredholm determinants (see for instance [34,IV.2], [69,Chapter 9] or [23,XI.9.21]) where p ∈ [2 max(q, q d ), ∞) is assumed to be an integer. The main property that we will use is that the f diag , f V are analytic functions in C \ [0, ∞) and have zeros (counted with multiplicity) exactly at the eigenvalues of H diag , H V , respectively. Moreover, by the generalized argument principle (see e.g. [28,Theorem 3.2] or [3, Theorem 6.7]), where H = H diag or H V and f = f diag or f V . This suggests a comparison between H diag and H V via Rouché's theorem (see e.g. [66] for related ideas). We set We will show that r δ < 1 if max j∈[n] s(L, ζ j ) is sufficiently small. Rouché's theorem then asserts that f diag and f have the same number of zeros in U δ .

Alternative proof of Proposition 17.
Without loss of generality we may assume that Σ contains exactly one eigenvalue ζ of H diag . We are going to use Lemma 32. For this purpose we set U 1 = U δ and let U 2 ⊂ C \ [0, ∞) be a precompact simply connected neighborhood of U 1 containing a point ζ 2 / ∈ σ(H diag ). This is possible by (3) applied to V j (i.e. with N = 1) since we can take ζ 2 = −A, where A ≫ 1. By (32) we find that where C 1 = C 1 (δ, Σ). We take A so large that This is possible since lim A→∞ ω q (−A) = 0 by (25). By Lemma 32 there exists a constant C 2 = C 2 (δ, Σ) such that Here we used that log |f diag (z 0 )| ≥ − log 2, which follows from (47), (25) and (31). Combining (46) and (48), we infer that r δ < 1 if s(L, ζ) is sufficiently small.

5.5.
Lower bounds: Quantitative results. In the following we establish quantitative versions of Proposition 17. We return to estimating the quantity r δ in (45) featuring in Rouché's theorem.
Proof. Again we may assume that Σ contains exactly one eigenvalue ζ of H diag , so that U δ = D(ζ, δ) and Γ δ = ∂D(ζ, δ). We clearly have δ(z) = δ and ω q (z) ≍ ω q (ζ) for z ∈ Γ δ . It is easy to see that (32) and Lemma 6 imply We have also used that L is strongly separating, hence s(L, z) s(L, ζ). In order to estimate r δ it remains to bound |f diag (z)| from below using (36).
As an immediate corollary we obtain an improvement of Proposition 17.
Proposition 19. Suppose Assumptions 1, 2 hold. Then provided that L is so large that r δ < 1 in (49). 6. From quasimodes to eigenvalues 6.1. Existence of a single eigenvalue. We record a useful corollary of Proposition 14. The proof is obvious.

6.2.
Existence of a sequence of eigenvalues. If instead of a single quasieigenvalue we consider a sequence (ζ j ) j with lim j→∞ Im ζ j = 0, an across-theboard assumption like the one in Corollary 20 is not feasible since the right hand side of (41) at z = ζ n tends to infinity as n → ∞. One possible solution would be to modify the previous arguments and select L as a function of the sequence (ζ j ) j . We will follow a similar, albeit slightly different strategy which we find more intuitive. It is also closer in spirit to the inductive argument in [5], which is based on strong resolvent convergence. Once more, the approach we will outline can be viewed as a quantitative version of that method. The strategy will be to first construct quasimodes of H V in a direct way (Lemma 21) and then use Corollary 20 to obtain existence of eigenvalues. The quasieigenvalues and quasimodes will be actual eigenvalues and eigenfunctions of H diag . We introduce a sequence of scales ε j and a j , where ε j has dimension of energy and a j has dimension of length, and assume that the eigenfunctions ψ j corresponding to ζ j decay exponentially away from Ω j in such a way that where q ≥ 2 and c 0 > 0. In the following applications we can take c 0 = 1. We will then choose L such that Im ζ n L n ≥ C log n log 2 n sup where C = C(d, q) is a large constant.
Lemma 21. Assume that V ∈ ℓ ∞ L q for some q ≥ 2 and that ζ j are eigenvalues of H Vj with normalized eigenfunctions ψ j satisfying (50). Then there exists an absolute constant C = C(d, q) such that if V (L) is separating and satisfies (51), then H V has a sequence of normalized quasimodes ψ j , Remark 4. Lemma 21 could be seen as a quantitative version of Lemma 2 in [5].
