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Proc. Indian Acad. Sci. (Math. Sci.) Vol. 116, No. 4, November 2003, pp. 375–391. Printed in India

Schr¨ odinger operators on the half line: Resolvent expansions and the Fermi golden rule at thresholds

ARNE JENSEN? and GHEORGHE NENCIU?,?

arXiv:0707.2146v1 [math-ph] 14 Jul 2007

? Department

of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, DK-9220 Aalborg ?, Denmark ? Department of Theoretical Physics, University of Bucharest, P. O. Box MG11, 76900 Bucharest, Romania ? Institute of Mathematics “Simion Stoilow” of the Romanian Academy, P. O. Box 1-764, RO-014700 Bucharest, Romania E-mail: matarne@math.aau.dk; nenciu@barutu.?zica.unibuc.ro; Gheorghe.Nenciu@imar.ro Dedicated to K B Sinha on the occasion of his sixtieth birthday

Abstract. We consider Schr¨ odinger operators H = ?d2 /dr2 + V on L2 ([0, ∞)) with the Dirichlet boundary condition. The potential V may be local or non-local, with polynomial decay at in?nity. The point zero in the spectrum of H is classi?ed, and asymptotic expansions of the resolvent around zero are obtained, with explicit expressions for the leading coef?cients. These results are applied to the perturbation of an eigenvalue embedded at zero, and the corresponding modi?ed form of the Fermi golden rule. Keywords. rule. Schr¨ odinger operator; threshold eigenvalue; resonance; Fermi golden

1. Introduction

This paper is a continuation of [5,7], where expansions of the resolvents of Schr¨ odinger type operators at thresholds, as well as the form of the Fermi golden rule (which actually goes back to Dirac), when perturbing a nondegenerate threshold eigenvalue, were obtained. While the methods and results in [5,7] are to a large extent abstract, the examples discussed were restricted to Schr¨ odinger operators in odd dimensions with local potentials. The aim of this paper is to show that the methods in [5,7] allow to treat the non-local potentials in exactly the same manner as the local ones, although the properties of the corresponding operators can be quite different. For example, one can have zero as an eigenvalue in one dimension, or eigenfunctions for the zero eigenvalue with compact support (in this connection see e.g. [2]). D denote ?d2 /dr2 on H = L2 ([0, ∞)) with Let us brie?y describe the results. Let H0 the Dirichlet boundary condition. Let V be a potential, which can be either local or nonlocal. We assume that V is a bounded selfadjoint operator on H . Let H s = L2,s ([0, ∞)) denote the weighted space. Then we assume that V extends to a bounded operator from H ?β /2 to H β /2 for a suf?ciently large β > 0. Since we are concerned with threshold phenomena, the ?rst step is to study the solutions of the equation H Ψ = 0. The result is that under the above conditions, for the solutions of H Ψ = 0 there are four possibilities: 375

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(i) No non-zero solutions. In this case zero is called a regular point for H . (ii) One non-zero solution in L∞ ([0, ∞)), but not in L2 ([0, ∞)). In this case zero is called an exceptional point of the ?rst kind for H . (iii) A ?nite number of linearly independent solutions, all belonging to L2 ([0, ∞)). In this case zero is called an exceptional point of the second kind for H . (iv) Two or more linearly independent solutions, which can be chosen such that all but one belong to L2 ([0, ∞)). In this case zero is called an exceptional point of the third kind for H . Let us note that if V is multiplication by a function, i.e. (V f )(r) = V(r) f (r) for some function V(r), then only cases (i) and (ii) occur. In all cases we obtain asymptotic expansions for the resolvent of H around the point √ zero. It is convenient to use the variable κ = ?i z in these expansions. We have (H + κ 2 )?1 =

j =?2

∑

p

κ j G j + O (κ p+1)

as κ → 0, in the topology of the bounded operators from H s to H ?s for a suf?ciently large s, depending on p and the classi?cation of the point zero for H . We compute a few of the leading coef?cients explicitly. These results on asymptotic expansion for the resolvent, and the explicit expressions for the coef?cients, are the main ingredients for the application of the results in [7], concerning the perturbation of an eigenvalue embedded at the threshold zero. The main result from [7] in the context of the Schr¨ odinger operators on the half line considered above is D + V , where V satis?es Assumption 3.3 for a suf?ciently large as follows. Let H = H0 β . Let W be another potential satisfying the same assumption. We consider the family H (ε ) = H + ε W for ε > 0. Assume that 0 is a simple eigenvalue of H , with normalized eigenfunction Ψ0 . Assume b = Ψ0 , W Ψ0 > 0, and that for some odd integer ν ≥ ?1 we have G j = 0, for j = ?1, 1, . . . , ν ? 2 and gν = Ψ0 , W Gν W Ψ0 = 0. (1.2) Then Theorem 3.7 in [7] gives the following result (the modi?ed Fermi golden rule) on the survival probability for the state Ψ0 under the evolution exp(?itH (ε )), showing that for ε suf?ciently small the eigenvalue zero of H becomes a resonance. There exists ε0 > 0, such that for 0 < ε < ε0 we have Ψ0 , e?itH (ε ) Ψ0 = e?it λ (ε ) + δ (ε , t ), Here λ (ε ) = x0 (ε ) ? iΓ(ε ) with x0 (ε ) = bε (1 + O (ε )), as ε → 0. The error term satis?es |δ (ε , t )| ≤ Cε p(ν ) , t > 0, p(ν ) = min{2, (2 + ν )/2}. (1.6) Γ(ε ) = ?iν ?1 gν bν /2 ε 2+(ν /2) (1 + O (ε )), (1.4) (1.5) t > 0. (1.3) (1.1)

