Resolvent estimates for one-dimensional Schr\"odinger operators with complex potentials

We study one-dimensional Schr\"odinger operators $\operatorname{H} = -\partial_x^2 + V$ with unbounded complex potentials $V$ and derive asymptotic estimates for the norm of the resolvent, $\Psi(\lambda) := \| (\operatorname{H} - \lambda)^{-1} \|$, as $|\lambda| \to +\infty$, separately considering $\lambda \in \operatorname{Ran} V$ and $\lambda \in \mathbb{R}_+$. In each case, our analysis yields an exact leading order term and an explicit remainder for $\Psi(\lambda)$ and we show these estimates to be optimal. We also discuss several extensions of the main results, their interrelation with some aspects of semigroup theory and illustrate them with examples.


Introduction
The structure of the pseudospectrum of non-self-adjoint operators can be very non-trivial and in general unrelated to the location of the spectrum. This fact is well-known to be responsible for typical non-self-adjoint effects such as spectral instabilities or long-time semigroup bounds unrelated to the spectrum, see e.g. [37,16,17,23] for details.
For Schrödinger operators H = −∆ + V with complex potentials V , the pseudospectral analysis was initiated in the seminal paper of E. B. Davies, cf. [15], where lower estimates for the resolvent norm inside the numerical range of H, Num(H), were obtained by a semi-classical pseudomode construction. The latter was subsequently generalised: in the semi-classical case in particular in [40,18] and in the non-semi-classical one in [30,4,29,20].
The upper estimates of the resolvent norm at the boundary of Num(H) were first obtained by L. Boulton in [12] for the quadratic potential. This work was followed up with several semi-classical generalisations in particular in [32,18,11,35,8,9,24,3] and also in [19] based on semigroup compactness or known behaviour of spectral projections.
In this paper, we study the behaviour of the resolvent norm at the boundary of Num(H) for non-semi-classical one-dimensional Schrödinger operators acting in L 2 (R + ) or in L 2 (R) for a wide class of unbounded complex potentials V ranging from iterated log functions to super-exponential ones (which are not accessible by previously used methods).
Our assumptions on V are compatible with those in [30] where lower resolvent norm estimates inside Num(H) were obtained. More precisely, restricting ourselves in this section to purely imaginary V , we assume that Im V is eventually increasing, unbounded at infinity and that the conditions (reflecting the growth of Im V ) x → +∞, (1.1) with some ν ≥ −1, are satisfied, see Assumption 3.1 for details. Moreover, the condition Υ(x) := x ν Im V (x) 1 3 = o(1), x → +∞, (1.2) is related to the separation property of the domain of H, see Sub-section 3.1.1, and the quantity Υ naturally enters the remainders in the derived asymptotic formulas (similarly to what happens e.g. for diverging eigenvalues in domain truncations in [33] or for asymptotics of eigenfunctions in [31]). It was established in [30] that (H − λ) −1 diverges as the spectral parameter λ = a + ib goes to infinity along a set of admissible curves determined by the potential. In particular, for operators in L 2 (R + ) the restriction on admissible curves is given by (with a, b ∈ R + ) Our first result (Theorem 3.2), specialised for purely imaginary potentials here, provides a two-sided estimate for the norm of the resolvent along the imaginary axis for operators on the half-line and it includes an exact leading order term and an explicit remainder estimate. Namely, (1.4) where A = −∂ 2 x + ix is the complex Airy operator in L 2 (R) (see Sub-section 2.3). In Section 5, we further explain how these results extend to operators in L 2 (R) as well as to multi-dimensional operators with radial potentials (see Sub-sections 5.3 and 5.5). Moreover, in Sub-section 5.6 we indicate how our strategy can be used in a semi-classical case where the problem substantially simplifies as only local properties of V are needed (similarly to the pseudomode construction in [30]). In Sub-section 5.1, we extend Theorem 3.2 (with Re V = 0) to describe the behaviour of the norm of the resolvent along general curves λ b = a(b) + ib inside the numerical range 2 3 . Precise resolvent estimates for semi-classical operators were found in [11]; in the special cases of the Davies operator and the imaginary cubic oscillator our construction allows us to recover those same curves (see the discussion for power-like potentials in Sub-section 7.1).
Under these conditions the resolvent norm of the operator where A β is the generalised Airy operator t a solves the equation t a Im V (t a ) = 2 √ a and ι(t) and l β,ε are determined by The additional smoothness and growth restrictions on V for this result stem from employing pseudo-differential operator techniques. The regular variation assumption arises naturally due to scaling (similarly to the analysis of the eigenfunctions' concentration in [31]). Basic properties of generalised Airy operators are described in Appendix A and a detailed spectral analysis can be found in [5], including precise resolvent norm estimates. The result (1.4) in particular relates the behaviour of V at infinity to the decay/growth of the resolvent along the imaginary axis, with the linear potential (i.e. the Airy operator) being the transition between the two cases. For sub-linear potentials, the resolvent norm diverges on the imaginary axis and the rate of divergence becomes very fast for slowly growing (e.g. iterated log) potentials (see Section 7 with several examples). The interest in such operators has been highlighted in recent research on one-parameter semigroups, e.g. [7,Thm. 1.5] relates the decay of solutions of the Cauchy problem to the growth of the resolvent norm along the imaginary axis. More precisely, if A is the generator of the bounded C 0 -semigroup (T (t)) t≥0 and σ(A) ∩ iR = ∅, then for fixed α > 0 we have For more general rates, see [36]. Inspired by the open problem presented by C. Batty [6], we note that Theorem 3.2 enables us to characterise the class of rates (e.g. |s| α ) for which we can construct potentials V such that the resolvent norm of the corresponding Schrödinger operator equals that given rate (see Section 6 for details). The proof of Theorem 3.2, originally inspired by [23,Prop. 14.13], revolves around a separate analysis of (H − ib)u depending on whether or not supp u is contained in a neighbourhood of the turning point x b designed so that Im V is approximately constant inside. More specifically, the proof consists of the following steps (several technical extensions are additionally needed for the case of potentials with non-zero real part).
(1) In Proposition 3.3, with Ω b representing a neighbourhood of x b chosen so that Im V (x) ≈ Im V (x b ) for x ∈ Ω b (see (3.10)), we use direct quadratic form estimates to find that asymptotically as b → +∞, with Υ as in (1.2).
(2) In Proposition 3.4, in a neighbourhood Ω b of x b (see (3.14)), appropriately shifted and scaled, we Taylor-approximate H − ib with the complex Airy operator A to yield as b → +∞. The norm resolvent convergence of (a localised realisation of) H − ib to the complex Airy operator A follows from the second resolvent identity and it makes use of certain graph-norm estimates introduced in Subsection 2.3.
(3) In Proposition 3.5, we show that our estimate for the norm of the resolvent of H cannot be improved by finding functions u b ∈ Dom(H) such that as b → +∞ The proof relies on exploiting the localisation technique used in step (2) and the fact that the operators involved have compact resolvent. Thus the norms of those resolvents can be obtained from the appropriate singular values and the corresponding eigenfunctions are used to find the u b family. (4) We combine the results from the previous steps with the aid of certain commutator estimates and a suitably constructed partition of unity.
The proof of Theorem 4.2, which describes the asymptotic behaviour of the resolvent norm along the real axis, follows the template outlined above but on the Fourier side and with substantial modifications at several stages. In particular, the commutator estimates in Step 4 are obtained using pseudo-differential operator techniques (see Lemma 4.4) resulting in additional smoothness and regularity assumptions.
The remainder of our paper is structured as follows. Section 2 introduces our notation and recalls some fundamental facts for the various tools used throughout (Fourier transform, pseudo-differential operators, Schrödinger operators with complex potentials, Airy operators and functions of regular variation). In Section 3 we formulate and prove Theorem 3.2 for the resolvent norm in RanV . Section 4 is devoted to the proof of Theorem 4.2 for the resolvent norm in the real line. Section 5 includes further extensions of the main theorems, in particular the resolvent estimates on more general curves in the numerical range. In Section 6 we deal with the inverse problem mentioned above and Section 7 illustrates our results on some concrete potentials. Finally, in Appendix A we show the key properties of the first order generalised Airy operators used in the proof of Theorem 4.2.
In the one-dimensional setting, we will refer to the first and second order differential operators with ∂ x and ∂ 2 x , respectively, reserving the symbols ∇ and ∆ for statements in higher dimensions.
If H denotes a Hilbert space, we shall use ·, · H and · H to represent the inner product and norm on that space. The L 2 inner product shall be denoted by ·, · 2 , or just by ·, · if there is no ambiguity, and the L 2 norm by · 2 or just by · . The other L p norms will be represented by · p with L ∞ denoting the space of essentially bounded functions endowed with the essential sup norm · ∞ .
Let ∅ = Ω ⊂ R d be open, k ∈ N and p ∈ [1, +∞]. We will denote the Sobolev spaces by W k,p (Ω) and W k,p 0 (Ω) (the latter representing as usual the closure of C ∞ c (Ω) in W k,p (Ω), see e.g. [21, Sub-sec. V.3] for definitions). We shall generally be concerned with the particular cases where Ω = R or R + , k = 1 or 2 and p = 2.
If T is a bounded operator on a Banach space X , we will denote by rad(T ) its spectral radius, i.e. rad(T ) := sup{|z| : z ∈ σ(T )} with σ(T ) denoting the spectrum of T . As usual, σ p (T ) will denote the set of eigenvalues of T and ρ(T ) its resolvent set.
If T , T b , b ∈ R + , are closed linear operators on the Banach space B, we say that T b converges to T in the norm resolvent (or generalised) sense, and we write To avoid introducing multiple constants whose exact value is inessential for our purposes, we write a b to indicate that, given a, b ≥ 0, there exists a constant C > 0, independent of any relevant variable or parameter, such that a ≤ Cb. The relation a b is defined analogously whereas a ≈ b means that a b and a b.

