Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Perturbation of embedded eigenvalues for
Schrödinger type operators
Arne Jensen
Department of Mathematical Sciences
Aalborg University

Himeji PDE Conference, 6 March 2015

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

BackgroundI will give a survey of results obtained in collaboration with
Horia Cornean (Aalborg) and Gheorghe Nenciu (Bucarest) during
the last ten years.
Two of the papers are. A. Jensen, G. Nenciu: Uniqueness results for transient
dynamics of quantum systems. Contemp. Math. 447(2007), 165–174.
. H. Cornean, A. Jensen, G. Nenciu, Metastable states when
the Fermi Golden Rule constant vanishes, Comm. Math.
Phys. 334 (2015), 1189–1218.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

The Problem

We consider a self-adjoint operator H on a Hilbert space H
such that H has an eigenvalue embedded in its continuous
spectrum.
We assume the eigenvalue is λ = 0, is non-degenerate, and has
normalized eigenfunction Ψ0 .

Assumption
There exists a0 > 0 such that for all 0 < a < a0 with
Ja = (−a, a) we haveJa ∩ σpp (H) = {0},Ja ∩ σsc (H) = ,andJa ∩ σac (H) ≠ .

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

The Problem
For the perturbation W we assume

Assumption
W is self-adjoint and bounded.
The family of operators is thenH(ε) = H + εW ,ε > 0.

The problem to be solved is to describe what happens to theembedded eigenvalue for sufficiently small ε.
I concentrate on the case where the eigenvalue is properly
embedded, Case 1 below.

Case 1Case 2
Case 3

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Methods to solve the problem
There are several different ways of getting results on thisproblem. Two of the major methods are:. Spectral deformation methods.
. Time-dependent methods.
The spectral deformation methods have a long history. I will
not review it, but mention two methods.
Very important is the dilation-analytic method of
Aguilar-Balslev-Combes (1971) with significant extensions by
Simon (1972).
The general spectral deformation technique was given afoundation by Hunziker (1996) and then many other authors
extended the results.
Main results include a unified perturbation theory forembedded eigenvalues and resonances.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Spectral deformation methods

General picture of spectral deformation. The are embedded
eigenvalues revealed as isolated eigenvalues by spectral
deformation. The are resonances revealed by spectral
deformation.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Spectral deformation methods

The method used is to continue matrix elements of theresolvent R(z) = (H − z)−1 , viz. hψ, R(z)ψi, from the upper
half plane Im z > 0 into a region in the lower half plane,crossing the absolutely continuous spectrum. This
continuation should exist for a dense set of vectors ψ.
Embedded eigenvalues give rise to poles on the real axis in thecontinued matrix elements. Poles of the continued matrix
element in the lower half plane are defined to be resonances.
To get a meaningful definition of a resonance one needs further
conditions. I will not enter into details on this point now.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Spectral deformation methods

Discussion of spectral deformation methods.
Advantages:. Covers many important quantum systems, including atoms
and molecules.
. Gives a consistent definition of a resonance and in manycases makes available the full power of analytic
perturbation theory for both resonances and embedded
eigenvalues.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Spectral deformation methods

Discussion of spectral deformation methods.
Advantages:. Covers many important quantum systems, including atoms
and molecules.
. Gives a consistent definition of a resonance and in manycases makes available the full power of analytic
perturbation theory for both resonances and embedded
eigenvalues.
Disadvantage:. Dilation analyticity (or other forms of analyticity) of
perturbations is a strong assumption.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
My interest is in time-dependent methods and I willconcentrate on this aspect from this point.
The time-dependent methods are well adapted to a family
H(ε) = H + εW ,ε > 0.

