Eigenvalues of Schr\"odinger operators near thresholds: two term approximation

We consider one dimensional Schr\"{o}dinger operators $H_\lambda=-\frac{d^2}{dx^2}+U+ \lambda V_\lambda$ with nonlinear dependence on the parameter $\lambda$ and study the small $\lambda$ behaviour of eigenvalues. The potentials $U$ and $V_\lambda$ are real-valued bounded functions of compact support. Under some assumptions on $U$ and $V_\lambda$, we prove the existence of a negative eigenvalue that is absorbed at the bottom of the continuous spectrum as $\lambda\to 0$. We also construct two term asymptotic formulas for the threshold eigenvalues.


Introduction
About forty years ago, Simon and Klaus [1][2][3][4] started studying the low energy behaviour of the so-called weakly coupled Hamiltonians −∆ + λV . The considerable interest has been in the study of negative-energy bound states and their small λ behaviour, as well as in the study of the absorption of the eigenvalues by the continuous spectrum. The main results here have been concerned with Schrödinger operators in one and two dimensions, because in three dimensions the weakly coupled Hamiltonians have no bound state if λ is small enough, i.e., if potential λV is a sufficiently shallow well. For the case of 1D Hamiltonians H λ = − d 2 dx 2 + λV , an suitable short-range potential V can produce a bound state for all small λ. Assuming that V is different from zero and R (1 + |x| 2 )|V (x)| dx < ∞, Simon [1] proved that the operator H λ has a negative-energy bound state e λ for all small positive λ if and only if R V (x) dx ≤ 0. If H λ does have an eigenvalue, then it is unique and simple, and obeys as λ → 0. This asymptotic formula is due to Abarbanel, Callan and Goldberger, but it was not published by them; (1) was firstly announced by Simon [1]. The eigenvalue e λ approaches zero as λ goes to zero and it is absorbed in the limit at the bottom of the continuous spectrum [0, +∞). Then we say that λ = 0 is a coupling constant threshold for H λ . Klaus [2] has extended this result to the class of potentials V obeying the condition R (1 + |x|)|V (x)| dx < ∞. In [5,6], the threshold behaviour has been studied as a general perturbation phenomenon and some general results on existence and asymptotic behaviour of eigenvalues for self-adjoint operators A+λB have been obtained. The main tool was the so-called Birman-Schwinger principle. Klaus [6] has also applied these results to several special cases. One of them has been concerned with the Hamiltonian − d 2 dx 2 + U + λV . If a certain relation between the potentials U and V holds, then the operator has a small negative-energy bound state (not necessarily a unique one) in the limit of weak coupling. Namely, it has been proved that the operator has the coupling constant threshold λ = 0, if the unperturbed operator − d 2 dx 2 + U possesses a zero-energy resonance with a half-bound state u and R V u 2 dx < 0. Among the negative eigenvalues there exists only one that is absorbed by the continuous spectrum as λ → 0. A unique threshold eigenvalue e λ is analytic at λ = 0 and obeys as λ → 0, where u ± = lim x→±∞ u(x). If R V u 2 dx = 0 and the support of V lies between two consecutive zeros of u, then there exists a bound state near zero for all small enough λ (positive and negative). Finally, if R V u 2 dx > 0, then the operator has no bound state and therefore λ = 0 is not a coupling constant threshold. We will give the precise definitions of the zero-energy resonances, half-bound states, and coupling constant threshold in the next section.
One of the motivations for writing this article was the desire to improve approximation (2). As another motivation for investigating the threshold behaviour of eigenvalues, we mention applications of this phenomenon to the study of the stability of solutions for the Korteweg-de Vries equation [21] and the existence of 'breathers' (the localized periodic solutions) for discrete nonlinear Schrödinger systems [22,23].
In this paper, we consider a more general class of Schrodinger operators with nonlinear dependence on the positive parameter λ. We analyse the existence of negative eigenvalues and their threshold behaviour. Here U and V λ are functions of compact support and V λ = V + λV 1 + o(λ) as λ → 0. The spectrum of H λ consists of the essential spectrum [0, ∞) and possibly a finite number of negative eigenvalues. Under certain conditions on the potentials U , V and V 1 the operator H λ has a negative eigenvalue e λ that is absorbed at the bottom of the essential spectrum as λ goes to zero. The threshold eigenvalue may or may not be the ground state. We examine the asymptotic behaviour of e λ as λ → 0 and compute the two term asymptotic formula which in particular improves the approximation (2). For the case U = 0 and V λ = V , our asymptotics turns into the Abarbanel-Callan-Goldberger formula. The threshold behaviour of eigenvalues for operators − d 2 dx 2 + U + λα λ V (α λ ·), where the positive sequence α λ converges to a finite or infinite limit as λ → 0, has recently been studied in [7]. These results gives us an example of the non-analytic threshold behaviour of negative eigenvalues.