2.1.
Fourier transform and pseudo-differential operators. For u ∈ S (R), the Fourier and inverse Fourier transforms read (with x, ξ ∈ R) we also useû := F u andǔ := F −1 u, and retain the same notations to refer to the corresponding isometric extensions to L 2 (R). We recall that the Schwartz space, S (R), is endowed with the family of seminorms |f | k,S := max When introducing pseudo-differential operators in Section 4, we follow [1, Part I]. Given m ∈ R, the symbol class S m 1,0 (R × R) is the vector space of smooth functions p : R × R → C such that for any α, β ∈ N 0 there exists C α,β > 0 satisfying This space is endowed with a natural family of semi-norms defined by Furthermore, for m, τ ∈ R, the space of amplitudes A m τ (R × R) consists of the smooth functions a : R × R → C such that for any α, β ∈ N 0 there exists C α,β > 0 satisfying |∂ α η ∂ β y a(η, y)| ≤ C α,β η τ y m , (η, y) ∈ R × R. This space is endowed with the family of semi-norms We associate a pseudo-differential operator with the symbol p ∈ S m 1,0 (R × R) via and it can be shown that this is a bounded mapping on S (R) (see [1,Thm. 3.6] the Dirichlet Laplacian in L 2 (Ω) is denoted by −∆ D and Under these assumptions on V one can find the (Dirichlet) m-accretive realization H = −∆ D + V by appealing to a generalised Lax-Milgram theorem [2,Thm. 2.2]. It is also known that the domain and the graph norm of H separate, i.e. Dom(H) = Dom(∆ D ) ∩ Dom(V ) and is a core of H. For details see [2,28,33] and [13,26], [21, Chap. VI.2] for cases with a minimal regularity of V .
2.3. Airy operators. An important class of objects in our analysis are complex Airy operators; details on the claims summarised here can be found in [23,Ch. 14] and in Section A of this paper for the more general case. The rotated Airy operator in L 2 (R) with r > 0 and θ ∈ (−π, π) is denoted by It is well-known that A r,θ has compact resolvent, its spectrum is empty, its adjoint satisfies A * r,θ = A r,−θ and A r,θ u 2 + u 2 In Section 4, we use operators in L 2 (R) of type (with β > 0) which we refer to as generalised Airy operators (on Fourier space). The motivation for this choice of terminology is as follows. By transforming the complex Airy operator, A 1,π/2 = −∂ 2 x + ix, to Fourier space (via F A 1,π/2 F −1 , where F denotes the Fourier transform, see Sub-section 2.1), one obtains A 2 = −∂ x + x 2 . Operators of type A β extend the simple structure of A 2 . Many properties of the usual complex Airy operators are preserved for A β . Namely, A β has compact resolvent, empty spectrum, and It is in fact possible to carry out this extension further to operators A = −∂ x + W with much more general W . See Appendix A for details and [5] for resolvent estimates.

Regular variation. A continuous function
is called regularly varying (at infinity) and β is called the index of regular variation. We can rewrite V as where L is a slowly varying function, i.e.
It is known (see [34,Sec. 1.5]) that, if L is slowly varying, then where a is positive and measurable, ε is continuous and In this paper, we shall be chiefly concerned with functions with index β > 0.
3. The norm of the resolvent in the range of V 3.1. Assumptions and statement of the result. We begin by describing the class of potentials encompassed by our estimate for the norm of the resolvent.
for some x 0 ≥ 0. With V 1 := Re V and V 2 := Im V , assume further that V 1 ≥ 0 a.e. in R + and that the following conditions are satisfied: (i) V 2 is unbounded and eventually increasing: (iii) we have: (iv) V 1 is sufficiently small w.r.t. V 2 : For a potential V satisfying Assumption 3.1, the Schrödinger operator in L 2 (R + ) is specified as in Sub-section 2.2; see also our comments in Sub-section 3.1.1 below.
To state our result, we introduce r := l 2 + 1, θ := arg(l + i) ∈ (0, π/2], (3.4) with l as in (3.2). Assuming that b > 0 is sufficiently large, we denote by x b ∈ R + the unique solution (see (3.1)) to the equation (sometimes called a turning point of V 2 ) and define Furthermore, noting that by Assumption (i) and (3.5) we have x b → +∞ as b → +∞, then from Assumption (iv) we deduce that 3.1.1. Remarks on the assumptions. Firstly, potentials V satisfying Assumption 3.1 obey the separation condition (2.1). To see this, consider a cut-off function φ ∈ C ∞ c ((−2x 0 , 2x 0 )) with 0 ≤ φ ≤ 1 and such that φ = 1 on [0, Thus it suffices to verify that (2.1) holds for large x. By Assumptions 3.1 (iv), (ii) and (iii), we get for x → +∞ Secondly, we note that, for ν > 0, Assumption 3.1 (iii) stipulates that V 2 (and hence V 2 ) must grow sufficiently fast as x → +∞. For ν = 0, it simply requires that V 2 be unbounded at infinity. Finally, when ν < 0, potentials where V 2 decays to zero as x → +∞ are supported provided that such decay is slower than that of x 3ν .
Our final observation is that Assumption 3.1 (ii) implies that, for any 0 < ε < 1, all sufficiently large x and |δ| ≤ εx −ν , we have (see e.g. [31,Lem. 4.1]). We can therefore control the variation of V 2 and that of V 2 in intervals whose length is of order x −ν .

