Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

We consider Schrödinger operators of the form HR=-d2/dx2+q+iγχ[0,R]\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H_R = - \,\text {{d}}^2/\,\text {{d}}x^2 + q + i \gamma \chi _{[0,R]}$$\end{document} for large R>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$R>0$$\end{document}, where q∈L1(0,∞)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q \in L^1(0,\infty )$$\end{document} and γ>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\gamma > 0$$\end{document}. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.


Introduction
There has recently been a surge of interest concerning bounds for the magnitude of eigenvalues and the number of eigenvalues of Schrödinger operators with complex potentials.In this paper, we consider Schrödinger operators of the form endowed with a Dirichlet boundary condition at 0, where γ > 0 and the background potential q ∈ L 1 (0, ∞) (which may be complex-valued) are regarded as fixed parameters.Perturbations of the form iγχ [0,R] are referred to as dissipative barriers and arise in spectral approximation, where they can be utilised as part of numerical schemes for the computation of eigenvalues [24,31,2,22,23,34].Our aim is to prove estimates for the magnitude and number of eigenvalues of H R for large R.
1.1.Existing Bounds for the Magnitude and Number of Eigenvalues.Let us first discuss some relevant existing results concerning the eigenvalues of (nonself-adjoint) Schrödinger operators and apply them to operators of the form H R .