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377

As an application of the results on asymptotic expansion of the resolvent of H near zero we explicitly compute the coef?cient gν in two cases. The contents of the paper is as follows: In §2 we introduce some notation used in the rest of the paper. Section 3 forms the core of the paper and contains our results on the resolvent expansions for the free Schr¨ odinger operator on the half line, and then for the Schr¨ odinger operator with a general class of potentials, including non-local ones. In §4 we illustrate the general results by giving an explicit example with a rank 2 operator as the perturbation. Finally, §5 contains the results on the modi?ed Fermi golden rule for the class of operators considered here. Let us conclude with some remarks on the literature. Resolvent expansions of the type obtained here are typical for Schr¨ odinger operators in odd dimensions, when the potential decays rapidly. Such results were obtained in [4,3,8]. More recently, a uni?ed approach was developed in [5,6]. It is this approach that we use here. Another approach to the threshold behavior is to use the Jost function. See for example [1,11]. See also the cited papers for further references to results on resolvent expansions around thresholds.

2. Notation

Let H be a self-adjoint operator on a Hilbert space H . Its resolvent is denoted by R(z) = (H ? z)?1 . In the sequel we will often look at operators with essential spectrum equal to [0, ∞), such that 0 is a threshold point. We will look at asymptotic expansions around this point for the resolvent. It is convenient to change the variable z by introducing z = ?κ 2 , with Re κ > 0. In the half line case there is a type of notation common in the physics literature that is very convenient. The resolvent will have an integral kernel k(r, r′ ), r, r′ ∈ [0, ∞). We introduce the two functions r> = max{r, r′ }, r< = min{r, r′ }. (2.1)

We note a few properties for future reference r> + r< = r + r′ , r> ? r< = |r ? r′ |, r> · r< = r · r′ . (2.2)

The weighted L2 -space on the half line is given by H s = L2,s ([0, ∞)) = f ∈ L2 loc ([0, ∞))

∞ 0

| f (r)|2 (1 + r2)s dr < ∞ , (2.3)

for s ∈ R. We write H = H 0 = L2 ([0, ∞)). We use the notation B (s1 , s2 ) for the bounded operators from H s1 to H s2 . The inner product ·, · on H is also used to denote the duality between H s and H ?s . We use the bra and ket notation for operators from H s to H ?s . For example, the operator ∞ f→ 0 f (r)dr · 1 from H s to H ?s for s > 1/2 is denoted by |1 1|. In the asymptotic expansions below there will be error terms in the norm topology of B (s1 , s2 ) for speci?ed values of the parameters s1 and s2 . Here κ ∈ {ζ | 0 < |ζ | < δ , Re ζ > 0} for a suf?ciently small δ . We will use the standard notation O (κ p ) for these error terms.

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3. Resolvent expansions

In this section we ?rst obtain the resolvent expansion of the free Schr¨ odinger operator on the half line, and then for the Schr¨ odinger operator with a general class of potentials, including non-local ones. 3.1 The free operator with the Dirichlet boundary condition

D the operator with the domain and action given by We denote by H0 D D (H0 ) = { f ∈ H | f ∈ AC2 ([0, ∞)), f (0) = 0}, D H0 f =?

d2 f. dr2

(3.1)

i D K0 (κ ; r, r′ ) = ? sin(iκ r< )e?κ r> , κ which can be rewritten as 1 D K0 (κ ; r, r′ ) = ? (e?κ (r> +r< ) ? e?κ (r> ?r< ) ). 2κ Using the Taylor expansion we can get the following result, as in [4,3,8]. PROPOSITION 3.1.

Here the space AC2 denotes functions f that are continuously differentiable on [0, ∞), with f ′ absolutely continuous (see [10]). It is well-known that this operator is self-adjoint. D ?1 has the integral kernel (using z = ?κ 2 as above) The resolvent RD 0 (z) = (H0 ? z) (3.2)

(3.3)

2 The resolvent RD 0 (?κ ) has the following asymptotic expansion. Let p ≥ 0 be an integer 3 and let s > p + 2 . Then we have 2 RD 0 (?κ ) = j p +1 ) ∑ GD j κ + O (κ p

(3.4)

j =0

in the norm topology of B (s, ?s). The operators GD j are given explicitly in terms of their integral kernels by GD j : (?1) j ((r> + r< ) j+1 ? (r> ? r< ) j+1 ). 2( j + 1)!

3 2

(3.5) we also have GD 0 ∈

Proof. The straightforward computations and estimates are omitted. Remark 3.2. For future reference we note the expressions GD 0 : r< ,

′ GD 1 : ?r< r> = ?r · r ,

1 Let s1 , s2 > 2 with s1 + s2 > 2. Then GD 0 ∈ B (s1 , ?s2 ). For s > s ∞ B (H , L ([0, ∞))). 1 , then GD If j ≥ 1 and s > j + 2 j ∈ B (s, ?s).