Proof of Theorem 3.2.
With λ as in (3.6), let The proof is structured in four steps. Firstly, we prove the claim "away" from the zero x b of V 2 − b. Then we study the behaviour of the norm of the resolvent locally (i.e. near x b ). Next we establish a lower bound for the norm. Our final step, the theorem proof proper, combines the previously derived estimates. Throughout the proof we are chiefly concerned with the behaviour as b → +∞ and will therefore assume b to be as large as needed for our assumptions to hold without further comment. Let where δ will be specified in Proposition 3.4 and ν ≥ −1 (see Assumption 3.1 (ii)). By remarks in Sub-section 3.1.1, the above choice for the width of Ω b implies that V 2 (x) is approximately equal to V 2 (x b ) inside that interval (see (3.8)) and this fact will be used in the proofs below. From (3.10) and the already noted fact that x b → +∞ as b → +∞, we deduce In what follows, we shall assume b to be large enough so that Step 1: estimate outside the neighbourhood of x b .

Proposition 3.3.
Let Ω b be defined by (3.10), let the assumptions of Theorem 3.2 hold and let H b be as in (3.9). Then we have as b → +∞ Proof. Define χ b (x) := sgn(V 2 (x) − b), x ∈ R + , and note that χ b ∞ ≤ 1 and Therefore (3.12) Next we find a lower bound for |V 2 (x) − V 2 (x b )| in R + \ Ω b . By Assumption 3.1 (i), V 2 is unbounded and increasing in (x 0 , +∞) and, since it is also bounded on [0, x 0 ], we have for large enough b and x ∈ R + \ Ω b Applying the mean-value theorem for the first term inside the min with ξ b ∈ (x b , x b + δ b ) and noting secondly that A similar result can be found for Hence by combining (3.13) and (3.12) we conclude that for all u ∈ Dom(H) with as required.
Proposition 3.4. Let the assumptions of Theorem 3.2 hold, let H b be as in (3.9) and define (3.14) Then as b → +∞ where s = s(x, b) and 0 < s < 1. Let and consider the operator in L 2 (R) If Ω b,ρ : where 0 <s < 1 and r b , θ b are as defined in (3.6). We are now in a position to define the value of ρ for the remainder of the proof Let us denote For any x ∈ Ω b,ρ , |sρx| ≤ 1 2 x −ν b by (3.10) and hence x −1 bs ρx + 1 ν ≈ 1, i.e. (sρx + Combining this fact with (3.8), we deduce For all x ∈ Ω b,ρ we have |ρx| δx −ν b and therefore Our next aim is to prove that S b nrc −→ S ∞ as b → +∞ with S ∞ := A r,θ from the statement of Theorem 3.2.
We begin by showing that there exists b 0 > 0 such that 0 ∈ ∩ b≥b0 ρ(S b ). Note as b → +∞. Note also that it follows from (2.3) that in the estimate of the second term we use the fact that (x(S * ∞ ) −1 ) * is bounded and therefore from the property of adjoint (AB) * ⊃ B * A * , if AB is densely defined, we get that S −1 ∞ x has a bounded extension. Hence, using (3.21) and (3.20), we obtain It therefore follows from (3.7) and an appropriate choice of sufficiently small δ > 0 (independent of b) that, for all large enough b, the operator I + (V b − re iθ x)S −1 ∞ is invertible and This shows that indeed 0 ∈ ρ(S b ), b → +∞, as claimed. Furthermore, using (3.21) and (3.22) we deduce We now prove that S b nrc −→ S ∞ as b → +∞. Using the second resolvent identity, (3.17), (3.21), (3.23) and (3.18), we obtain We therefore conclude that Let b ≥ b 0 and u ∈ Dom(H) such that supp u ⊂ Ω b . Then u ∈ Dom( H b ) and H b u = H b u (we view a function from L 2 (R + ) as belonging to L 2 (R) using the natural embedding). Finally, with v := H b u ∈ L 2 (R), we conclude that
Proposition 3.5. Let the assumptions of Theorem 3.2 hold and let H b be as in (3.9). Then there exist functions 0 = u b ∈ Dom(H) such that Proof. We retain the notation introduced in the proof of Proposition 3.4; in particular, S ∞ := A r,θ and S b is as defined in (3.19). With a sufficiently large b 0 > 0, the operators , on L 2 (R), are compact, self-adjoint and non-negative. Let 0 < ς 2 b := rad(B b ) = max{z : z ∈ σ(B b )} and let g b ∈ L 2 (R) be a corresponding normalised eigenfunc- and it is straightforward to verify that (3.25) Moreover, from (3.24), we obtain Note also that arguing as in the justification of (3.23) and recalling (2.4), we obtain Using (3.10) and (3.15), we find by Assumption 3.1 (iii). As a consequence, The last two terms can be estimated using (3.25), (3.26), (3.27), (3.28) and (3.29) Recalling from the proof of Proposition 3.
from which the claim follows.
4. The norm of the resolvent in the real axis 4.1. Assumptions and statement of results. We begin by describing the class of potentials covered by our estimate for the norm of the resolvent in the real axis.
(ii) V 2 is eventually increasing: (iv) V 2 has controlled derivatives: For potentials V satisfying Assumption 4.1, we consider the Schrödinger operator To state the result, we define the positive real numbers t a via the equation notice that t → tV 2 (t) is eventually increasing by Assumption (4.1), thus a → t a is well-defined for all sufficiently large a > 0. Moreover, it follows that t a → +∞ as a → +∞. Finally, let Lemma 4.7 shows that ι(t) → 0 as t → +∞.
Theorem 4.2. Let V = iV 2 satisfy Assumption 4.1 and let H be the Schrödinger operator (4.5) in L 2 (R). Furthermore let A β be the generalised Airy operator (2.5), let t a be as in (4.6) and let ι be as in (4.7). Then as a → +∞ with 0 < ε < β arbitrarily small and l β,ε := 1 − ε, β > 1/2, Moreover, by Assumption 4.1 (iv) with n = 1, for any arbitrarily small ε > 0 and it follows that V satisfies condition (2.1). Hence the graph norm of H separates Finally, the following estimates for the derivatives of W t shall be used in Steps 2 and 3 of the proof of Theorem 4.2. . Then for each n ∈ N , there exists a constant D n , independent of t, such that for all t > t 0 , with a sufficiently large t 0 > 0, independent of n, and all |x| ≥ 1 (4.12) Proof. The claim follows from (4.4), (4.10) and |x| ≥ 1, namely

Proof of Theorem 4.2.
We transform the problem to Fourier space and implement there the strategy of Sub-section 3.2. To this end, we introduce the operators in L 2 (R) (4.13) Notice that H = V − i ξ 2 , Hu = Hǔ for all u ∈ Dom( H) and V u = Vǔ for all u ∈ Dom( V ). Thus the separation of the graph norm of H, see (4.11), yields (4.14) The proof has an analogous structure to that of Theorem 3.2 but nonetheless some steps are more technical. In particular, our simple estimate of the commutator of −∂ 2 x and a cut-off partition of unity in Step 4 of Theorem 3.2 (see Sub-section 3.2.4) requires more effort here (see Step 0 below).