Proof. Let ψ j be the eigenfunctions of H Vj corresponding to ζ j , i.e. (H Vj −ζ j )ψ j = 0. For n ∈ [N ] we make the following (stronger) induction hypothesis P (n): (recall (17) for the definition of H (n) ). The base case n = 1 is true by assumption. Assume now that P (n − 1) holds. By the exponential decay (50), where we have set η n := Im √ ζ n . Moreover, by induction hypothesis, for j ∈ [n−1], Hence P (n) would hold if L n satisfied the estimates for j ∈ [n]. By the mean value theorem and it is easy to check that (53)-(54) are satisfied for the choice (51). This completes the induction step. For N < ∞ the claim now follows from (52) with n = N . Now consider the case N = ∞. Since L is separating, Together with (52) this yields the claim for N = ∞.
Remark 5. The factor n log 2 n in (51) comes from the induction hypothesis and should not be taken too seriously. One could of course replace log n by any other slowly varying sequence tending to infinity. However, this would not change the bound (51) significantly.
6.3. Quasimode construction. We now construct the potential W j having ζ j as an eigenvalue.
Lemma 22. Given ζ ∈ Σ 0 and x 0 ∈ R d there exists a potential W = W (ζ, x 0 ) ∈ L ∞ comp (R d ) such that the following hold.
(1) H W has eigenvalue ζ; (5) The normalized eigenfunction ψ = ψ(ζ, x 0 ) of H W corresponding to ζ satisfies the exponential decay estimate Proof. By scaling it suffices to prove this for |ζ| ≍ 1. In view of the results of Section 7 (and ζ = E in the notation of that section) we can then simply choose W as a shifted step potential. The shift of course does not affect the eigenvalues nor the L q norms. The latter are trivial to compute using the size bound |W | = O(ǫ) and the formula (55) for the width of the step. The estimate (57) follows from a direct computation. The exponential decay follows from Lemma 23 or the explicit form of the wavefunction for the step potential.
Remark 6. Similar results involving complex step potentials are contained in [5], [14], [15], albeit in a less quantitative form. A technical detail that distinguishes our proof from these is that we first pick the eigenvalue, then find the potential. This avoids the use of Rouché's theorem in [14], [15].
6.4. Exponential decay. We prove that the exponential decay bound (50) holds for a class of compactly supported potentials that will be relevant in the next section. The important point here is that the constant C in (59) is independent of W .
Lemma 23. Assume that supp W ⊂ B(0, R) and ζ ∈ [0, ∞), Im √ ζ ≤ 1 2 R −1 log R, |ζ| 1/2 ≥ KR −1 for a large absolute constant K. Assume that ψ is a normalized eigenfunction of H W with eigenvalue ζ. Then there exists an absolute constant C = C(d) such that for |x| > R, Proof. Since ψ is normalized in L 2 it has units l −d/2 . By homogeneity, we may thus assume that |ζ| = 1. Since ψ solves the Helmholtz equation This would imply (59) if we could show that A has an upper bound independent of W . Since ψ is normalized, For sufficiently large K we estimate the integral from below by which proves that A ≤ 2/c d .
Corollary 24. Assume that V j (x) = W j (x − x j ) and that the assumptions of Lemma 23 are satisfied for W j , ζ, ψ j , R j . Then (50) holds for any q ≥ 2 and with Proof. Let 1 q + 1 r = 1 2 . By Hölder, and by (59), The claim follows. 6.5. A quantitative version of Boegli's example. In view of Corollary 20, given ζ j ∈ σ HV j and δ j > 0 we would like to choose ε j = ǫ j (ζ j , δ j , L) as and require that (51) holds with a j as in (60). This gives a sufficient condition on the sequence L ensuring that d(ζ j , σ(H V )) ≤ δ j . The following proposition follows immediately from Corollary 20, Lemma 21 and Corollary 24.
