[PPT] - Dissipative quantum systems: scattering theory and J er emy PowerPoint Presentation

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

ptical

model The abstract framework

f

dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Dissipative quantum systems: scattering theory and spectral singularities

J´ er´ emy Faupin

Universit´ e de Lorraine, Metz

Conference: The Analysis of Complex Quantum Systems CIRM, October 2019 J.F. and J. Fr¨

hlich, Asymptotic completeness in dissipative scattering theory,
Adv. Math., 340, 300-362, (2018).

J..F. and F. Nicoleau, Scattering matrices for dissipative quantum systems,

J. Funct. Anal., 9, 3062-3097, (2019).

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

1 Introduction: The nuclear optical model 2 The abstract framework of dissipative scattering theory 3 Spectral singularities 4 Asymptotic completeness 5 Applications

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Introduction The nuclear optical model

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The nuclear optical model (I)

Quantum system

Neutron targeted onto a complex nucleus
Either the neutron is elastically scattered off the nucleus
Or it is absorbed by the nucleus =

⇒ Formation of a compound nucleus

Concept of a compound nucleus was introduced by Bohr (’36)

Model

Feshbach, Porter and Weisskopf (’54): nuclear optical model describing both

elastic scattering and absorption

“Pseudo-Hamiltonian” on L2(R3)

H = −∆ + V (x) − iW (x) with V and W real-valued, compactly supported, W ≥ 0

Widely used in Nuclear Physics, refined versions include, e.g., spin-orbit

interactions

Empirical model

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The nuclear optical model (II)

Interpretation

−iH generates a strongly continuous semigroup of contractions
e−itH

t≥0

Dynamics described by the Schr¨
dinger equation
i∂tut = Hut

u0 ∈ D(H) If the neutron is initially in the normalized state u0, after a time t ≥ 0, it is in the unnormalized state e−itHu0

Probability that the neutron, initially in the normalized state u0 (supposed to be
rthogonal to bound states), eventually escapes from the nucleus:

pscatt(u0) = lim

t→∞

e−itHu0
2
Probability of absorption:

pabs(u0) = 1 − lim

t→∞

e−itHu0
2
If pscatt(u0) > 0 (and u0 is orthogonal to bound states), one expects that there

exists an (unnormalized) scattering state u+ such that u+2 = pscatt(u0) and lim

t→∞

e−itHu0 − eit∆u+
= 0

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The nuclear optical model (III)

Aim

Explicit expression of H rests on experimental scattering data
Nuclear optical model generalizes to any quantum system S interacting with

another system S′ and susceptible of being absorbed by S′

Need to develop the full scattering theory of a class of models

References: mathematical scattering theory for dissipative operators in Hilbert spaces

Abstract framework: Lax-Phillips [’73], Martin [’75], Davies [’79,’80], Neidhardt

[’85], Exner [’85], Petkov [’89], Kadowaki [’02,’03], Stepin [’04], . . .

Small perturbations: Kato [’66], Falconi-F-Fr¨
hlich-Schubnel [’17], . . .
Schr¨
dinger operators: Mochizuki [’68], Simon [’79], Wang-Zhu [’14], . . .

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The abstract framework of dissipative scattering theory

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Abstract model

The model

H complex Hilbert space
Pseudo-Hamiltonian

H = H0 + V − iC ∗C = HV − iC ∗C, with H0 ≥ 0, V symmetric, C ∈ L(H) and V , C relatively compact with respect to H0

HV is self-adjoint, H is closed and maximal dissipative, with domains

D(H) = D(HV ) = D(H0)

−iH generates a strongly continuous semigroup of contractions {e−itH}t≥0.

More precisely, −iH generates a group {e−itH}t∈R s.t.

e−itH

≤ 1, t ≥ 0,

e−itH

≤ eC∗C|t|, t ≤ 0

σess(H) = σess(H0) and σ(H) \ σess(H) consists of an at most countable

number of eigenvalues of finite algebraic multiplicities that can only accumulate at points of σess(H)

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Figure: Form of the spectrum of H.