4.2.1.
Step 0: commutator estimate. The proof of our next lemma specialises that of [1,Thm. 3.15] for the operators that we are interested in.
, we define the operators (with P := P (0) and Q := Q (0) ) Then, for any N ∈ N 0 , we have where R N +1 is a pseudodifferential operator with symbol r N +1 ∈ S m−N −1 Moreover, for every N ∈ N with N > m, there exist l = l(N ) ∈ N and K N > 0, independent of F and φ, such that and therefore both symbols define continuous mappings on S (R) (see [1,Thm. 3.6]). An analogous claim holds for P (j) u, Q (j) , j ∈ N. Furthermore, by the composition theorem [1,Thm. 3.16], P Q is a pseudo-differential operator with symbol p#q ∈ S m 1,0 (R × R) determined by Thus the composition formula (4.16) follows by simple manipulations. In the following, x, x , ξ, ξ ∈ R and α = ( with a ξ,x given by (4.20). Using the assumption (4.15), we obtain where in the last step we have used the fact Notice that C N,β is independent of ξ, x, ξ , x and θ and therefore (4.21) shows that . Applying Fubini's theorem for oscillatory integrals [1, Thm. 3.13] to (4.19), we deduce that for any Moreover, by Peetre's inequality (see [1,Lem. 3.7]) Therefore (4.21) also implies that, for any ξ, Hence by [1,Thm. 3.9], for a sufficiently large l ∈ N (depending on N ) , it follows that, for any N > m, ξ ∈ R and β 1 ∈ N 0 , ∂ β1 x r N +1 (ξ, ·) ∈ L 1 (R) and therefore is well-defined. Moreover, by (4.22), for large enough l ∈ N and some C N,l > 0 (independent of F and φ) and The claim (4.18) follows by Young's inequality and (4.23).

4.2.2.
Step 1: estimate outside the neighbourhoods of ±ξ a . For a ∈ R + , we shall denote where the parameter δ will be specified in Proposition 4.9 and Let Ω a,± be defined by (4.24), let the assumptions of Theorem 4.2 hold and let H a be as in (4.25). Then as a → +∞ Proof. In what follows, we shall assume a to be large and positive. Let 0 = u ∈ Dom( H) with supp u ∩ (Ω a,+ ∪ Ω a,− ) = ∅ and consider appealing to Assumption 4.1 (iv) with n = 1 for the last estimate. Furthermore, for any ε > 0, there exist C ε > 0 such that Noting also that, for any ξ ∈ supp u, there exists C δ > 0 such that Hence, with an appropriate choice of ε, we conclude that there exists C δ > 0 such that which proves the claim.

4.2.3.
Step 2: estimate near ±ξ a . We start with three lemmas used in the proof of Proposition 4.9 below.
Proof. Because of Assumption 4.1 (i), it suffices to consider a ≥ 0. Assume firstly that a > 0 and let L be the slowly varying function such that where we have used the assumption that V 2 is increasing in [x 0 , +∞). Therefore, by (4.10) and Assumption 4.
Let ε > 0, then there exists M 1 > 1 such that (4.28) Let L be the slowly varying function such that V 2 = ω β L (see (2.7)-(2.8)) and consider γ ∈ (0, β). Using the representation of L in (2.10) and properties of a and (see (2.11)), there exists τ 1 > 1 such that for all t > τ 1 and x > 1, we have Therefore by (2.7) and we conclude that there exists M 2 ≥ M 1 such that Combining (4.28) and (4.31), we find that Notice that for any x ≥ 0 and t > 0 We now apply Lemma 4.6 to [0, M 2 ] to deduce that there exists which, in conjunction with (4.32) and (4.33), yields the desired claim.
and there exists C > 0, independent of t, such that The same statements hold true for (S 0 t ) * . Proof. First observe that (4.4) with n = 1 and (4.10) imply that and therefore for every t > 1 and all sufficiently large x Hence for every t > 1, Dom(W t ) = Dom(V ). Next, consider φ ∈ C ∞ c ((−2, 2)), 0 ≤ φ ≤ 1 such that φ = 1 on (−1, 1) and denoteφ := 1 − φ. We split W t as W t = φW t +φW t and show that φW t is uniformly bounded andφW t satisfies (A.7) uniformly in t. The claims then follow from Proposition A.2.
Firstly, by the locally uniform convergence of W t to ω β (see Lemma 4.6) Secondly, since suppφ is bounded and W t converges to ω β locally uniformly. Moreover the last term in (4.35) is estimated using (4.12) with n = 1 and the fact that suppφ is outside (−1, 1). Thus altogether we obtain thus (A.7) is indeed satisfied (uniformly for all sufficiently large t). Define with ξ a , δ a as in (4.24). Let the assumptions of Theorem 4.2 hold and let H, H a , A β , t a and ι be as in (4.13), (4.25), (2.5), (4.6) and (4.7), respectively. Then as a → +∞ (4.37) Proof. We shall derive estimate (4.37) for u such that supp u ⊂ Ω a,+ . The procedure when supp u ⊂ Ω a,− is similar (see our comments at the end of the proof). Clearly With V as in (4.13), let us define the following operator in L 2 (R) Given t > 0 to be chosen below, we define a unitary operator on L 2 (R) by Then with Ω a,t := (−2δ a t, 2δ a t) In what follows, we select t as t := t a , where t a is defined by equation (4.6), i.e. t a V 2 (t a ) = 2ξ a , and we recall that t a → +∞ as a → +∞. We denote and, from (4.39) and δ a = δξ a , we obtain We further denote , where (with an abuse of notation) S 0 a := S 0 ta and W a := W ta from Lemma 4.8. Our next aim is to show that S a := S 0 a + R a (4.41) converges to S ∞ := F A β F −1 in the norm resolvent sense as a → +∞. The spectra of A β and S 0 a , and hence those of S ∞ and S 0 a , are empty, see Lemma 4.8 and Proposition A.1. Moreover Take φ 1 , φ 2 ∈ S (R) and define with ϕ 1 := (1 + W a )ψ 1 and ϕ 2 := (1 + ω β )ψ 2 . From the graph norm estimates (4.34) and (2.6), we obtain Therefore, with ι from (4.7) and ι a := ι(t a ), Hence by Lemma 4.7, the density of S (R) in L 2 (R) and a standard resolvent identity argument, see e.g. the proof of [14, Lem. 2.6.1], we arrive at (employing (4.42)) as a → +∞. We transport the graph-norm estimate (4.34) to the Fourier side and thus in particular (similarly as in the justification of (3.21)) Combining (4.45) and (4.40), we deduce that R a ( S 0 a ) −1 δ as a → +∞. It follows, by choosing a sufficiently small δ > 0, independently of a, that the bounded operator I + R a ( S 0 a ) −1 is invertible, for all large enough a, and This shows that 0 ∈ ρ( S a ) for a → +∞ and furthermore, using (4.45), we deduce By the second resolvent identity, we have as a → +∞ where, for the last estimate, we have applied  Noticing that S a = V 2 (t a ) −1 U a,ta H a U −1 a,ta (see (4.38)) and that H a u = H a u for 0 = u ∈ Dom( H) such that supp u ⊂ Ω a,+ , we arrive at as required.
For the case supp u ⊂ Ω a,− , we repeat the above arguments but defining instead
where for any arbitrarily small 0 < ε < β l β := 1, β > 1/2, Proof. We retain the notation introduced in the proof of Proposition 4.9; in particular, S ∞ = F A β F −1 and S a from (4.41). The proof follows the steps of that of Proposition 3.5.
With a sufficiently large a 0 , let g a ∈ Dom( S * a S a ), g a = 1, a ∈ (a 0 , +∞], such that S a g a = ς −1 a = S −1 a −1 . Note that from (4.47) we obtain a t a , 2δ a t a )), 0 ≤ ψ a ≤ 1, ψ a = 1 on (−δ a t a , δ a t a ) and such that with N := β + 1 and sufficiently large l ∈ N (see the statement of Lemma 4.4 and, in particular, (4.18)). Recall that t a → +∞ as a → +∞ (see (4.6)), hence ψ a → 1 pointwise in R as a → +∞. Next, we justify that ψ a g a ∈ Dom(F W a F −1 ) and therefore ψ a g a ∈ Dom( S a ). Similarly to (4.29)-(4.30) (but estimating instead an upper bound), and using the locally uniform convergence of W a to ω β (see Lemma 4.6), we find that W a (x) x β+γ , x ∈ R, with any arbitrarily small 0 < γ < β, for all sufficiently large a. Moreover, as in the proof of Lemma 4.8, consider φ ∈ C ∞ c ((−2, 2)), 0 ≤ φ ≤ 1 such that φ = 1 on (−1, 1) and denoteφ := 1 − φ. Then the estimate (4.12) and Leibniz rule show that there exist C n , C n > 0, independent of a, such that for all sufficiently large a, Thus for sufficiently large a, F :=φW a satisfies the assumptions of Lemma 4.4 (with constants independent of a). Hence, for all u ∈ S (R), we have with C N > 0 independent of a. Hence by (4.49) Since W a converges to ω β uniformly on bounded sets (see Lemma 4.6), we have Moreover, S (R) is a core for F W a F −1 and we conclude that [F W a F −1 , ψ a ] is relatively bounded w.r.t. F W a F −1 . Hence we have indeed ψ a g a ∈ Dom( S a ). Next, we write S a ψ a g a = S a g a + (ψ a − 1) S a g a + [Fφ W a F −1 , ψ a ]g a and we proceed to estimate all the above terms but the first one. Employing (4.48), (4.46) as well as the graph norm separation as in (4.44) for S 0 a (and analogously for the adjoint S * a ), we obtain as a → +∞ in the last two estimates we have also used (4.51) and (4.52), respectively. Since For β > 1/2, we have where in the last step we use (φ ω β ) 1, φ ω β ∞ 1 and ǧ a ∞ g a 1 ξ g a 1, ǧ a ξg a 1.