The results I present are for small ε. So it is an asymptotic
theory.
First some general results. The set-up is: H is self-adjoint on
H with an eigenvalue λ0 and eigenprojection P0 assumed to beof finite rank. We assume that we have constructed an effective
Hamiltonian h(ε) on P0 H .
We assume that we haveP0 e−itH(ε) P0 = e−ith(ε) P0 + δ(ε, t)

(1)

with
supkδ(ε, t)k ≤ Cεp ,t≥0

p > 0.

(2)

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
This framework is general enough to cover both the case whenthe eigenvalue remains (e.g. if λ0 is isolated) and the casewhere it becomes a resonance.
Thus h(ε) describes the short time behaviour of the evolution
in the subspace P0 H . If h(ε) has an asymptotic expansion as
ε → 0 we have a uniqueness result.

Theorem(i) Assume that h1 (ε) and h2 (ε) both satisfy (1) and (2) with thesame p > 0. Assume that for some ε0 > 0 h1 (ε) satisfies
h1 (ε) = λ0 P0 + εh11 + εf 1 (ε),

0 ≤ ε < ε0 ,such that h11 = (h11 )∗ , Im f 1 (ε) ≤ 0, and f 1 (ε) = o(1). Then forε0 sufficiently small we have
kh1 (ε) − h2 (ε)k ≤ Cεp+1 ,0 ≤ ε < ε0 .

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Continuation:

Theorem(ii) Assume that h1 (ε) and h2 (ε) both satisfy (1) and (2) with
p = 2. Assume that for some ε0 > 0 h1 (ε) satisfies
h1 (ε) = λ0 P0 + εh1 + ε2 h2 + o(ε2 ),0 ≤ ε < ε0 ,1such that h1 = h∗
1 and Im h (ε) ≤ 0. Then there exists a familyof invertible operators U(ε) on P0 H with U(ε) = P0 + O(ε2 ) suchthat for sufficiently small ε0 we have

kh1 (ε) − U(ε)h2 (ε)U(ε)−1 k ≤ Cε4 ,0 ≤ ε < ε0 .

Reference:
Rank P0 = 1: Cattaneo-Graf-Hunziker 2006
General case: J.-Nenciu 2007.
If Im h2 ≠ 0 we speak of resonance behaviour.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods

The quantity of interest for a simple eigenvalue is thetransition probability amplitude
A0 (ε, t) = hΨ0 , e−itH(ε) Ψ0 i.
Let us consider the case when λ0 = 0 is an isolated eigenvalue.
and take h(ε) = λ(ε)P0 , with λ(ε) from eigenvalue perturbation
theory. Then
|A0 (ε, t) − e−itλ(ε) | ≤ Cε2 .

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
In general, if we restrict to P0 H for the effective Hamiltonian,then we can do no better. This holds even in the case of anembedded eigenvalue, if an eigenvalue moves but persists
under perturbation.

Proposition
Assume that for sufficiently small ε there exists exactly one
eigenvalue λ(ε) (λ(ε) → 0 for ε → 0), with eigenfunction Ψ (ε).
Assume that there exists a Ψ1 ≠ 0 such thatkΨ (ε) − Ψ0 − εΨ1 k = o(ε) and hΨ0 , Ψ1 i = 0. Then there exists
C > 0 such that
supkA0 (ε, t) − e−itλ(ε) k ≥ Cε2 .
t≥0Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

We now state the first result on perturbation of embedded
eigenvalues. This requires some preparation.

Assumption
Assume there exists a Hilbert space K and bounded operators
A ∈ B(H , K) and D ∈ B(K) such that W = A∗ DA. Here D = D ∗and D 2 = I is assumed.
Let Q0 = I − P0 . Define
G(z) = AQ0 (H − z)−1 Q0 A∗ ,and
F 0 (z, ε) = εhΨ0 , W Ψ0 i − z− ε2 hΨ0 , A∗ D{G(z) − εG(z)[D + εG(z)]−1 G(z)}DAΨ0 i.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

Using the Stone formula and the Schur-Livsic-Feshbach-Grushin(SLFG) formula we obtain our starting point:
Z


1
1
1
dx e−ixt 0− 0.
A0 (ε, t) = lim
η&0 2π i R
F (x + iη, ε) F (x − iη, ε)We need conditions on G(z). Let
Da = {z = x + iη ∈ C | x ∈ Ja = (−a, a), 0 < η < 1}.
For 0 < θ < 1 we use the Hölder space C 1,θ (Da , B(K)).