Literature
Our Results Magnitude Bound Number of Eigenvalues (q Compactly Supported) 2020) Table 1.A summary of the large R asymptotic estimates for the eigenvalues of H R implied by our results compared to estimates obtained by applying various results in the literature.on L 2 (R + ), endowed with a Dirichlet boundary condition at 0, satisfies (2) |λ| Note that the right hand side of the bound presented in [14] depends on arg λ and is sharper than (2).An application of this result to operators of the form H R gives an estimate |λ R | = O(R) as R → ∞ for any eigenvalue λ R of H R .
Proving bounds for the number of eigenvalues of a Schrödinger operator is often regarded a more difficult problem.A sufficient condition for the potential V to ensure that the number of eigenvalues of a Schrödinger operator on L 2 (R + ) is finite is the Naimark condition [25]: There exist other such sufficient conditions and it is known that the number of eigenvalues may not be finite for certain potentials decaying only sub-exponentially [3,27,28].
Quantitative bounds for the number of eigenvalues of a Schrödinger operator on L 2 (R d ) were proved by Stepin in [32,33] for dimensions d = 1, 3. Bounds for arbitrary odd dimensions were later proved by Frank, Laptev and Safronov in [13], which give better large R estimates when applied to operators H R of the form (1). [13,Theorem 1.1] states that the number of eigenvalues N (counting algebraic multiplicity) of a Schrödinger operator − d 2 / dx 2 + V on L 2 (R + ) endowed with a Dirichlet boundary condition at 0 satisfies for any ε > 0. With the assumption that the background potential q satisfies the Naimark condition, applying this inequality to H R with ε = 1/R gives an estimate where C 1 , C 2 > 0 are some numerical constants.With the assumption that the background potential q is compactly supported, applying this inequality to H R gives an estimate N (H R ) = O(R 2 ) as R → ∞.We mention also other estimates for numbers of eigenvalues in [5,18,30].
1.2.Summary of Results.Table 1 summarises our results for the large R behaviour of the eigenvalues of H R and compares them to the application of the existing results to operators of the form H R .Let H R denote the operator H R for the case q ≡ 0. The semi-infinite strip (6) plays an important role throughout the paper and has the property that its closure Γ γ is equal to the numerical range of the operator R for any R > 0. An open ball in C of radius r > 0 about a point z 0 ∈ C is denoted by B r (z 0 ).Note that in this paper we make no attempt to optimise numerical constants.
Our first result gives a uniform in R enclosure for the eigenvalues of H R : (A) (Theorem 4 (a)) There exists X = X(q, γ) > 0 such that, for any R > 0, the eigenvalues of H R lie in B X (0) ∪ Γ γ .
In particular, the imaginary and negative real components of the eigenvalues are bounded independently of R.
Our next result is a bound for the magnitude of eigenvalues of H R for sufficiently large R. The bound gives the estimate |λ R | = O(R/ log R) as R → ∞ for any eigenvalue λ R of H R providing a logarithmic improvement to the application of the result [14] of Frank, Laptev and Seiringer to this system.
The first inequality in (9) shows that the estimate is obtained by considering an analytic function whose zeros are the eigenvalues of H R and applying large-|λ| Levinson asymptotics.The enclosure that results from combining (A) and (B) is illustrated in Figure 1.
The fact that large eigenvalues of H R for large R must be contained in the numerical range of H (0) R and the right hand side of inequality ( 7) is independent of q indicates that the effect of the background potential q on the large eigenvalues is dominated by effect of the dissipative barrier iγχ [0,R] for large R.
Our first estimate for the number of eigenvalues N (H R ) for H R is for the case that the background potential q is compactly supported.It gives the estimate which offers a logarithmic improvement to the application of the result [19,Theorem 1.6] of Korotyaev to this system.
(C) (Theorem 8) If q is compactly supported then there exists R 0 = R 0 (q, γ) > 0 such that for every R R 0 , The second inequality in (9) shows that the estimate N (H R ) = O(R 2 / log R) is sharp.The proof consists in an application of Jensen's formula.The case in which the background potential q merely satisfies the Naimark condition requires more sophisticated techniques compared to the compactly supported case.Our result gives the estimate N (H R ) = O(R 3 /(log R) 2 ) as R → ∞, providing a more significant improvement to the application of the result [13, Theorem 1.1] of Frank, Laptev and Safronov to this system, which gives N (H R ) = O(R 4 ).The reasons for the more significant improvement are discussed below.
(D) (Theorem 10) If there exists a > 0 such that then there exists R 0 = R 0 (q, γ) > 0 such that for every R R 0 , ( 8) where X = X(q, γ) > 0 is the constant appearing in (A) and C = 88788.The proof of (D) involves first obtaining a bound which counts the number of zeros in a strip for an arbitrary analytic function in the upper half plane (Proposition 9).This bound can be applied to the estimation of N (H R ) thanks to the uniform in R enclosure (A), which implies that the square-roots of the eigenvalues of H R are contained in a strip, uniformly in R. Without the uniform enclosure, we would have to use the magnitude bound (B) in place of the uniform enclosure with which the best we could obtain is inequality (8) with ).This indicates that the more significant improvement in (D) is due to the combination of a bound for the quantity √ λ of the eigenvalues λ with the bound Proposition 9 for analytic functions.
Operators of the form R , corresponding to the special case q = 0, have been studied by Bögli and Štampach in [4], by Golinskii in [16] and by Cuenin in [7].A consequence of [7,Theorem 4] is that there exists constants C 1 , C 2 > 0 such that for all large enough R > 0, (9) sup and .
Note that although this result was formulated for the Schrödinger operator on R, it applies to Schrödinger operators on R + endowed with a Dirichlet or Neumann boundary condition since the author constructs both odd and even eigenfunctions of R in [7, Section 7.1].As already mentioned, the inequalities (9) show that Theorem 4 (b) and Theorem 8 provide optimal large R estimates.
The reader is referred to [31,Section 5] for numerical illustrations of the eigenvalues of operators of the form H R for large R.
1.3.Notations and Conventions.Throughout the paper, C > 0 denotes a constant, whose dependencies are generally indicated, that may change from line to line.ψ (x, λ) will denote d dx ψ(x, λ) throughout.The branch cut of shall denote the number of eigenvalues of H R , counting algebraic multiplicities (as above).Finally, note that f R will always denote an analytic function but will be redefined in each section.