(3.6) (3.7) (3.8) (3.9)

1 1 3 2 GD 2 : r< r> + r< , 2 6 1 3 ′ 1 3 3 ′ 3 GD 3 : ? (r> r< + r> r< ) = ? (r · r + r · (r ) ). 6 6

Schr¨ odinger operators on the half line 3.2 The potential and the factorization method

379

D and ?nd the asymptotic expansion of the resolvent We now add a potential V to H0 D of H = H0 + V around zero. We will allow a rather general class of potentials, so we introduce the following assumption. We consider only bounded perturbations, however it is possible to extend the results to potentials with singularities.

Assumption 3.3. Let V be a bounded self-adjoint operator on H , such that V extends to a bounded operator from H ?β /2 to H β /2 for some β > 2. Assume that there exists a Hilbert space K , a compact operator v ∈ B (H ?β /2 , K ), and a self-adjoint operator U ∈ B (K ) with U 2 = I , such that V = v?Uv. Remark 3.4. The factorization leads to a natural additive structure on the potentials. Assume that V j = v? j U j v j , j = 1, 2, satisfy Assumption 3.3. Let K = K1 ⊕ K2 . Using matrix notation we de?ne v= v1 , v2 U= U1 0 0 . U2 (3.10)

Then it follows that V = V1 + V2 has the factorization V = v?Uv with the operators v and U de?ned in (3.10) and the space K = K1 ⊕ K2. Example 3.5. We give two examples, the ?rst one a local perturbation, and the second one a non-local perturbation. (i) Let V be multiplication by a real-valued function V(r). Assume that |V(r)| ≤ C(1 + r)?β for some β > 2. Take K = H and let v = v? denote multiplication by |V(r)|1/2 . Let U denote multiplication by 1, if V(r) ≥ 0, and by ?1, if V(r) < 0. Then all conditions in Assumption 3.3 are satis?ed. (ii) Let ? ∈ H β /2 and γ ∈ R, γ = 0. Let V = γ |? ? |. It has the following factorization. Let K = C. Let v : H ?β /2 → K be given by v( f ) = |γ |1/2 ? , f , and U multiplication by sign(γ ). Then v? (z) = z|γ |1/2 ? , and we have V = v?Uv. The generalization to an operator of rank N follows from Remark 3.4.

D + V with V satisfying Assumption 3.3. We note the following result. Write H = H0 D -compact. Lemma 3.6. Let V satisfy Assumption 3.3. Then V is H0

Proof. We have

D D + i)?1 ]. V (H0 + i)?1 = [V (1 + r)β /2][(1 + r)?β /2(H0

The ?rst factor [· · · ] is bounded by the assumption and the second factor [· · · ] is compact by well-known arguments. 2 We now brie?y recall the factorization method, as used in [5], but here extended to cover the non-local potentials. The starting point is the operator

D M (κ ) = U + v(H0 + κ 2 )?1 v? ,

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which is now a bounded operator on K . The factored second resolvent equation is given by

?1 D 2 D 2 ? 2 R(?κ 2 ) = RD 0 (?κ ) ? R0 (?κ )v M (κ ) vR0 (?κ ).

(3.11)

The ?rst step in obtaining an asymptotic expansion for R(?κ 2) is to study the invertibility of M (κ ) and the asymptotic expansion of the inverse. Inserting the asymptotic expansion (3.4) we get M (κ ) =

j =0

∑ κ j M j + O (κ p+1),

? and M j = vGD jv ,

p

(3.12)

provided β > 2 p + 3. Here

? M0 = U + vGD 0v

j = 1, . . . , p.

(3.13)

3.3 Analysis of ker M0 We analyze the structure of ker M0 and the connection with the point zero in the spectrum of H . Lemma 3.7. Let Assumption 3.3 be satis?ed with β > 3.

? (i) Let f ∈ ker M0 . De?ne g = ?GD 0 v f . Then Hg = 0, with the derivatives in the sense of distributions. We have that g ∈ L∞ ([0, ∞)) ∩ C([0, ∞)), with g(0) = 0. We have g ∈ H , if and only if

vr , f

K

= 0.

(3.14)

(ii) Assume g ∈ H ?s ∩ C([0, ∞)), s ≤ 3/2, satis?es g(0) = 0 and Hg = 0, in the sense of distributions. Let f = Uvg. Then f ∈ ker M0 . (iii) Assume additionally that V is multiplication by a function. Let f ∈ ker M0 , f = 0. Then vr , f = 0, and dim ker M0 = 1.

? Proof. Let f ∈ ker M0 , and de?ne g = ?GD 0 v f . Then we have

g(r) = ?

∞

0

r′ (v? f )(r′ )dr′ ?

∞

r

(r ? r′ )(v? f )(r′ )dr′ .

Since v? f ∈ H s for some s > 3/2, the second term belongs to H . The ?rst term is a constant. Thus part (i) follows. For part (ii), assume g ∈ H ?s ∩ C([0, ∞)), s ≤ 3/2, satis?es g(0) = 0 and Hg = 0, in the sense of distributions. Then f = Uvg ∈ K . By assumption and de?nition we have d2 g = V g = v? f . dx2 The mapping properties of v? imply that v? f ∈ H s for some s > 3/2. Thus we can de?ne h(r) = ?