Assumption
For some 0 < θ < 1 we have G(·) ∈ C 1,θ (Da , B(K)).
This assumption can be verified using Mourre theory,
Agmon-Herbst-Skibsted 1989.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

Theorem (J.-Nenciu 2006)
Under the above Assumptions and 0 < ε < ε0 taken sufficiently
small we have F 0 (·, ε) ∈ C 1,θ (Da ; C).
In particular, this function has an extension to the real axis withthe same smoothness properties
F 0 (x, ε) := limη&0 F 0 (x + iη, ε) ∈ C 1,θ (Ja ; C).
Let R 0 (x, ε) and I 0 (x, ε) be the real and imaginary part ofF 0 (x, ε), respectively, viz. F 0 (x, ε) =: R 0 (x, ε) + iI 0 (x, ε).
For a fixed ε sufficiently small the equation R 0 (x, ε) = 0 has aunique solution x 0 (ε) in Ja/2 , which obeys the estimate|x 0 (ε)| Ü ε. Define λ0 (ε) := x 0 (ε) + iI 0 (x 0 (ε), ε). Then forsufficiently small ε we have:0|A0 (ε, t) − e−itλ (ε) | Ü ε2 .

(3)Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

We call λ0 (ε) the resonance position. The uniqueness results
show that it is asymptotically well defined.
Comments. The estimate (3) has exactly the same form as for the caseof eigenvalues, with the perturbed eigenvalue replaced bythe ‘resonance position’, λ0 (ε). In particular, theProposition above shows that in general, the error in (3)cannot be made smaller.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

We call λ0 (ε) the resonance position. The uniqueness results
show that it is asymptotically well defined.
Comments. The estimate (3) has exactly the same form as for the caseof eigenvalues, with the perturbed eigenvalue replaced bythe ‘resonance position’, λ0 (ε). In particular, theProposition above shows that in general, the error in (3)cannot be made smaller.
. The computation of I 0 (x 0 (ε), ε), using |x 0 (ε)| Ü ε, leads toI 0 (x 0 (ε), ε) = −ε2 ΓFGR + O(ε3 ), with
ΓFGR := π hΨ0 , W δ(Q0 HQ0 )W Ψ0 i,
which coincides with the result given by the Dirac
computation. HereW δ(Q0 HQ0 )W = limη&0 Im W (Q0 HQ0 − iη)−1 W .

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues. Note that no condition ΓFGR > 0 or I 0 (x 0 (ε), ε) < 0 is
required for this result, in contrast to other authors. Orth
1990 has proved that I 0 (x 0 (ε), ε) = 0 if and only if x 0 (ε) is
an eigenvalue.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues. Note that no condition ΓFGR > 0 or I 0 (x 0 (ε), ε) < 0 is
required for this result, in contrast to other authors. Orth
1990 has proved that I 0 (x 0 (ε), ε) = 0 if and only if x 0 (ε) is
an eigenvalue.
. In the analytic case, the resonance position is spectrally
defined as a pole of the analytically continued resolventand coincides with the zero zr = xr + iyr of the analytic
continuation of F 0 (z, ε). In our case its definition also
involves Ψ0 since it is given via the limit values of F 0 (z, ε).
Comparing the decay law given by the Theorem with the
one given by Hunziker 1990 in the analytic case, one canshow that |λ0 (ε) − zr | Ü ε2 |yr |, i.e. up to some order in ε,
λ0 (ε) is indeed a spectral object of the family H(ε), due tothe uniqueness result.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