Extensions and further remarks
5.1. The norm of the resolvent inside the numerical range. A simple application of the triangle inequality allows us to obtain estimates for the resolvent norm in regions adjacent to the imaginary and real axes as well as to include further bounded perturbations. In detail, for an operator H (as in Sections 3, 4), λ, µ ∈ C and a bounded operator W , we get In particular, for H as in Section 3 with purely imaginary V satisfying Assumption 3.1, Theorem 3.2 and (5.1) with λ = ib, µ = a ≥ 0, W = 0 yield as b → +∞. Thus assuming that V 2 does not grow too slowly (e.g. V 2 is bounded below by a strictly positive constant), that b is large enough and that ε, ε > 0 are sufficiently small, the region in the first quadrant of C (which contains the numerical range of the operator and its spectrum, if any) determined by In both cases, bounded perturbations W can be included in an analogous way.
even outside the region determined by (5.2). Let for simplicity V = iV 2 obey Assumption 3.1 and, with ρ and Υ as defined in (3.15) and in Assumption 3.1 (iii), respectively, let a : R + → R + satisfy In our analysis, we shall be mainly concerned with two types of curves: for b → +∞, corresponding asymptotically to the critical region (5.2), i.e. where µ b satisfies , b → +∞, and therefore λ b grows away from the critical region, i.e. where µ b satisfies Note that, in the first case, we have Φ b = O(Υ(x b )) due to the fact that (A 1,π/2 − z) −1 is bounded on compact sets in C and therefore, by Assumption 3.1 (iii), condition (5.5) holds automatically. We further observe that, for any z ∈ C, it can be shown that ( [23,Sec. 14.3.1]) and that there exists a precise asymptotic estimate for z ∈ C + (see [11,Cor. 1.4]) For any µ ≥ 0, applying standard arguments it is also possible to extend the graphnorm estimate (2.3) , and to deduce from this (see e.g. (3.21), (3.27)) Then Sketch of proof. We shall sketch the proof of this result by closely following the steps in Sub-section 3.2, keeping the notation introduced there but omitting details whenever the arguments used earlier remain valid.
Step 1 Repeating the reasoning in Proposition 3.
Step 2 With H b and S b as in Proposition 3.4, it is clear that (recall S ∞ = A 1,π/2 ) We shall prove next that ∞ is bounded and invertible and moreover by (5.9) we have Recalling from Proposition 3.4 that 0 ∈ ρ(S b ) for large enough b and defining . Moreover, by (5.13), (3.24) and (5.5), we have It follows that K b is invertible and and, using (3.23), (3.27) and (5.13), we deduce as b → +∞ Furthermore, we have → +∞, and therefore by (3.24) and (5.13) and using the fact that µ b satisfies (5.6) or (5.7) It follows that (5.16) and hence from (5.12) as b → +∞ Arguing as in the last stage of Proposition 3.4, this yields as b → +∞ (5.17) Step 3 We follow the proof of Proposition 3.5, replacing Recalling the cut-off functions ψ b , we write and, applying (5.14) and (5.18) (refer also to (3.28) and (3.29)), we deduce Hence, noting that ς b,∞ is bounded below by a positive constant when µ b ∈ R + , we have As before, we set Step 4 Repeating the commutator calculations in the proof of Lemma 3.6 for H b − a b , we find for all u ∈ Dom(H) and k ∈ {0, 1} We remark that as expected formula (5.20) indicates that the level curves of a sublinear potential (where ρ −2 → 0 as b → +∞) will cross the imaginary axis into C − . We also note that, when V 2 (x) = x 2 (i.e. H is the Davies operator), then x b = b 1 2 and the above equation becomes (compare these curves with (7.4) for n = 2 and with the known formulas derived in [11,Prop. 4.6]).