∞ r

(r ? r′ )(v? f )(r′ )dr′ .

Schr¨ odinger operators on the half line Hence d2 h = v? f . dx2

381

d (h ? g) = 0 in the sense of distributions, and thus for some a, b ∈ C We conclude that d r2 we have g(r) = h(r) + a + br. Since g ∈ H ?s , s ≤ 3/2, and h ∈ H , we conclude that b = 0. Since g(0) = 0 by assumption, we have

2

a = ?h(0) = ? Thus we have shown that g(r) = ? such that

∞ 0

∞ 0

r′ v(r′ ) f (r′ )dr.

r′ v(r′ ) f (r′ )dr ?

∞ r

? (r ? r′ )(v? f )(r′ )dr′ = ?(GD 0 v f )(r),

? U f = UUvg = vg = ?vGD 0 v f,

or M0 f = 0. Assume now that V is multiplication by a function V, and that the factorization is chosen as above in Example 3.5. To prove part (iii), assume that f ∈ ker M0 and that ? vr, f = 0. Let g = ?GD 0 v f . Then M0 f = 0 implies f = Uvg. Using vr, f = 0, we ?nd that g satis?es the homogeneous Volterra equation g(r) = ?

∞ r

(r ? r′ )V(r′ )g(r′ )dr′ .

It follows by a standard iteration argument that g = 0, and then also f = 0. To prove the ?nal statement, assume that we have f j ∈ ker M0 , and f j = 0, j = 1, 2. De?ne g j = ? ? GD 0 v f j . Then we can ?nd α ∈ C, such that vr, f 1 + α vr, f 2 = 0. Thus we get (g1 + α g2 )(r) = ?

∞ r

(r ? r′ )V(r′ )(g1 + α g2 )(r′ )dr′ .

It follows again by the iteration argument that g1 + α g2 = 0, and then as above also f1 + α f2 = 0. This concludes the proof of part (iii). Remark 3.8. Let us note that for a local potential it suf?ces to assume β > 2 for the results in Lemma 3.7 to hold, since in this case we can use the mapping property of GD 0 given in Proposition 3.1. We need the following result, which is analogous to Lemma 2.6 of [4]. We include the proof here. Lemma 3.9. Assume that f j ∈ K , such that (3.14) holds for f j , j = 1, 2. Then we have that

? D ? D ? f1 , vGD 2 v f 2 = ? G0 v f 1 , G0 v f 2 .

(3.15)

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? 2 Proof. Let g j = ?GD 0 v f j . Since (3.14) holds, we have that g j ∈ L ([0, ∞)). Furthermore, we have

d2 g j = v? f j dr2

(3.16)

in the sense of distributions. We denote the Fourier transform on the line by ·. From (3.16) it follows that we have

ξ 2 g j (ξ ) = ?(v? f j ) (ξ ).

Since v? f j ∈ H s for some s > 3/2, the Fourier transform (v? f j ) is continuously differentiable, by the Sobolev embedding theorem. Since g j ∈ L2 (R), we must have (v? f j ) (0) = 0, d ? (v f j ) (0) = 0. dξ (3.17)

? It follows from (3.7) that GD 1 v f j = 0. Thus we have ? f1 , vGD 2 v f 2 = lim

1 ? ? D + κ 2 )?1 ? GD v f1 , ((H0 0 )v f 2 . κ →0 κ 2

Now compute using the Fourier transform: 1 ? ? D v f1 , ((H0 + κ 2)?1 ? GD 0 )v f 2 κ2 = = 1 κ2

∞ ?∞ ∞ ?∞

(v? f1 ) (ξ )

1 1 ? ξ 2 + κ2 ξ 2 ?1

(v? f2 ) (ξ )dξ .

(v? f1 ) (ξ )

(ξ 2 + κ 2)ξ 2

(v? f2 ) (ξ )dξ .

It follows from (3.17) that 1 ? (v f j ) (ξ ) ∈ L2 (R). ξ2 Thus we can use dominated convergence and take the limit κ → 0 under the integral sign above, to get the result. 3.4 Resolvent expansions: Results Let us now state the results obtained. We use the same terminology as in [4], since we have the same four possibilities for the point zero. We say that zero is a regular point for H , if dim ker M0 = 0. We say that zero is an exceptional point of the ?rst kind, if dim ker M0 = 1, and there is an f ∈ ker M0 with vr, f = 0. We say that zero is an exceptional point of the second kind, if dim ker M0 ≥ 1, and all f ∈ ker M0 satisfy vr, f = 0. In this case zero is an eigenvalue for H of multiplicity dim ker M0 . Finally, we say that zero is an exceptional point of the third kind, if dim ker M0 ≥ 2, and there is an f ∈ ker M0 with vr , f = 0. We introduce the following notation. Let S denote the orthogonal projection onto ker M0 . Then M0 + S is invertible in B (K ). We write J0 = (M0 + S)?1. (3.18)

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383

Theorem 3.10. Assume that zero is a regular point for H. Let p ≥ 1 be an integer. Assume that β > 2 p + 3 and s > p + 3 2 . Then we have the expansion R(?κ 2 ) =

j =0

∑ κ j G j + O (κ p+1)

p

(3.19)

in the topology of B (s, ?s). We have

?1 D G0 = (I + GD 0 V ) G0 ,

(3.20) (3.21)

?1 D D ?1 G1 = (I + GD 0 V ) G1 (I + V G0 ) . D The kernels of the operators GD 0 and G1 are given in (3.6) and (3.7), respectively.