Can one improve the above results with respect to the error
term by replacing Ψ0 by another choice?
In the dilation analytic framework Hunziker 1990 proved that if
the eigenvalue λ0 (= 0) initially is isolated and the formal
Rayleigh-Schrödinger (R-S) expansion for the perturbed
eigenfunction is well defined up to order εN then taking as theresonance function the truncated R-S series one can prove thatboth Im λ(ε) and the error term are of order ε2N+2 . The result
applies to the Stark Hamiltonian to any order N.
We want to obtain similar results in the framework of finite
regularity instead of analyticity.
Under the assumption that ΓFGR = 0 it turns out that one candeal with the case N = 1 in the R-S expansion.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

The detailed construction of a better resonance function israther technical. I will not give the details. The result is afamily Ψ 1 (ε). We define
A1 (ε, t) = hΨ 1 (ε), e−itH(ε) Ψ 1 (ε)iand we define F 1 (z, ε) in a manner analogous to F 0 (z, ε).
The required assumptions are

Assumption(i) For some 0 < θ < 1 we have G(·) ∈ C 1,θ (Da , B(K)).
(ii) We have ΓFGR = 0.
Under these assumptions we have for the boundary values
F 1 (x, ε) := lim F 1 (x + iη, ε) ∈ C 1,θ (Ja/2 ; C)η&0

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Results on embedded eigenvalues

Theorem (Cornean-J.-Nenciu 2015)
Under the above assumptions we have the following results. Let
F 1 (x, ε) =: R 1 (x, ε) + iI 1 (x, ε). For a fixed ε the equation
R 1 (x, ε) = 0 has a unique solution x 1 (ε) in Ja/2 , with |x 1 (ε)| Ü ε. Define
λ1 (ε) := x 1 (ε) + iI 1 (x 1 (ε), ε).
Then for sufficiently small ε we have1|A1 (ε, t) − e−itλ (ε) | Ü ε4 .

(4)Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
High order R-S

One obvious question is whether we can continue theconstruction of better Ψ N (ε) in the finite regularity case for
perturbation of a simple embedded eigenvalue. Looking at theformula in Kato’s book it looks hopeless with the present
approach.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Time-dependent methods
Eigenvalue at threshold

We recall the illustration

Case 1Case 2
Case 3A question we have also dealt with is what happens in Case 2
above.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

I will briefly review some of the results obtained with G. Nenciu
on this problem some time ago.
Resolvent smoothness is replaced by asymptotic expansion of
the resolvent around threshold. Results in odd dimensions.
Definition of zero resonance for Schrödinger operator.
H = −∆ + V on L2 (Rd ), V (x) real-valued, |V (x)| ≤ C|x|−β ,β  2. Assume (−∆ + V )Ψ = 0. Ψ is a zero resonance function,if. d = 1: Ψ (x) = c0 + f , c0 ≠ 0, f ∈ L2 (R1 ).

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

I will briefly review some of the results obtained with G. Nenciu
on this problem some time ago.
Resolvent smoothness is replaced by asymptotic expansion of
the resolvent around threshold. Results in odd dimensions.
Definition of zero resonance for Schrödinger operator.
H = −∆ + V on L2 (Rd ), V (x) real-valued, |V (x)| ≤ C|x|−β ,β  2. Assume (−∆ + V )Ψ = 0. Ψ is a zero resonance function,if. d = 1: Ψ (x) = c0 + f , c0 ≠ 0, f ∈ L2 (R1 ).
c0. d = 3: Ψ (x) =+ f , c0 ≠ 0, f ∈ L2 (R3 ).
|x|

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

I will briefly review some of the results obtained with G. Nenciu
on this problem some time ago.
Resolvent smoothness is replaced by asymptotic expansion of
the resolvent around threshold. Results in odd dimensions.
Definition of zero resonance for Schrödinger operator.
H = −∆ + V on L2 (Rd ), V (x) real-valued, |V (x)| ≤ C|x|−β ,β  2. Assume (−∆ + V )Ψ = 0. Ψ is a zero resonance function,if. d = 1: Ψ (x) = c0 + f , c0 ≠ 0, f ∈ L2 (R1 ).
c0. d = 3: Ψ (x) =+ f , c0 ≠ 0, f ∈ L2 (R3 ).
|x|. d ≥ 5: No zero resonance function exists.