Proposition 25. Suppose Assumption 1 holds, V j ∈ ℓ ∞ L q for some q ≥ 2 and that supp V j (· + x j ) ⊂ B(0, R j ) for some positive R j . Let ζ j , δ j be sequences satisfying Im ζ j ≤ 1 2 R −1 j log R j , |ζ j | 1/2 ≥ KR −1 j for some large absolute constant K and δ j ∈ (0, 1/2). Assume that L satisfies (51) with a j as in (60) and ǫ j as in (61). If ζ j ∈ Σ 0 is an eigenvalue of H Vj of multiplicity m j , then D(ζ j , δ j ) contains at least m j eigenvalues of H V , counted with multiplicity.
In the following we apply Proposition 25 with V j = W (ζ j , x j ), where W is the complex step potential in Lemma 21. Clearly, V j ∈ L q (R d ) for every q ∈ [1, ∞], with |Im ζ n |.
We will also take q = ∞, so that a −d/ q j = 1. For the remainder of this section we assume the following. Assumption 3. Let q > d, and assume that (2) holds. Without loss of generality we may and will also assume that Im ζ n is monotonically decreasing.
Proof. Condition (2) states that the right hand side of (62) with p = q is bounded by ǫ 1 . Since p > q and the embedding ℓ q ⊂ ℓ p is contractive, the first claim in (i) follows. Since |Im ζ n | ≤ |ζ n | and | log |Im ζ n /ζ|| ≥ 1 for ζ n ∈ Σ 0 , Condition (2) also implies Since q > d the first bound implies (ii) and thus the second claim in (i) follows from (63). The claim (iii) follows from the second bound in (64) and (ii). Using (iii), we find This yields (iv) since M p,q (ζ n ) ≥ ω q (ζ n ) p . It also follows from (ii) that Combining (iii), (65) and (66) with the trivial lower bound |ζ n | ≥ |Im ζ n | in (34) yields (v).
Remark 7. From the first equality in (65) and the definition of M p,q (z) in (34) it is easy to see that for |ζ n | ≍ 1, we have better bounds Lemma 27. Suppose that L k k α . Then, under Assumption 3, Proof. Combining (66) with the estimate where the first bound holds since ζ n ∈ Σ 0 and the second bound follows from Lemma 26 (iii), we obtain .
The claim thus follows from Proposition 35 and Example a) following it.
Remark 8. For |ζ n | ≍ 1, we again have better bounds We assume now that for some γ > 0. This lower bound is motivated from the corresponding upper bound that results from the Ghershgorin estimate (42) and a posteriori by (70).
The following lemma is obvious.
Lemma 29. If ǫ n is defined by (61), L k k α and δ n satisfies (68), then under Assumption 3, Lemma (26) (i) and Lemma 29 imply that the right hand side of (51) (with q = ∞) is bounded by |Im ζ n | −κα−γ log n . We will show that for all but finitely many n ∈ N, which will then give a sufficient condition for the choice of L in Proposition 25, namely Here we have used (67) to estimate Im √ ζ n from below. In order to be consistent with our assumption L k k α we actually choose where, in view of (69), it suffices to take The exact choice of α is not important for us and we choose α = 1 for convenience.

Lemma 30.
Under Assumption 3 we have (69) for all but finitely many n ∈ N.
Proof. Suppose the claim is false. Then there exists a subsequence, again denoted by (ζ n ) n , such that n > |Im ζ n | − d 2 −q+1 . Since |ζ n | ≥ |Im ζ n |, (2) implies n n −1 < n |Im ζ n | 6.6. Proof of Theorem 1. We now specialize Proposition 25 to the step potential V j = W (ζ j , x j ) and the explicit choice (70), which will prove Theorem 1. Since we already know that the exponential decay bound is true for these potentials (see (58)) we do not need to check the conditions Im ζ j ≤ 1 2 R −1 j log R j , |ζ j | 1/2 ≥ KR −1 j , but it is easy to see from (55) that they do hold. Proposition 31. Suppose Assumption 3 holds, δ n > 0 satisfies (68) for some γ > 0, and let V = V (L) be the potential whose bumps V n = W (ζ n , x n ) are separated by L n in (70). Then D(ζ n , δ n ) contains an eigenvalues of H V . Moreover, V L q ≤ ǫ 1 and V decays polynomially, where κ is given by (71) for some arbitrary α > 0.