5.1.2.
The norm of the resolvent along curves adjacent to the real axis. We can similarly estimate (H − λ) −1 , for H as in (4.5) and potential V := iV 2 satisfying Assumption 4.1, along general curves adjacent to the real axis , and A β , ι and l β,ε are as defined in (2.5), (4.7) and (4.9), respectively. We are interested in two types of curves: (1) λ a with b a V 2 (t a ) for a → +∞, corresponding asymptotically to the critical region (5.3), i.e. where µ a satisfies µ a 1, a → +∞; (5.24) (2) λ a with V 2 (t a ) = o(b a ), a → +∞, and therefore λ a grows away from the critical region, i.e. where µ a satisfies µ a → +∞, a → +∞. (5.25) Note that, in the first type above, we have Φ a = O(ι(t a ) + (a 1 2 t a ) −l β,ε ) due to the fact that (A β −z) −1 is bounded on compact sets in C and therefore, by Lemma 4.7 and the fact that t a → +∞ as a → +∞, condition (5.23) holds automatically.
In [5,Ex. 4.3], the following asymptotic estimate was found Before formulating our result, we also note that, for any µ ≥ 0, it is possible to extend the graph-norm estimates (A.10) applying standard arguments to and it follows (see e.g.  Sketch of proof. We shall sketch the proof of this result by closely following the steps in Sub-section 4.2, keeping the notation introduced there but omitting details whenever the arguments used earlier remain valid. We introduce the operators (refer also to (4.13) and (4.25)) Step 1 Repeating the reasoning in Proposition 4.5 (replacing H a with H λa ) and applying (4.27), we find that for all u ∈ Dom( H) such that supp u ∩ (Ω a,+ ∪ Ω a,− ) = ∅ (with ε > 0 arbitrarily small and some C ε > 0) using assumption (5.22) in the last step. This proves that (4.26) continues to hold when we replace H a with H λa .
Step 2 We retain the notation in Proposition 4.9. From (4.38), (4.39) and (4.41), we have 1 Our next aim is to prove that µ a ∈ ρ( S a ) as a → +∞. To do this, we argue as in Step 2 of Proposition 5.1. For any µ a > 0, the operator K a,∞ : ∞ is bounded and invertible and moreover by (5.27) we have Recalling from Proposition 4.9 that 0 ∈ ρ( S a ) for large enough a and defining K a := I − µ a S −1 a = S −1 a ( S a − µ a ) = ( S a − µ a ) S −1 a , we find K a = K a,∞ (I − µ a K −1 a,∞ ( S −1 a − S −1 ∞ )). Moreover, by (4.47), (5.32) and (5.23), we have It follows that K a is invertible and K −1 a K −1 a,∞ as a → +∞. Since S a − µ a = K a S a = S a K a , we conclude that µ a ∈ ρ( S a ) for a → +∞, as claimed. Moreover, and, using (4.46) and (5.32), we deduce as a → +∞ Furthermore, we have (see the argument in (5.15)) a , a → +∞, and therefore by (4.47) and (5.32) and using the fact that µ a satisfies (5.24) or (5.25) It follows that , a → +∞, (5.34) and hence from (5.31) and (5.34) as a → +∞ Arguing as in the last stage of Proposition 4.9, this yields as a → +∞ Step 3 We follow the proof of Proposition 4.10, replacing S a with S a − µ a , to find g a ∈ Dom(( S * a − µ a )( S a − µ a )) such that Recalling the cut-off functions ψ a , we write and we proceed to estimate the terms in the right-hand side, except the first one, as we did in Proposition 4.10. Applying (5.33), (5.36) and (5.37), we have as a → +∞ a,∞ . From the proof of Proposition 4.10, the fact that S a = K −1 a ( S a − µ a ) for large enough a > 0 (refer to Step 2 above, noting also (5.32)), (5.36) and (5.37), we find as a → +∞ Using again the proof of Proposition 4.10, estimate (4.46) and the above arguments, we have as a → +∞ To estimate the last term in the right-hand side of (5.38), we adapt the corresponding section of the proof of Proposition 4.10.
(ψ a − 1)F φ W a F −1 g a ι(t a ) + (ψ a − 1)F φ ω β F −1 g a . (5.41) For β > 1/2, using (4.44) and arguing as in (5.39), we have as a → +∞ For β ∈ (0, 1/2], using (φω βǧa ) = φ ω βǧa + φω βǧ a + φω βǧ a , estimating each term in turn as in the proof of Proposition 4.10 and then proceeding as in the β > 1/2 case, we obtain as a → +∞ Hence, going back to (5.41), we have as a → +∞ and therefore, returning to (5.38) with the individual term estimates, we obtain Writing ψ a g a = g a +(ψ a −1)g a , we deduce that ψ a g a = 1+O(ς −1 a,∞ Φ a ) as a → +∞ (see (5.40)) and hence, applying (5.42) and (5.37), we arrive at Recalling that S a − µ a = (V 2 (t a )) −1 U a,ta H λa U −1 a,ta (see (5.31)) and letting u a := U −1 a,ta ψ a g a , then u a ∈ Dom( H) with supp u a ⊂ Ω a,+ . It follows Step 4 We begin by updating the commutator estimate (4.56) provided in Lemma 4.12. Starting with (4.59) and applying (5.29) followed by (5.22), we find As in the proof of Lemma 4.12, we consider firstly the case β < 2. The initial commutator estimate remains valid Turning to the first term in the right-hand side and applying (4.4) with n = 1, (5.45), (5.30) from Step 1, (5.22) and the fact that t 2 Furthermore using (4.16), (4.4) with n = 1 and (5.45), we have a + 1) u and, returning to (5.46), this yields the estimate (1), a → +∞. We note that estimate (4.60) plays no role in the proof of Lemma 4.12 for the case β ≥ 2. On the other hand, estimate (4.59) is indeed used but its replacement here (5.44) is simply a matter of substituting H a u with H λa u ; the same substitution happens between (4.27) (from Proposition 4.5), which is also used in Lemma 4.12 when β ≥ 2, and (5.30) (Step 1 above). We can therefore repeat the proof for β ≥ 2 to derive where for any arbitrarily small ε > 0 (5.48) We also need to revise estimate (4.71) in Lemma 4.13. We repeat the original argument, with H λa instead of H a , to arrive at (4.72). Taking N ≥ 5 in (4.53), we have R N +1,k u a −3 u for k ∈ {1, 2} and a → +∞. Moreover, expanding B N,k u as in (4.73) and using our new commutator estimates, we have for any arbitrarily small ε > 0 and k ∈ {1, 2} Θ(a, ε)) u , a → +∞. We can now estimate the first term in the right-hand side of (4.72) as a → +∞ where in the last step we have used the fact that as a → +∞ we have β < 2, Θ(a, ε)a (refer back to the definition (5.47)). We can similarly estimate the other terms in the right-hand side of (4.72) to find as a → +∞ Finally, we combine the above results to prove the proposition. As in the proof of Theorem 4.2, we have as a → +∞ with small ε > 0 and Θ(a, ε) as in (5.47). By (5.35), (5.49) and (5.50), we obtain for a → +∞ Combining (5.51) and (5.52), we find that as a → +∞ Recalling (5.47) and a −1 t 2 a = o(1), we find as a → +∞ where in the last line we have applied (5.48). It follows This result combined with the lower bound (5.43) yields (5.28).
We conclude this sub-section with a general construction for the level curves of the resolvent similar to that in Sub-section 5.1.1 but considering now those adjacent to the real axis. Letting ζ a := µ 3β+1 2β a (A β − µ a ) −1 , we re-write (5.26) as This is an equation in µ a which we can solve using the Lambert function W 0 (x) (refer to Sub-section 5.1.1 for further details) Using (5.28) and substituting (H − λ a ) −1 = ε −1 , with ε > 0, we obtain (recall t a V 2 (t a ) = 2a Finally, we note that, when V 2 (x) = x 2 (i.e. H is the Davies operator), then t a = 2 (compare these curves with (7.5) for n = 1 and with the known formulas derived in [11,Prop. 4.6]).