Magnitude Bound
Since q ∈ L 1 (0, ∞), we can employ Levinson's asymptotic theorem which states that the solution space of the Schrödinger equation −u + qu = λu on [0, ∞) is spanned by solutions ψ + and ψ − , which admit the decomposition [26, Appendix II, Theorems 1 and 3] [9, Theorem 1.3.1]: Here, E ± and E d ± are some functions such that, (11) While the error E + (x, λ) tends to 0 as x → ∞ uniformly for λ ∈ C\B δ (0), δ > 0, the error E − does not have this property.For this reason, we will need to utilise large-|λ| asymptotics of ψ ± in this section.
Here, E 1 , E 2 are defined, for any R > 0 and λ ∈ C\{0, iγ}, by λ is an eigenvalue of H R if and only if there a solution to the boundary value problem ( 16) Any solution to ( 16) on [0, R] must be of the form and C 1 ∈ C is independent of x.Any solution to the boundary value problem ( 16) on [R, ∞) must be of the form Hence λ is an eigenvalue if and only if there exists C 1 , C 2 ∈ C\{0} independent of x such that the function is continuously differentiable which holds if and only if The required expression for f R holds by a direct computation, using expressions (10) for ψ ± .
For λ ∈ C with |λ| 1 + γ we have |λ| 1 and |λ − iγ| 1.Therefore, estimates (12) apply to all the terms in ( 13) and ( 14) involving E ± or E d ± .The O(1/ |λ|) decay of the terms involving E ± or E d ± as |λ| → ∞ cancel the growth of the square roots hence estimate (15) In the special case q ≡ 0, f R is denoted by f (0) R and we have that: The terms E ± and E d ± in Levinson's asymptotic theorem are simply zero for this case, so (19) f Lemma 2. There exists a constant for all R > 0 and all λ ∈ C with |λ| 1 + γ.
Recall that Γ γ is an open strip defined by equation (6).We shall need the following elementary inequalities: The inequality for √ λ + √ λ − iγ follows from the identity ( 22) Using the function f R for the eigenvalues of H R , combined with the large-|λ| asymptotics of ψ ± , we can estimate the location of the eigenvalues of H R : Theorem 4. (a) There exists X = X(q, γ) > 0 such that, for any R > 0, the eigenvalues of H R lie in B X (0) ∪ Γ γ .(b) There exists R 0 = R 0 (q, γ) > 0 such that for every R R 0 , any eigenvalue λ of H R in Γ γ satisfies (23) |λ − iγ| 5γR log R .
Proof.(a) Let R > 0. H R has no eigenvalues in [0, ∞) (indeed, this follows from the Levinson asymptotic formulas (10)) so it suffices to show that any zero of f R in C\(Γ γ ∪ [0, ∞)) must lie in an open ball in the complex plane, whose radius is independent of R. Let λ ∈ C\(Γ γ ∪ [0, ∞)) be such that |λ| X, where X = X(q, γ) > 0 is a large enough constant to be further specified.Let X > 0 be large enough so that |λ| 1 + γ.By the expression for f R in Lemma 1, By the boundedness of E 1 and E − (Lemma 1 and estimates ( 12)), as well an inequality in Lemma 3 (b), there exists Then, using Lemma 3 (b), (26) ψ + (0, λ − iγ) Combining ( 24), ( 25) and ( 26), we have Consequently, λ is not an eigenvalue of H R proving that there are no eigenvalues of H R in C\Γ γ with magnitude greater than X.
(b) Let R R 0 , where R 0 = R 0 (q, γ) > 0 is a large enough constant to be further specified.Let λ ∈ Γ γ be such that We aim to prove that λ is not an eigenvalue of H R .
Let W denote the Lambert-W -function (also known as the product log function).W satisfies Hence (31) can be written as from which (23) follows.
Remark 1.The constant X = X(q, γ) > 0 in Theorem 4 (a) satisfies This can be seen by noting that L 1 ) and C 1 = O( q 3 L 1 ).