Proof. By (56), we have the bound |V (x)| |Im ζ n | for |x − x n | ≤ R(ζ n ) and zero elsewhere. Since |Im ζ n | 1 by Lemma 26 (i), it follows that V is bounded. Since κ ≥ 2, a comparison between L n and |Ω n | = R(ζ n ) in (55) shows that V is sparse. Therefore, by (16), we have L n |x n |. Hence, (70) yields from which the decay bound follows.

Complex step potential
In this section we will establish precise estimates for eigenvalues of the sperically symmetric complex step potential V = V 0 1 B(0,R) , where V 0 ∈ C and R > 0. The bound state problem for V 0 < 0 and d = 1, 3 is treated in virtually any quantum mechanics textbook (see e.g. Problem 25 and Problem 63 in [26]). We adopt the notation Here E ∈ C is the eigenvalue parameter, i.e. we consider the stationary Schrödinger equation which becomes −∆ψ − κ 2 ψ = 0 inside the step and −∆ψ − χ 2 ψ = 0 outside the step.
7.1. One dimension. We start with one-dimensional case. The solution space to (73) then splits into even and odd functions, while in higher dimensions it splits into functions with definite angular momentum ℓ. We consider odd functions as these also provide a solution for the case d = 3 and ℓ = 0 (s-waves). The standard procedure to solving the square well problem reduces the task to finding zeros of the nonlinear scalar function F (V 0 , κ) := iχ − κ cot(κR), where χ = √ κ 2 + V 0 by (72). A complete study of all the complex poles of this equation was initiated by Nussenzveig [56] for V 0 ∈ R\{0}. Subsequent articles in the physics literature [39], [18], [17], [35] investigated the case of complex potentials. The solution κ = κ(V 0 ) is not single-valued as there are branch points where ∂F/∂χ = 0. The viewpoint endorsed by [35] is to regard the equation F (V 0 , κ) = 0 as the definition of a Riemann surface. This approach treats the complex variables κ and V 0 on equal footing. In fact, it is easy to see that one can always use κ as a coordinate, i.e. one can solve for V 0 , For the purpose of the construction of the sparse potential in Subsection 6.5 we do not need to solve for κ. Instead, we pick κ first and then define V 0 by (74). To get an eigenvalue (i.e. a resonance on the physical sheet) we simply need to take care of the condition Im χ > 0, i.e.
We are only interested in complex eigenvalues E with |E| ≍ 1 (the general case can be obtained by scaling). We will try to make V 0 in (74) small, i.e. we postulate that V 0 = ǫ V 0 , where ǫ > 0 is a small parameter and V 0 ∈ C is of unit size. By (72) this implies that |κ| ≍ 1, and (74) then reveals that | sin 2 (κR)| ≍ ǫ −1 , which means that Since we are free to choose κ, set κ = ±1 + iǫσ with σ > 0, which will yield an eigenvalue with Re E = 1 + O(ǫ) and |Im E| = O(ǫ). Going back to (76) we see that we must have for some constant C. It is quickly checked that this is consistent with the bound of Abramov et al. [1] since V L 1 log 1 ǫ and |E| ≍ 1. In view of (76) we may write e 2iκR = ǫu, where u = Ce 2iRe κR . Using the Taylor approximation we obtain from (74) that which provides the desired smallness |V 0 | = O(ǫ). As already mentioned, we need to make sure that (75) holds. Using the Taylor approximation cot(κR) = i(1 + 2ǫu + O(ǫ 2 )), we find that (75) holds if In particular, for u ∈ iR + , we find that (80) forces us to choose Re κ = −1.
Adopting this choice for u, it is then easy to check that we get an eigenvalue with Re E = 1 + O(ǫ) and Im E = ǫ(4|u| − 2σ) + O(ǫ 2 ) as desired. By simple scaling arguments this proves the one-dimensional case of Lemma 22. We observe that the result is consistent with the trivial numerical range bound Im E ≤ Im V 0 ; in fact, by choosing σ small, E can be taken arbitrarily close to the boundary of the numerical range Im z = 4ǫ|u|, up to errors of order ǫ 2 .