5.2.
Optimality of the pseudomode construction in [30]. In this paper, the curves in C along which the norm of the resolvent diverges are found by a nonsemi-classical pseudomode construction. As a corollary of (5.2), using Assumption 3.1 (ii), we find that for any ε > 0, the norm of the resolvent is uniformly bounded inside the region determined by a b Similarly using (5.3) we obtain optimality of the upper edge of the condition [30,Eq. (5.5)] (with ν = −1). Denoting the regular variation index of V 2 by β > 0, we obtain from (4.6) and (2.7) that t a = (2a Hence, recalling that t a → +∞ as a → +∞ and using (2.9), we get (with any Then, using (5.55), inequality (5.3) can be rewritten as (the constant C β,ε > 0 can be given explicitly) 5.3. Extension of Theorem 3.2 to operators in L 2 (R). We outline a procedure to extend Theorem 3.2 to operators defined on the real line.
Since u + ⊥ u − and H b u + ⊥ H b u − in L 2 , by combining (5.62) and (5.63) we find , then arguing as in Proposition 3.3 we deduce that for large enough b reasoning as in the proof of Proposition 3.3 and applying Assumption 5.3 (ii). Hence Furthermore, arguing as in the proof of (3.32), we are able to derive upper estimates as and Combining the lower and upper estimates (5.64), (5.65), (5.66) and (5.67) and noting as above b −1 ≈ α b,± γ b,± , we have as b → +∞ Hence, by Assumption 5.3 (iii), for any u ∈ Dom(H) we obtain If max{α b,± } = α b,+ , using Proposition 3.5 we can find a family of functions u b ∈ Dom(H) such that as b → +∞ and it therefore follows as b → +∞ We can similarly argue when max{α b,± } = α b,− .
Our strategy to prove Theorem 3.2 can be re-deployed, with minimal and obvious changes, when Assumption 5.3 (i) is replaced with where we have used the fact that A 1,−π/2 = A * 1,π/2 and therefore A −1 1,−π/2 = A −1 1,π/2 . One can analogously treat the case where the potential is bounded on one of the half-lines and unbounded on the other one.
Finally, without going into details, we remark that our analysis for general curves in the numerical range (see Subsection 5.1) can be extended, using the above methodology, to the whole line. For example, with V satisfying Assumption 5.3,

5.4.
Extension of Theorem 4.2 to operators in L 2 (R + ). We briefly indicate how Theorem 4.2 can be adapted for operators H + = −∂ 2 x + V + in L 2 (R + ) subject to a Dirichlet boundary condition at 0 and with V + := iV 2,+ satisfying the conditions in Assumption 4.1 for x > 0. The even extension V 2 of V 2,+ to R, and the corresponding complex potential V := iV 2 , satisfy Assumption 4.1 up to a possible lack of smoothness at 0, which can however be removed by a compactly supported perturbation W , similarly as in Subsection 5.1. Then Theorem 4.2 can be applied to H = −∂ 2 x + V in L 2 (R). Since the odd extension of each u + ∈ Dom(H + ) belongs to Dom(H) and for each odd u ∈ Dom(H), we have (Hu) R+ = H + (u) R+ , we arrive at the desired inequality for any u + ∈ Dom(H + ) (see (4.77) in the proof of Theorem 4.2)

Extension of Theorem 3.2 to radial operators.
Let v : R + → R + and consider the Schrödinger operator in L 2 (R d ) with d ≥ 2 We justify below that the claim of Theorem 3.2 remains valid in this case; a relatively small real part of the potential (in the sense of Assumption 3.1) can be added in a straightforward way but we omit the details.
Proposition 5.5. Let H be the radial Schrödinger operator in L 2 (R d ) as in (5.69) with d ≥ 2 and with v such that V := iv satisfies Assumption 3.1. Then Sketch of proof. The first step of the proof (see Sub-section 3.2.1) can be performed in the same way using the multidimensional operator H, i.e. we split , and the rest.
Since λ b,l ≥ 0 for all l ≥ l d and all large b and A is m-accretive, we get for finitely many l ∈ N 0 , l < l d , the same claim follows by treating T b,l as a perturbation. The rest of the proof of this step is the same as the one in Subsection 3.2.2 and can be reformulated as an estimate for the full operator H. The third step (see Sub-section 3.2.3) can be performed for S b,l with a fixed l and so it requires only minor and straightforward adjustments.
The last step (see Sub-section 3.2.4) is completely analogous.
5.6. Remarks on semi-classical operators. We indicate how the strategy of Theorem 3.2 applies in the semi-classical case for the operator H h = −h 2 ∂ 2 x + V in L 2 (R + ) subject to Dirichlet boundary condition at 0 with h > 0, h → 0 and V := iV 2 , noting that resolvent norm bounds for semi-classical Schrödinger operators with purely imaginary potential have been derived in e.g. [32,18,11,35,8,9,24,3]. We assume that 0 ≤ V 2 ∈ C 2 (R + ) satisfies the conditions in Sub-section 2.2 so that H h is m-accretive. Suppose in addition that x 0 ∈ R + is such that V 2 (x 0 ) = 0 and there is δ 0 > 0 such that inf |x−x0|≥δ0 In Step 2 (see Sub-section 3.2.2), for functions u supported in I := (x 0 −2δ h , x 0 + 2δ h ), taking out the factor h 2 , the shift x → x + x 0 , rescaling x → σx and Taylor's theorem lead to operators in L 2 (R) Selecting σ so that σ 3 h −2 = 1, we obtain 3 +ε with ε > 0, we readily obtain the norm resolvent convergence to the Airy operator A r,θ , with r = |V 2 (x 0 )| and θ = sgn(V 2 (x 0 ))π/2, see Sub-section 2.3, Thus, rewritting (5.74) for H h , we arrive at (for the considered functions u) Following the strategy in Step 4 (see Sub-section 3.2.4), we combine the estimates (5.73), (5.75) above. To this end we employ a cut-off φ satisfying φ(x) = 1 for Moreover, a simple estimate of the quadratic form yields u 2 ≤ h −2 (H h − λ)u u , u ∈ Dom(H h ). By following the steps in Step 4, we obtain Finally, it is straightforward to adapt the reasoning in Proposition 3.5 (see Subsection 3.2.3) to prove that the bound (5.76) is optimal and we omit the details.
6. An inverse problem In [7, Thm. 1.5], the authors relate the rate of time-decay of the norm of a one-parameter semigroup to the rate of growth of the norm of the resolvent of its generator along the positive part of the imaginary axis. Inspired by the presentation on this topic in [6], we consider the following problem. Which assumptions must a non-negative, unbounded function r : R + → R + satisfy for there to exist a potential iV 2 such that the Schrödinger operator H = −∂ 2 x + iV 2 verifies (H − ib) −1 = r(b) as b → +∞? Theorem 3.2 enables us to answer this question as follows.
Proposition 6.1. Let r ∈ C 1 (R + ; R + ) and r(y) → +∞ as y → +∞. Assume furthermore that r satisfies the following conditions as y → +∞: Then the potential V := iV 2 , where V 2 is a real function determined by the equation with A 1,π/2 as in (2.2), satisfies Assumption 3.1 with ν = −1 and with x b as in (3.5).
Proof. Note that (6.4) can be indeed solved as the left-hand side is an increasing function in y := V 2 (x). It is immediate that V 2 determined by (6.4) satisfies (6.5). Moreover such V 2 is unbounded and increasing. It remains to verify Assumptions 3.1 (ii) and (iii). Firstly, by differentiating (6.4) and employing (6.1), we have Similarly using (6.1) and (6.2) Lastly, by (6.3) As for the second statement in the Proposition, let r be regularly varying with index β > 0 (see Sub-section 2.4) and eventually increasing and assume furthermore that it satisfies (6.6). From the facts that r is bounded on compact subsets of R + and that it is eventually increasing, we have as y → +∞ Finally, calling W y (t) = r(yt)/r(y), ω β = t β , t ≥ 0, and arguing as in Lemma 4.6, it is possible to show that (W y − ω β ) χ [0,1] ∞ → 0 as y → +∞, and we have for y → +∞, as required.