Theorem 3.11. Let p ≥ 0 be an integer, and let V satisfy Assumption 3.3 for some β > 2 p + 7. Assume that zero is an exceptional point of the ?rst kind for H. Assume that s > p+ 7 2 . Then we have an asymptotic expansion R(?κ 2 ) =

j =?1

∑

p

κ j G j + O (κ p+1)

(3.22)

in the topology of B (s, ?s). We have G?1 = |Ψc Ψc |, Ψc = (3.23) where f , vr GD v f , | f , vr |2 0

Theorem 3.12. Let p ≥ 1 be an integer, and let V satisfy Assumption 3.3 for some β > 2 p + 11. Assume that zero is an exceptional point of the second kind for H. Assume that s > p + 11 2 . Then we have an asymptotic expansion R(?κ 2 ) =

for f ∈ ker M0 , f = 1.

j =?2

∑

p

κ j G j + O (κ p+1)

(3.24)

in the topology of B (s, ?s). We have G?2 = P0 , G? 1 = 0 ,

D ? D D ? D D ? D G0 = GD 0 ? G0 v J0 vG0 ? G0 v J0 vG2 V P0 ? P0V G2 v J0 vG0 D D + P0V GD 4 V P0 + P0V G2 + G2 V P0 , D ? D D ? D D D G1 = GD 1 ? G1 v J0 vG0 ? G0 v J0 vG1 + G3 V P0 + P0V G3 ? D D ? D + GD 1 v J0 vG2 V P0 + P0V G2 v J0 vG1 .

(3.25) (3.26)

(3.27)

(3.28)

Here P0 denotes the projection onto the zero eigenspace of H , and the operator J0 is de?ned by (3.18).

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Theorem 3.13. Let p ≥ 0 be an integer, and let V satisfy Assumption 3.3 for some β > 2 p + 11. Assume that zero is an exceptional point of the third kind for H. Assume that s > p + 11 2 . Then we have an asymptotic expansion R(?κ 2 ) =

j =?2

∑

p

κ j G j + O (κ p+1)

(3.29)

in the topology of B (s, ?s). We have G?2 = P0 , (3.30) (3.31) G?1 = |Ψc Ψc |.

Here P0 is the orthogonal projection onto the zero eigenspace, and Ψc is the canonical zero resonance function de?ned in (3.50). Remark 3.14. It is instructive to compare the results above with the results in the case of dimension d = 3 (see [4]). The operator we consider here is the angular moment component ? = 0 of ?? + V on L2 (R3 ), provided V commutes with rotations. In particular, we can only get zero as an eigenvalue for non-local V , and the expansion in the second exceptional case has coef?cient G?1 = 0 (and in the third exceptional case this coef?cient only contains the zero resonance term), consistent with the result in [4], where in the radial case this term lives in the ? = 1 subspace (see [4] Remark 6.6). 3.5 Resolvent expansions: Proofs We now give some details on the proofs of the resolvent expansions. Proof of Theorem 3.10. We give a brief outline of the proof. Since by assumption M0 is invertible in K , and since we assume β > 2 p + 3, we can compute the inverse of M (κ ) up to an error term O (κ p+1) by using the Neumann series and the expansion (3.12). This expansion is then inserted into (3.11), leading to the existence of the expansion up to terms of order p, and to the two expressions

D ? ?1 D G0 = GD 0 ? G0 v M0 vG0

and

? ?1 D ? ?1 D G1 = (I ? GD 0 v M0 v)G1 (I ? v M0 vG0 ).

Now we carry out the following computation:

? ?1 D ? D ? ?1 I ? GD 0 v M0 v = I ? G0 v (U + vG0 v ) v ? D ? ?1 = I ? GD 0 v U (I + vG0 v U ) v ? D ? ?1 = I ? GD 0 v Uv(I + G0 v Uv) ?1 D = I ? GD 0 V (I + G0 V ) ?1 = (I + GD 0V) .

Using this result, and its adjoint, we get the expressions in the theorem. It is easy to check that the above computations make sense between the weighted spaces.

Schr¨ odinger operators on the half line

385

Proof of Theorem 3.11. We assume that zero is an exceptional point of the ?rst kind. Thus we have that dim ker M0 = 1. Take f ∈ ker M0 , f = 1. Let S = | f f | be the orthogonal projection onto ker M0 . Assume β > 2 p + 7. Let q = p + 2. Then by Proposition 3.1 we have an expansion M (κ ) =

j =0

∑ κ j M j + O (κ q+1) = M0 + κ M1(κ ).