Introduction

Methods

Time-dependent methods

High order R-S

Case 2 our results I

H Schrödinger operator on H . Assumptions are. There exists a > 0, such that (−a, 0) ⊂ ρ(H) and[0, a] ⊂ σess (H)Eigenvalue at thresholdIntroduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

H Schrödinger operator on H . Assumptions are. There exists a > 0, such that (−a, 0) ⊂ ρ(H) and[0, a] ⊂ σess (H). Assume that zero is a non-degenerate eigenvalue of H:
HΨ0 = 0, with kΨ0 k = 1, and there are no other eigenvalues
in [0, a]. Let P0 = |Ψ0 ihΨ0 | be the orthogonal projection
onto the one-dimensional eigenspace.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

H Schrödinger operator on H . Assumptions are. There exists a > 0, such that (−a, 0) ⊂ ρ(H) and[0, a] ⊂ σess (H). Assume that zero is a non-degenerate eigenvalue of H:
HΨ0 = 0, with kΨ0 k = 1, and there are no other eigenvalues
in [0, a]. Let P0 = |Ψ0 ihΨ0 | be the orthogonal projection
onto the one-dimensional eigenspace.
. Assume hΨ0 , W Ψ0 i = b > 0.

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I√. For Re κ ≥ 0 and z ∈ C \ [0, ∞) we let κ = −i z, z = −κ 2 .
There exist N0 ∈ N and δ0 > 0, such that forκ ∈ {κ ∈ C | 0 < |κ| < δ0 , Re κ ≥ 0} we have (recall
G(z) = AQ0 (H − z)−1 Q0 A∗ )G(z) =N0X

ej κ j + κ N0 +1 G
eN0 (κ),
G

j=−1

where
ej
G

are bounded and self-adjoint,e−1
G

is of finite rank and self-adjoint,eN0 (κ) is uniformly bounded in κ.
GIntroduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I. From the expansion we gethΨ0 , A∗ DG(z)DAΨ0 i =N0X

gj κ j + κ N0 +1 gN0 (κ),

j=−1

where
ej DAΨ0 i,
gj = hΨ0 , A∗ D G
eN0 (κ)DAΨ0 i.
gN0 (κ) = hΨ0 , A∗ D G
Notice that we have gj = g j .

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I. From the expansion we gethΨ0 , A∗ DG(z)DAΨ0 i =N0X

gj κ j + κ N0 +1 gN0 (κ),

j=−1

where
ej DAΨ0 i,
gj = hΨ0 , A∗ D G
eN0 (κ)DAΨ0 i.
gN0 (κ) = hΨ0 , A∗ D G
Notice that we have gj = g j .
. There exists an odd integer, −1 ≤ ν ≤ N0 , such that gν ≠ 0,
ej = 0 for j = −1, 1, . . . , ν − 2.
G

Introduction

Methods

Time-dependent methods

High order R-S

Eigenvalue at threshold

Case 2 our results I

Theorem
Let the above assumptions hold. Then for sufficiently small ε > 0
we have|hΨ0 , e−itH(ε) Ψ0 i − e−it(x0 (ε)−iΓ (ε)) | ≤ Cεp(ν) .
Here p(ν) = min{2, (2 + ν)/2}, andΓ (ε) = −iν−1 gν bν/2 ε2+(ν/2) (1 + O(ε)),
x0 (ε) = bε(1 + O(ε)).
Recall that ν = −1, 1, 3, 5, . . ..