Main Results
We start with some definitions. Let A and B λ be self-adjoint operators and B λ be relatively A-compact for all λ > 0; then σ ess (A + B λ ) = σ ess (A). Suppose that the interval (a, b) is a gap in the spectrum of A. If we can find an eigenvalue e λ of A + B λ in (a, b) for all λ > 0 with the property that e λ → a or e λ → b as λ → 0, then we call λ = 0 the coupling constant threshold. So the eigenvalue e λ is absorbed by the continuous spectrum at "time" λ = 0.
We say operator − d 2 dx 2 + U possesses a zero-energy resonance if there exists a non trivial solution u of the equation that is bounded on the whole line. We then call u the half-bound state. Any halfbound state u possesses finite limits lim x→±∞ u(x), because u is constant outside the support of U ; both the limits are different from zero. Since a half-bound state is defined up to a scalar multiplier, we say a half-bound state u is normalized if lim x→−∞ u(x) = 1. Let θ hereafter denote the limit of the normalized half-bound state as x → +∞, i.e., θ := lim x→+∞ u(x). We also introduce the function Assume u 1 is a solution of (4) such that u 1 (x) = x to the left of the support of U . Then u and u 1 are linearly independent solutions of (4) and we will show below that there exists a constant θ 1 such that u 1 (x) = θ −1 x + θ 1 for all x large enough (see Fig. 1). Let v * be a solution of −v + U v = −V u which vanishes to the left of the supports of U and V . Here and subsequently, · stands for the norm in L 2 (R).
then operator H λ = − d 2 dx 2 + U + λV λ possesses the coupling constant threshold λ = 0, i.e., for all small positive λ there exists a negative eigenvalue e λ of H λ such that e λ → 0 as λ → 0. Moreover the threshold eigenvalue e λ has the asymptotic The threshold phenomenon is also possible if inequality (5) turns into the equality. In this case the absorption of the eigenvalue at the bottom of σ ess (H λ ) occurs with the rate O(λ 4 ) as λ → 0.
Theorem 2. Under the assumptions of Theorem 1, we suppose that Then the operator H λ has the coupling constant threshold Moreover the threshold eigenvalue e λ admits the asymptotics Return now to operator family − d 2 dx 2 + U + λV studied in [6].
then − d 2 dx 2 + U + λV possesses the coupling constant threshold λ = 0 and a negative eigenvalue e λ admits the asymptotics where ω 0 is given by (6) and then the operator − d 2 dx 2 + U + λV has a negative eigenvalue e λ with the asymptotics where E U is the fundamental solution for d 2 dx 2 − U which vanishes to the left of supp U .
Proof. Most of the proof follows from the previous theorems, assuming V λ = V for all λ. We are left with the task of deriving (12). If (11) holds, then ω 0 = 0 and Recall that v * solves equation v * −U v * = V u and vanishes to the left of the supports of U and V . Then v * can be represented as the convolution Substituting (14) into (13) finishes up the proof.
Remark 1. Klaus did not use the notion of a normalized half-bound state. To agree the asymptotic formulas, we rewrite ω 0 and ω 1 in (10) in terms of an arbitrary halfbound state u for which lim x→±∞ u(x) = u ± . Then in notation of [6] we obtain Let us compare our results with those of Simon when the unperturbed operator is the free Schrödinger operator.
Proof. The trivial potential U = 0 has a zero-energy resonance with half-bound state u = 1; then θ = 1 and Θ(x) = 1 for all x ∈ R. In addition, we have θ 1 = 0, because equation u = 0 possesses the solution u 1 = x. Therefore condition (5) becomes (15), and (6), (7) simplify to read The fundamental solution E 0 (x) = 1 2 (|x| + x) for the differential operator d 2 dx 2 vanishes for x < 0. As in Corollary 1, we derive dx dy = 0 for any f , for which the integral exists. This gives the second equality in (16), and the proof is complete.
then for all nonzero λ, positive or negative, the operator H λ = − d 2 dx 2 +λV λ possesses an eigenvalue e λ having the asymptotics as λ → 0. This asymptotic formula can be also written in the form This assertion will be proved in Section 4.