Before we conclude the one-dimensional case we note that the same result could have been obtained with an even wavefunction, in which case sec replaced by csc in (74) and cot is replaced by tan in (75). The Taylor approximations csc 2 (κR) = 4ǫu(1 + O(ǫ)), tan(κR) = i(1 − 2ǫu + O(ǫ 2 )), and the freedom to choose the signs and the imaginary part of u yields a proof of Lemma 22 using odd solutions. 7.2. Higher dimensions. By symmetry reductions we are led to consider the radial Schrödinger equation It can be shown (see e.g. [10, (5.12)]) that an eigenvalue E corresponds to a zero of the function (Wronskian) We recall that χ, κ, E, V 0 are related by (72). Computations of resonances for spherically symmetric potentials can be found in [55], [82], [74], [10]. The last three papers use uniform asymptotic expansion of Bessel functions for large order. Here we only consider s-waves, i.e. ℓ = 0. Then we have the asymptotics With the same choice of κ as in the one-dimensional case and with u ∈ C such that e 2i(κR− πν 2 − π 4 ) = ǫu, we then obtain that the zeros of F (V 0 , κ) coincide with the zeros of a function κ sin(ω(κR)) − iχ cos(ω(κR)) + O(R −1 ), where ω(κR) = κR − πν 2 − π 4 and the κ-derivative of the error term is O(1). Recall that χ = χ(V 0 , κ) is given by (72). The zeros of the function without the error term are found exactly as in the one-dimensional case and can be parametrized by κ, v.i.z.
7.3. Proof of Theorem 4. We return to one dimension. We first prove the upper bound (11). Since |V 0 |R 0 is of order one, the bound in [29] yields that the total number of eigenvalues of H Vj is also of order one. By Proposition 17 (it is clear that the assumption on the norm of V can be dropped), given N ≫ 1, we can find L = L(N ) such that H s has the same number of eigenvalues in Σ = Σ(N ) as H diag , which is just the N -fold orthogonal sum of the H Vj and hence has less than O(N ) eigenvalues by the first part of the argument.
To prove the lower bound (10) we return to the formulas (74), (75), but we now fix V 0 = i. We also set R = N R 0 and R 0 ≍ 1, so that the dimensionless parameter |V 0 |R is of size N . We first solve an approximate equation and then use Rouché's theorem to show that the exact equation (74) has solutions close to the approximate ones. Finally, we use (75) to check that we have found a pole on the physical plane (i.e. an eigenvalue). The approximate equation is G 1 (κ) = 0, where and the approximation will be valid in the regime Im κR ≫ 1. Since G 1 can be factored, we only look for zeros of the first factor. These zeros κ n are expressed by means of the Lambert W function, where n ∈ Z and W n are the branches of the Lambert W function. According to in [11, (4.19)] the asymptotic expansion of W n (z) as |z| → ∞ is W n (z) = log(z) + 2πin − log(log(z) + 2πin) + O( log(log(z) + 2πin) log(z) + 2πin ), where log is the principal branch of the logarithm on the slit plane with the negative real axis as branch cut. For z = i √ V 0 R/2 this gives where, for N ≫ 1, the error satisfies |E n (V 0 , R)| log(log N + |n|)/(log N + |n|), where we recalled that |V 0 |R ≍ N . For the assumption Im κR ≫ 1 made before to be consistent with the formula for κ n we require Since N ≫ 1 we can neglect the logarithm in the second expression and deduce the condition |n| ≫ N , which we will assume henceforth. This gives us the error bound |E n (V 0 , R)| log |n|/|n|, which implies that in agreement with the assumption Im κR ≫ 1. We also obtain the more precise formulas Re κ n R = 2πn + 5π 4 where we assumed that n < 0. To justify this assumption, we recall from the discussion at the end of Subsection 7.1 that (79) and (80), together with (82) and the assumption V 0 = i made at the beginning of this subsection, imply that n must be negative.