∞ j +1

q

(3.32)

We now use Corollary 2.2 of [5]. Thus M (κ ) is invertible, if and only if m(κ ) =

j =0

∑ (?1) j κ j S

M1 (κ )J0

S,

(3.33)

is invertible as an operator on SK . We also recall the formula for the inverse from Corollary 2.2 of [5], M (κ )?1 = (M (κ ) + S)?1 + 1 (M (κ ) + S)?1Sm(κ )?1 S(M (κ ) + S)?1. (3.34) κ

It is easy to see that we have an expansion m(κ ) = where m0 = SM1 S, m1 = SM2 S ? SM1J0 M1 S, m2 = SM3 S ? SM1J0 M2 S ? SM2J0 M1 S + SM1J0 M1 J0 M1 S. Using (3.7) we see that m0 = SM1 S = ?|Svr Svr| = ?| f , vr |2 S. (3.38) (3.35) (3.36) (3.37)

q ?1 j =0

∑ κ j m j + O (κ q),

Since f , vr = 0, it follows that m0 is invertible in SK . The Neumann series then yields an expansion

j q 1 m(κ )?1 = m? 0 + ∑ κ A j + O (κ ). j =1 q ?1

The coef?cients A j are in principle computable, although the expressions rapidly get very complicated. This expansion is inserted into (3.34). We also use the Neumann series to expand (M (κ ) + S)?1 = J0 + ∑ κ j M j + O (κ q+1).

j =1 q

This leads to an expansion M (κ )?1 =

q ?2 1 ?1 Sm0 S + ∑ κ j B j + O (κ q?1), κ j =0

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Arne Jensen and Gheorghe Nenciu

where we also used that SJ0 = J0 S = S. We now use (3.11) together with the expansion 2 above and the expansion of RD 0 (?κ ) from Proposition 3.1 to conclude that we have an expansion

q ?2 1 ? ?1 D v Sm SvG + R(?κ 2 ) = ? GD ∑ κ j G j + O (κ q?1). 0 0 κ 0 j =0

This concludes the proof of the theorem. Proof of Theorem 3.13. Assume that zero is an exceptional point of the third kind for H . Thus dim ker M0 ≥ 2, and there exists an f ∈ ker M0 with vr, f = 0. We repeat the computations in the proof of Theorem 3.11, although the assumptions are different. As above, S denotes the orthogonal projection onto ker M0 . Given p ≥ 0, assume β > 2 p + 11, and let q = p + 4. Then β > 2q + 3, and for this q we have the expansion (3.32). We also have the expansion (3.33) and the expressions for the ?rst three coef?cients given in (3.35), (3.36) and (3.37), respectively. We have m0 = SM1 S = ?|Svr Svr|, which by our assumption is a rank 1 operator. The orthogonal projection onto ker m0 is given by S1 = S + 1 |Svr Svr|, α

α = Svr

2 K

,

and by assumption S1 = 0. Now we use the main idea in [5], the repeated application of Corollary 2.2. Applying it once more, we get m(κ )?1 = (m(κ ) + S1)?1 + with q(κ ) = q0 + κ q1 + · · · + O (κ q?1) = S1 m1 S1 + κ [S1m2 S1 ? S1 m1 (m0 + S1)?1 m1 S1 ] + · · · + O (κ q?1). (3.40) Here the · · · are terms, whose coef?cients can be computed explicitly. We must have that q0 is invertible in S1 K . Otherwise, we can iterate the procedure, leading to a singularity in the expansion of R(?κ 2 ) of type κ ? j with j ≥ 3, contradicting the self-adjointness of H . Thus we have

q ?1 1 ?1 ?1 q(κ )?1 = q? ). 0 ? κ q0 q1 q0 + · · · + O (κ

1 (m(κ ) + S1)?1 S1 q(κ )?1 S1 (m(κ ) + S1)?1 κ

(3.39)

(3.41)

It remains to perform the back-substitution, and to compute the coef?cients. The backsubstitution leads to R(?κ 2 ) = 1 1 G?2 + G?1 + · · · + O (κ q?4), 2 κ κ

Schr¨ odinger operators on the half line with expressions

?1 D G? 2 = ? GD 0 vS1 q0 S1 vG0 , ?1 ?1 D G? 1 = GD 0 vS1 q0 S1 m2 S1 q0 S1 vG0 ?1 ?1 ?1 D ? GD 0 v(S ? S1 q0 S1 m1 )(m0 + S1 ) (S ? m1 S1 q0 S1 )vG0 .

387

(3.42)

(3.43)

These expressions can be simpli?ed. The computations are similar to the ones in [7], although there are some differences. Let P0 denote the projection onto the eigenspace for eigenvalue zero for H . Let us start by reformulating the result in Lemma 3.7. Let The operator T is a priori only bounded from K to H for s > 1/2, but Lemma 3.7 shows that it is actually bounded from K to H , with Ran T = P0 H . We also have that T is bounded from H to K , with Ran T = S1 K . Now Lemma 3.9 implies that T T = P0 and T T = S1 . (3.45) The adjoint T ? is the closure of the operator ?S1 vGD 0 . These observations lead to the result Now insert into (3.42) to get Then we note that

?1 ?1 D GD 0 vS1 q0 S1 m2 S1 q0 S1 vG0 = 0. ? S 1 m2 S 1 = S 1 GD 3 v S1 1 ? S1 q? 0 S1 = ?T T . ? T = ? GD 0 v S1

and T = UvP0 .