Number of Eigenvalues
In this section, we estimate the number of eigenvalues for H R , for which we necessarily need to add additional assumptions on the background potential q.
3.1.Preliminaries.Let ψ ± denote the solutions (10) for the Schrödinger equation and ϕ(x, z) . ϕ is commonly referred to as the Jost solution.For each R > 0, define function where, for any z ∈ C, θ(•, z) is defined as the solution to the initial value problem By the same arguments as in Lemma 1, we have the following.
Lemma 5. f R is analytic on C + and any z ∈ C + satisfies ϕ can be decomposed in a similar way to ψ ± , for some functions E and E d whose properties will be later specified, for the different assumptions on the background potential q that we consider.We shall need the following facts concerning f R and θ.Note that in Lemma 6, E 1 and E 2 are defined in a different way than in Lemma 1. Lemma 6. Suppose that, for each R > 0, ϕ(R, •) and ϕ (R, •) admits an analytic continuation from and Proof.Analytic continuation holds by the fact that θ(R, •) is entire [35,Lemma 5.7] for each R > 0. If z = ± √ iγ then the functions ψ ± (•, z 2 − iγ) span the solution space of the Schrödinger equation where ψ 1 denotes the function defined by ( 17) in Lemma 1.The lemma follows by a direct computation, similar to one in Lemma 1.
Assumption 1. q is compactly supported, that is, there exists Q > 0 such that If Assumption 1 holds and then the Jost solution ϕ satisfies hence, for each x ∈ [0, ∞), ϕ(x, •) can be analytically continued to C. Consequently, for R > Q, f R can be analytically continued to C and can be written as Theorem 8. Suppose that Assumption 1 holds.Then there exists R 0 = R 0 (q, γ) > 0 such that for every R R 0 , Proof.Let z 0 ∈ C + be such that (42) ψ + (0, z 2 0 − iγ) = 0, z 2 0 − iγ 1 and |z 2 0 − iγ| 2. In fact, by choosing z 0 to be the minimiser of some suitable total order on C in the set of points that maximise z → ψ + (0, z 2 − iγ) while satisfying the latter two inequalities of (42), z 0 can be determined uniquely by q and γ, z 0 = z 0 (q, γ).Define r = r(R) > 0 by (43) By the triangle inequality, Let R > Q > 0 be large enough so that estimate (23) of Theorem 4 (b) holds.Since the zeros of f R in C + have a bijective correspondence with the eigenvalues of H R , the set S R contains all the zeros of f R in C + and hence the number of eigenvalues of H R is bounded by the number of zeros for f R in the ball B r/2 (z 0 ), ( 46) Since R > Q, the terms E 1 (R, z) and E 1 (R, z), defined by (37) and (38) respectively, vanish.Hence, by Lemma 6 and the fact that Note that z 2 0 − iγ 1 implies that z 0 = ± √ iγ so Lemma 6 is indeed applicable here.Then, since ψ + (0, z 2 0 − iγ) = 0, |z 2 0 − iγ| 2 and z 0 = z 0 (q, γ), for large enough R. By expression (40) for f R and the estimates in Lemma 7 for θ and θ , for all z ∈ ∂B r (z 0 ), Furthermore, by the triangle inequality and expression (43) for r, for all z ∈ ∂B r (z 0 ), (50) Noting that for z ∈ ∂B r (z 0 ), the factor (1 + |z|) in (49) is o(R), combining (46) -(50) gives us as R → ∞.Estimate (41) follows.

Exponentially Decaying Potentials. Assumption 2 (Naimark Condition
).There exists a > 0 such that If Assumption 2 is satisfied then for each x > 0 the functions ϕ(x, •) and ϕ (x, •) admit analytic continuations from C + into { z > −2a}.For each x > 0, the functions E and E d appearing in the decomposition (34) of the Jost solution ϕ satisfy Combining (79) and (80), we have which gives estimate (62) when substituted into (77), with Y = √ X + a and η = a.

Figure 1 .
Figure 1.Illustration the enclosure for the eigenvalues of H R provided by Theorem 4.

See [ 25 ,Proposition 9 .Remark 2 . 4 η
Theorem 2.6.1] and [32, Lemma 1] for proofs of the above claims.The next proposition allows us to utilise the uniform enclosure of Theorem 4 (a) in the estimation of the number of eigenvalues of H R .Suppose that f is an analytic function defined on an open neighbourhood of the closed semi-disc D r := B r (0) ∩ C + for some r > 0. Let α and β be any numbers in the interval (0, 1) satisfying let N (αr) denote the number of zeros in the region (54) D αr,η,Y := {z ∈ C : η z Y, |z| αr} where Y, η > 0 are given parameters satisfying η < Y < r.Then, One can always guarantee that condition (53) for α and β is satisfied by choosing, for instance, (57) α = β = 1 2Y + η .