Preliminaries
We first record some technical facts. Assume, without loss of generality, the supports of potentials U and V λ lie within I = (− , ) for λ small enough. Then a half-bound state of operator − d 2 dx 2 + U is constant outside I and its restriction to I is a non-trivial solution of the problem −u + U u = 0, t ∈ I, u (− ) = 0, u ( ) = 0.
Proposition 1. Assume that h belongs to L 2 (I) and γ is a real number. Let w be a solution of the Cauchy problem If − d 2 dx 2 + U has a zero-energy resonance with normalized half-bound state u, then In addition, this solution obeys the estimate for some positive C being independent of γ and h.
Proof. Since u(− ) = 1 and u( ) = θ, (20) can be easily obtained by multiplying the equation in (19) by u and integrating by parts. Next, application of the variation of parameters method yields where k(x, s) = u(x)u 1 (s)−u(s)u 1 (x). Under the assumptions made on potential U , u and u 1 belong to W 2 2 (I); consequently u, u 1 ∈ C 1 (I) by the Sobolev embedding theorem. From this and the representation of the first derivative ) for x ∈ I, which completes the proof.
Proposition 2. Let u 1 be the solution of (4) as described in Section 2. Then for some constant θ 1 we have u 1 (x) = θ −1 x + θ 1 for all x > .
Our method is different from that of Simon and Klaus. We don't use the Birman-Schwinger principle. To prove the main results, we use the asymptotic method of quasimodes or in other words of "almost" eigenvalues and eigenfunctions. Let A be a self-adjoint operator in a Hilbert space L. We say a pair (µ, φ) ∈ R × dom A is a quasimode of A with accuracy δ, if φ L = 1 and (A − µI)φ L ≤ δ.
Proof. If µ ∈ σ(A), then λ = µ. Otherwise the distance d µ from µ to the spectrum of A can be computed as where ψ is an arbitrary vector of L. Taking ψ = (A − µI)φ, we deduce from which the assertion follows.
Let us first show that constants ω 0 , ω 1,λ , ω 2,λ , a λ and b λ in (22) can be chosen so that ψ λ will belong to dom H λ . First of all, the L 2 (R)-norm of ψ λ is finite if and only if ω λ < 0; therefore we must impose the conditions ω 0 < 0 (the case ω 0 = 0 will be treated in Theorem 2). Note that u and v k belong to the Sobolev space W 2 2 (I) as solutions of the equation −y +U y = f with f ∈ L 2 (I). By construction, ψ λ and its first derivative are continuous at x = − , then it is enough to ensure the continuous differentiability of ψ λ at x = .

4.3.
Proof of Corollary 3. This statement differs from all earlier proved by the fact that here the threshold eigenvalue exists for both positive and negative λ small enough. For the case V λ = V this result has been proved by Simon [1]. Since R V dx = 0, from (16) we observe ω 0 = 0 and Proof. From R V dx = 0 we immediately deduce because integrating by parts yields The proof is completed by showing that In view of this proposition, if potential V is different from zero, then ω 1 < 0. Hence ω λ = λ 2 ω 1,λ + λ 3 ω 2,λ is negative for λ small enough, positive or negative.