?s

(3.44)

(3.46)

G?2 = T T T ? T ? = P0 . (3.47)

This result holds, since = 0, as can be seen from the kernel (3.9) and the condition (3.14), which holds for all functions in the range of S1 . As for the last term in (3.43), from (3.38) and (3.40) it follows that 1 (3.48) (m0 + S1 )?1 = S1 ? 4 |Svr Svr|, α De?ne

1 (S ? S1q? 0 S1 m1 )S1 = 0.

(3.49)

Ψc =

1 Svr

2

D (GD 0 v|Svr ? P0V G2 v|Svr ).

(3.50)

Then a computation shows that we have This concludes the proof of Theorem 3.13. Proof of Theorem 3.12. We will not give the details of the proof of this theorem. It follows along the lines of the previous proofs. More precisely, if as above S is the orthogonal projection onto ker M0 , then (3.34)–(3.37) hold true with m0 = 0, and the argument leading to the invertibility of q0 , (see (3.40)), gives the fact that M1 is invertible. Then expanding in (3.34) and carrying the computation far enough, one ?nds the expressions in (3.25)– (3.28) for the ?rst four coef?cients explicitly, which are of interest in connection with the Fermi golden rule results below. G?1 = |Ψc Ψc |. (3.51)

388

Arne Jensen and Gheorghe Nenciu

4. A non-local potential example

We will illustrate Theorem 3.13 by giving an explicit example, using a rank 2 perturbation. The example is constructed such that H has zero as an exceptional point of the third kind. Let us de?ne two functions in L2 ([0, ∞)) as follows: ? ? ?0, for 0 < r ≤ 3 φ1 (r) = 1, for 3 < r < 4 , ? ? 0, for 4 ≤ r < ∞ ? 0, for 0 < r ≤ 1 ? ? ? ?1 , for 1 < r < 2 φ2 (r) = . 3 ? ? , for 2≤r≤3 ? 5 ? ? 0, for 3 < r < ∞

∞ 0

We have

rφ1 (r)dr = 0 and

∞ 0

rφ2 (r)dr = 0.

(4.1)

As our potential we take V =? 3 75 |φ1 φ1 | ? |φ2 φ2 |. 10 28 (4.2)

For the factorization we take K = C2 , and de?ne v ∈ B (H , K ) by ? ? 3 , f φ 1 10 ?. v( f ) = ? 75 φ , f 28 2

(4.3)

We let U = ?I , where I is the identity operator on K . Then we have V = v?Uv. Next we compute M0 . Direct computation shows that

? vGD 0 v = I.

The constants in V were chosen to obtain this result. Thus M0 = 0. Take f1 = Then vr, f1 = 0 and vr, f2 = 0, 1 0 and f2 = 0 . 1

due to (4.1). Thus zero is an exceptional point of the third kind for H with this potential. We can also ?nd the resonance function and an eigenfunction explicitly. An eigenfunction ? is given by ?GD 0 v f 2 . Carrying out the computations, one ?nds after normalization ? 2 ? 5 r, for 0 < r ≤ 1 ? ? ? ? 7 1 1 2 ? for 1 < r < 2 375 2 r ? 5 r + 2 , . (4.4) Ψ0 (r) = 9 27 3 98 ? ? 10 r2 + 5 r ? 10 , for 2 ≤ r ≤ 3 ? ? ? ? 0, for 3 < r < ∞

Schr¨ odinger operators on the half line

1

389

0.8

0.6

0.4

0.2

0 0 1 2 3 4 5 6

Figure 1. Canonical zero resonance function Ψc .

0.8

0.6

0.4

0.2

0 0 1 2 3 4 5 6

Figure 2. Normalized zero eigenfunction ?Ψ0 .

Using this function and the expression (3.50) one gets ? ? ? 52 r, for 0 < r ≤ 1 ? ? 343 ? ? ? ? 375 r2 ? 61 r + 375 , for 1 < r < 2 ? 49 686 ? ? 686 773 2025 2 . Ψc (r) = ? 225 686 r + 343 r ? 686 , for 2 ≤ r ≤ 3 ? ? ? ?? 1 r 2 + 8 r ? 9 , ? for 3 < r ≤ 4 ? 7 7 7 ? ? ? ? 1, for 4 < r < ∞

(4.5)

The plots of the two functions are shown in ?gures 1 and 2, respectively. The computations in this example have been made using Maple, the computer algebra system.