Having found the large zeros of G 1 (κ) we proceed to find those of which determines the eigenvalues of the step potential (see the beginning of Subsection 7.1). We define ǫ n := exp(−2Im κ n R), so that e 2iκnR = ǫ n u n for some u n on the unit circle. Note that, by (82) Moreover, for |κ − κ n | = ǫ n we have |G ′ 1 (κ)| |κ| 2 Re −2Im κR N n 2 ǫ n , |G ′′ 1 (κ)| |κ| 2 R 2 e −2Im κR N 2 n 2 ǫ n . Using G 1 (κ n ) = 0 and Taylor expanding, it follows that |G 1 (κ)| N n 2 ǫ n ǫ n + O(N 2 n 2 ǫ n ǫ 2 n ). For this to be meaningful we must of course assume ǫ n ≪ 1/N , which we do. Then we have |G 1 (κ)| N n 2 ǫ n ǫ n . Comparing this with (83) we see that sup |κ−κn|= ǫn |G 1 (κ)| −1 |G 2 (κ) − G 1 (κ)| < 1, provided ǫ n ≫ ǫ n /N . Adopting the choice ǫ n = C N n 2 , where C is a large constant, we see that there exists a zero κ n ∈ D(κ n , C N n 2 ) of G 2 . By the smallness of N/n 2 , it follows that κ n also satisfies (82). We drop the tilde, i.e. we now denote the zeros of G 2 by κ n . Summarizing what we have done so far, we have found infinitely many resonances κ n , |n| ≫ N , of of the step potential satisfying (82). The last step is to check which of the resonances lie on the physical sheet, i.e. are actual eigenvalues. For this we need to check condition (75). By Hence, recalling that n < 0, the condition (75) is nonvoid and is satisfied whenever |n| log |n| N ≪ N 2 ; this holds for |n| ≪ N 2 / log N . Recalling (72) we obtain the complex energies E = E n , Re E n ≍ n 2 N 2 , Im E n ≍ |n| N 2 log |n| N , and those energies with c N 2 log N ≤ |n| ≤ C N 2 log N lie in the rectangle Σ (see Theorem 4). This completes the proof of the lower bound (10).

Technical tools
8.1. Lower bounds on moduli of holomorphic functions. We collect some well known results about the modulus of holomorphic functions away from zeros, based on Cartan's bound for polynomials (see e.g. [50]).
Let U 1 ⋐ U 2 ⋐ C, where U 2 is simply connected. Assume that f is holomorphic in a neighborhood of U 2 and ζ 2 ∈ U 2 . Let z 1 , z 2 . . . , z n , be the zeros of f in U 2 . Define Z f,δ,U2 := n j=1 D(z j , δ).
The following version can be found in [24,Appendix D].
We need also use a more precise version, where f is holomorphic in a neighborhood of U 3 , U j = D(0, r j ), j = 1, 2, 3, with r 1 < r 2 < r 3 and ζ 2 = 0.
for z ∈ U 2 . Denoting the numbers r j in Lemma 33 by ρ j instead (with ρ 3 = 1), we then find, using (88), where in the second line we used the triangle inequality and the second inequality in (86). By the third inequality in (86) we can Taylor expand Lemma 33 now yields the claim. is an arbitrary length scale. Note that h L is decreasing and tends to infinity as s → 0. In fact, h L is the distribution function of the sequence (η 0 L k ) −1 . Since we assume that L k is increasing, we also have h L (s) = min{k ∈ Z + : η 0 L k+1 > 1/s}. We will show that, under the assumption ∃λ ∈ (0, 1) such that lim sup the potential V (L) is strongly separating in the sense of Definition 2.  Proof. We may restrict our attention to the case δ < 1 as the case δ ≥ 1 is trivial. By (89) there exist λ ∈ (0, 1) and s 0 > 0 such that h L (λs) < e h L (s) holds for all s ∈ (0, s 0 ]. Now let n be the smallest integer such that λ n ≤ δ. Iterating (91) n times, we get h L (δs) < e n h L (s) for all s ∈ (0, s 0 ]. For s > s 0 , the inequality holds trivially. Proof of Proposition 35. Without loss of generality we may assume that η = η 0 . We first consider the case ηL 1 ≤ 1, and hence h L (1) ≥ 1. Then ∞ k=1 exp(−ηL k ) ≤ h L (1) exp(−ηL 1 ) + ∞ k=hL (1) exp(−ηL k ).
It remains to show that the second term is bounded by the right hand side of (90).