Examples
We illustrate the behaviour of the norm of the resolvent in several examples where the numerical range, Num(H), and the spectrum, if any, lie in the first quadrant of the complex plane. In the sequel we denote Ψ(λ) := (H − λ) −1 .
For V (x) = ix 2n , n ∈ N, we find similar formulas with improved remainder term for the real axis (in this case, ι(t a ) = 0 and moreover we can take ε = 0 in (4.8)) We can also derive estimates for odd potentials V (x) = ix 2n−1 , n ∈ N, along both the positive and negative parts of the imaginary axis (see (5.68) in our closing remarks in Sub-section 5.3), namely as b → +∞, 2n−1 )), Ψ(−ib) = Ψ(ib). (7.3) From (7.1), we note that, for power-like potentials with degree p > 1, Ψ(ib) decays progressively faster as p → +∞ with limit Ψ(ib) ≈ b − 2 3 , the decay rate for V (x) = ie x . As we consider potentials that grow super-exponentially, the asymptotic behaviour of Ψ(ib) changes, and an additional factor (a negative power of log b) comes into play (see Example 7.3). At the other end of the range for p, as p → 0+, p < 1, we observe the growth rate of Ψ(ib) along the imaginary axis increasing ever faster. The transition from power-like potentials to (more slowly growing) logarithmic ones also determines a change in asymptotics for Ψ(ib), with growth along the imaginary axis becoming exponential (see Example 7.2).
Two cases of particular interest are the operators with potentials V (x) = ix 2 (the Davies operator) and V (x) = ix 3 (the imaginary cubic oscillator). They have been studied in detail in the literature using both semi-classical and non-semi-classical methods: see e.g. [15,12,18,11,30] for the Davies example and [10,11,35,19] for the cubic case. The behaviour of the norm of the resolvent for each of them is illustrated in Fig. 7.1 which shows the regions of uniform boundedness of Ψ(λ) described in Sub-section 5.1 (see (5.2) and (5.3)). Furthermore we observe that the level curves determined by (7.4) with n = 2 and n = 3 match those found using semi-classical methods in [11,Prop. 4.6,Prop. 4.2] and that, as expected, the level curves determined by (7.4) with n = 2 and those determined by (7.5) with n = 1 (see (5.54)) are symmetrical with respect to the bi-sector y = x. We also show the behaviour of Ψ(λ) for the operator with sub-linear potential V (x) = i x 2 3 in Fig. 7.1, remarking that the completeness of the eigensystem for this operator with Dirichlet boundary conditions in L 2 (R + ) was proved in [38]. As in the sub-linear potential case, the fact that Ψ(λ) grows along the imaginary axis leads to an ε-shifted critical curve that intersects it at some b > 0. 7.3. Fast growing potential. Let H = −∂ 2 x + ie x 2 with Dom(H) = W 2,2 (R) ∩ Dom(e x 2 ). Then 3 )), b → +∞, (7.6) which is as before illustrated in Fig. 7.1. Since the decay of Ψ(λ) on the imaginary axis is faster than for any polynomial potential, the region for uniform boundedness of Ψ(λ) adjacent to the imaginary axis is correspondingly wider. Note that Theorem 4.2 on the behaviour of Ψ(λ) for λ ∈ R + is not applicable in this case, see also

Proof.
i) It is clear that C ∞ c (R) ⊂ Dom(A) and therefore that A is densely defined. Moreover, a standard cut-off argument, using a sequence u n := φ(x/n)u for 0 = φ ∈ C ∞ c (R) and any u ∈ Dom(A), see e.g. [28,Lem. 3.6], shows that C A := {u ∈ W 1,2 (R) : supp u is bounded} (A.5) is a core of A. Thus for all u ∈ C A , we have Au, u = − u , u + W u, u , hence Re Au, u = Re W u, u = (Re W ) This shows that A + λ is injective, that (A + λ) −1 : Ran(A + λ) → Dom(A) is bounded and that (A + λ) −1 ≤ 1/λ. Moreover, Ran(A + λ) is closed.
iii) By simple adjustments of the arguments to prove i), we can show that B := d/dx + W with the maximal domain Dom(B) := {u ∈ L 2 (R) : u + W u ∈ L 2 (R)} is m-accretive. Moreover, for all u ∈ C A and v ∈ Dom(B), we have which shows that B ⊂ A * . However, the fact that A is m-accretive implies that A * is also m-accretive (see e.g. [21, Thm. 6.6]) and therefore it must be the case that B = A * , as claimed. iv) If λ ∈ σ p (A), there is 0 = u λ ∈ Dom(A) such that −u λ + W u λ − λu λ = 0. Then u λ must have the form u λ (x) = C exp( x 0 W (t) dt − λx), x ∈ R, for some C ∈ C \ {0}. Therefore |u λ (x)| = |C|e and we have Au 2 + u 2 ≥ C A u 2 + Re W u 2 + u 2 , u ∈ Dom(A), A * u 2 + u 2 ≥ C A * u 2 + Re W u 2 + u 2 , u ∈ Dom(A * ); (A.10) the constants C A , C A * > 0 depend only on ε, M , C W and W χ [−x0,x0] ∞ .
Let A 1 be the operator determined by (A.1) with potential W 1 . We show that the separation (A.9) and (A.10) holds for A 1 . The latter remain valid for A = A 1 + W 2 since W 2 is bounded.
Using (A.7) for W 1 (see remarks above), Young inequality with δ ∈ (0, 1) and the assumption (A.8) in the second step, we arrive at We select δ so that C W / (1 − ε + C W ) < δ < 1, thus 1 − ε − C W δ −1 − 1 > 0. Therefore for all u ∈ C A1 (and hence for all u ∈ Dom(A 1 )) Since the opposite inequality is immediate, we conclude with (A.9) for A 1 and hence for A since W 2 is bounded. The reasoning for A * is completely analogous.