390

Arne Jensen and Gheorghe Nenciu

5. Application to the Fermi golden rule at thresholds

We recall the main result from [7] in the context of the Schr¨ odinger operators on the half D + V , where V satis?es Assumption 3.3 for a suf?line considered above. Let H = H0 ciently large β . Let W be another potential satisfying the same assumption. We consider the family H (ε ) = H + ε W for ε > 0. Assume that 0 is a simple eigenvalue of H , with normalized eigenfunction Ψ0 . Assume b = Ψ0 , W Ψ0 > 0. (5.1)

The results in [7] show that under some additional assumptions the eigenvalue zero becomes a resonance for H (ε ) for ε suf?ciently small. Here the concept of a resonance is the time-dependent one, as introduced in [9]. The additional assumption needed is that for some odd integer ν ≥ ?1 we have G j = 0, for j = ?1, 1, . . . , ν ? 2 and gν = Ψ0 , W Gν W Ψ0 = 0. (5.2)

Here G j denotes the coef?cients in the asymptotic expansion for the resolvent of H around zero, as given in either Theorem 3.12 or Theorem 3.13. The main result in [7] gives the following result on the survival probability for the state Ψ0 under the evolution exp(?itH (ε )). There exists ε0 > 0, such that for 0 < ε < ε0 we have Ψ0 , e?itH (ε ) Ψ0 = e?it λ (ε ) + δ (ε , t ), Here λ (ε ) = x0 (ε ) ? iΓ(ε ) with x0 (ε ) = bε (1 + O (ε )), as ε → 0. The error term satis?es |δ (ε , t )| ≤ Cε p(ν ) , t > 0, p(ν ) = min{2, (2 + ν )/2}. (5.6) Γ(ε ) = ?iν ?1 gν bν /2 ε 2+(ν /2) (1 + O (ε )), (5.4) (5.5) t > 0. (5.3)

We state two corollaries to the results in this paper and in [7]. COROLLARY 5.1.

D + V be a Schr¨ Let H = H0 odinger operator on the half line, with V satisfying Assumption 3.3 for some β > 17. Assume that zero is an exceptional point of the second kind for H. The zero eigenfunction is denoted by Ψ0 and is assumed to be simple. Let W also satisfy Assumption 3.3 for some β > 17. Assume that

b = Ψ0 , W Ψ0 = 0, g1 = Ψ0 , W G1W Ψ0 = 0. Let H (ε ) = H + ε W , ε > 0. The results (5.3)–(5.6) hold with ν = 1. We note that an expression for g1 can be obtained from (3.28). COROLLARY 5.2.

(5.7) (5.8)

Schr¨ odinger operators on the half line

391

D + V be a Schr¨ Let H = H0 odinger operator on the half line, with V satisfying Assumption 3.3 for some β > 9. Assume that zero is an exceptional point of the third kind for H. The zero eigenfunction is denoted by Ψ0 and is assumed to be simple. The canonical resonance function is denoted by Ψc . Let W also satisfy Assumption 3.3 for some β > 9. Assume that

b = Ψ0 , W Ψ0 = 0, g?1 = Ψ0 , W G?1W Ψ0 = | Ψ0 , W Ψc |2 = 0. Let H (ε ) = H + ε W , ε > 0. The results (5.3)–(5.6) hold with ν = ?1.

(5.9) (5.10)

This second Corollary is particularly interesting, since we can check the conditions (5.9) and (5.10) in the example given in § 4.. It is easy to see that one can get both Ψ0 , W Ψc = 0 and Ψ0 , W Ψc = 0, for both local and non-local perturbations W . Only in the ?rst case can one apply directly the results from [7], due to the condition (5.2). The other case has not yet been investigated in detail. One can also use the results on resolvent expansions to give examples using two channel models, as in [7]. We omit stating these results explicitly.

Acknowledgments

The ?rst author (AJ) was partially supported by a grant from the Danish Natural Sciences Research Council. The second author (GN) was partially supported by Aalborg University and was also supported in part by CNCSIS under Grant 905-13A/2005.

References

[1] Aktosun T, Factorization and small-energy asymptotics for the radial Schr¨ odinger equation, J. Math. Phys. 41 (2000) 4262–4270 [2] Amrein W O, Berthier A-M and Georgescu V, Lower bounds for zero energy eigenfunctions of Schr¨ odinger operators, Helv. Phys. Acta 57 (1984) 301–306 [3] Jensen A, Spectral properties of Schr¨ odinger operators and time-decay of the wave functions, Results in L2 (Rm ), m ≥ 5, Duke Math. J. 47 (1980) 57–80 [4] Jensen A and Kato T, Spectral properties of Schr¨ odinger operators and time-decay of the wave functions, Duke Math. J. 46 (1979) 583–611 [5] Jensen A and Nenciu G, A uni?ed approach to resolvent expansions at thresholds, Rev. Math. Phys. 13 (2001) 717–754 [6] Jensen A and Nenciu G, Erratum to the paper: A uni?ed approach to resolvent expansions at thresholds, Rev. Math. Phys. 16 (2004) 675–677 [7] Jensen A and Nenciu G, The Fermi golden rule and its form at thresholds in odd dimensions, Comm. Math. Phys. 261(3) (2006) 693–727 [8] Murata M, Asymptotic expansions in time for solutions of Schr¨ odinger-type equations, J. Funct. Anal. 49(1) (1982) 10–56 [9] Orth A, Quantum mechanical resonance and limiting absorption: the many body problem, Comm. Math. Phys. 126(3) (1990) 559–573 [10] Reed M and Simon B, Methods of Modern Mathematical Physics I: Functional Analysis, revised and enlarged edition (New York: Academic Press) (1980) [11] Yafaev D R, The low energy scattering for slowly decreasing potentials, Comm. Math. Phys. 85 (1982) 177–196

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