Example

Example to keep in mind: H = L2(R3), H0 = −∆, HV = −∆ + V (x) = H∗

V ,

C = W (x)

1 2

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Spectral subspaces

Space of bound states

Hb(H) = Span

u ∈ D(H), ∃λ ∈ R, Hu = λu
Generalized eigenstates corresponding to non-real eigenvalues
For λ ∈ σ(H) \ σess(H), Riesz projection defined by

Πλ = 1 2iπ

γ

(zId − H)−1dz, where γ is a circle centered at λ, of sufficiently small radius

Ran(Πλ) spanned by generalized eigenvectors of H associated to λ, u ∈ D(Hk)

s.t. (H − λ)ku = 0

Space of generalized eigenstates corresponding to non-real eigenvalues:

Hp(H) = Span

u ∈ Ran(Πλ), λ ∈ σ(H), Im λ < 0
“Dissipative space”

Hd(H) =

u ∈ H, lim

t→∞ e−itHu = 0

⊃ Hp(H)

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The adjoint operator H∗

Properties of H∗

H∗ = H0 + V + iC ∗C
λ ∈ σ(H∗) if and only if ¯

λ ∈ σ(H)

iH∗ generates the strongly continuous contraction semigroup {eitH∗}t≥0
Spectral subspaces

Hb(H∗) = Span

u ∈ D(H), ∃λ ∈ R, H∗u = λu
,

Hp(H∗) = Span

u ∈ Ran(Π∗

λ), λ ∈ σ(H∗), Im λ > 0

,

Hd(H∗) =

u ∈ H, lim

t→∞ eitH∗u = 0

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The wave operators

The wave operator W−(H, H0)

Defined by

W−(H, H0) = s-lim

t→∞ e−itHeitH0

If it exists, W−(H, H0) is a contraction
W−(H, H0)H0 = HW−(H, H0)

The wave operator W+(H0, H)

Defined by

W+(H0, H) = s-lim

t→∞ eitH0e−itHΠb(H)⊥

where Πb(H)⊥ denotes the orthogonal projection onto Hb(H)⊥

If it exists, W+(H0, H) is a contraction
H0W+(H0, H) = W+(H0, H)H
Under some conditions, W+(H0, H) = W+(H∗, H0)∗
For u ∈ Hb(H)⊥,

u+ = W+(H0, H)u ⇐ ⇒ lim

t→∞

e−itHu − e−itH0u+
= 0

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

The scattering operator and matrices

The scattering operator S(H, H0)

Defined by

S(H, H0) = W+(H∗, H0)∗W−(H, H0) = W+(H0, H)W−(H, H0)

If it exists, S(H, H0) is a contraction
S(H, H0) commutes with H0

The scattering matrices S(λ)

Defined by the fiber decomposition

S(H, H0) = ⊕

Λ

S(λ)dλ in H = ⊕

Λ

H(λ)dλ, Λ = σ(H0)

If H0 has a purely absolutely continuous spectrum of constant multiplicity, then

H(λ) = M and S(H, H0) acts in H = L2(Λ; M) as [S(H, H0)u](λ) = S(λ)u(λ)

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Basic assumptions

(H1) Spectrum of H0

The spectrum of H0 is purely absolutely continuous and has a constant multiplicity (which may be infinite)

(H2) Spectrum of HV

HV has finitely many eigenvalues counting multiplicity, no embedded eigenvalues, and σsc(HV ) = ∅

(H3) Wave operators for HV and H0

The wave operators W±(HV , H0) = s-lim

t→±∞eitHV e−itH0,

W±(H0, HV ) = s-lim

t→±∞eitH0e−itHV Πac(HV )

exist and are asymptotically complete, i.e., Ran(W±(HV , H0)) = Hac(HV ) = Hpp(HV )⊥, Ran(W±(H0, HV )) = H

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Regularity of C w.r.t. HV

(H4) Relative smoothness of C with respect to HV

There exists a constant cV > 0, such that

R
Ce−itHV Πac(HV )u
2dt ≤ c2

V Πac(HV )u2,

for all u ∈ H

Remarks

Estimates of this form considered in [Kato ’66]
The following estimate is always satisfied

∞

Ce−itHu
2dt ≤ 1

2 u2

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Basic results

Existence and properties of the wave operators

Suppose (H1)–(H4). Then ∗ W−(H, H0) and W+(H0, H) exist ∗ W−(H, H0) is an injective contraction and Ran(W−(H, H0)) =

Hb(H∗) ⊕ Hd(H∗)

⊥ ∗ W+(H0, H) is a contraction with dense range and Ker(W+(H0, H)) = Hb(H) ⊕ Hd(H) ∗ S(H, H0) exists and is a contraction

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Definition of asymptotic completeness and consequences

Definition

W−(H, H0) is said to be asymptotically complete if

Ran(W−(H, H0)) =

Hb(H∗) ⊕ Hp(H∗)

⊥

Main issues: prove that Hd(H∗) = Hp(H∗) and that Ran(W−(H, H0)) is closed

Consequences of asymptotic completeness

∗ Direct sum decomposition H = Hb(H) ⊕ Hp(H) ⊕

Hb(H∗) ⊕ Hp(H∗)

⊥ and the restriction of H to (Hb(H∗) ⊕ Hp(H∗))⊥ is similar to H0 ∗ W+(H0, H) : H → H is surjective and Ker(W+(H0, H)) = Hb(H) ⊕ Hp(H) ∗ S(H, H0) : H → H is bijective

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Spectral singularities

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Regular spectral point and spectral singularity

Notation

Recall that Λ = σ(H0) = σess(H) and set R(z) = (H − z)−1, RV (z) = (HV − z)−1, R0(z) = (H0 − z)−1

Definition

λ ∈ ˚ Λ is a regular spectral point of H if there exists a compact interval Kλ ⊂ R whose interior contains λ and such that the limit CR(µ − i0+)C ∗ = lim

ε↓0 CR(µ − iε)C ∗

exists uniformly in µ ∈ Kλ in the norm topology of L(H). If λ is not a regular spectral point of H, we say that λ is a “spectral singularity” of H

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Equivalent possible definitions of a regular spectral point

Theorem [F., Nicoleau]

Suppose

(H1)–(H4)
V is strongly smooth w.r.t. H0 and C is strongly smooth w.r.t. HV

Let λ ∈ ˚ Λ. The following conditions are equivalent: ∗ λ is a regular spectral point of H ∗ λ is not an accumulation point of eigenvalues of H located in λ − i(0, ∞) and the limit CR(λ − i0)C ∗ := lim

ε↓0 CR(λ − iε)C ∗

exists in the norm topology of L(H) ∗ I − iCRV (λ − i0)C ∗ is invertible in L(H) ∗ S(λ) is invertible in L(M)

Proposition [F., Nicoleau]

{Spectral singularities of H} = closed set of Lebesgue measure 0

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Properties of the scattering matrices

Theorem [F., Nicoleau]

Under the previous assumptions, for all λ ∈ ˚ Λ, ∗ S(λ) is a contraction ∗ S(λ) − I is compact ∗ If in addition dim M = +∞, then S(λ) = 1

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Consequences: Properties of the scattering

perator

Theorem [F., Nicoleau]

Under the previous assumptions, if dim M = +∞, S(H, H0) = 1 Suppose in addition that

Λ \ ˚

Λ is finite

All λ ∈ Λ \ ˚

Λ are regular in a suitable sense (if Λ is right-unbounded, we assume in addition that +∞ is regular). Then ∗ S(H, H0) is invertible in L(H) ⇐ ⇒ H has no spectral singularities in ˚ Λ ∗ If the previous equivalent conditions hold, then Ran(W−(H, H0)) =

Hb(H∗) ⊕ Hd(H∗)

⊥ ∗ In particular, H has a spectral singularity = ⇒ W−(H, H0) is not asymptotically complete

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Spectral singularity of finite order

Definition

We say that λ ∈ ˚ Λ is a spectral singularity of H of finite order if λ is a spectral singularity of H and there exists ν ∈ N∗ and a compact interval Kλ, whose interior contains λ, such that the limit lim

ε↓0 (µ − λ)νCR(µ − iε)C ∗

exists uniformly in µ ∈ Kλ in the norm topology of L(H)

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Asymptotic completeness

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Further conditions

(H5) Pure point spectrum of H

H has at most finitely many eigenvalues

(H6) Spectral singularities of H

H has at most finitely many spectral singularities in ˚

Λ

Each spectral singularity is of finite order
If Λ is right-unbounded, +∞ is regular

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Dissipative space

Theorem [F., Fr¨

hlich]

Suppose (H1)–(H6). Then Hd(H) = Hp(H)

Remarks

Finding conditions implying this result quoted as open in [Davies ’80]
For small perturbations, the theorem follows from similarity of H and H0 [Kato

’66], implying that Hd(H) = Hp(H) = {0}

Interpretation for the nuclear optical model: unless the initial state is a linear

combination of generalized eigenstates corresponding to non-real eigenvalues of H, the probability that the neutron eventually escapes from the nucleus is always strictly positive

Idea

Generalization of spectral projections for non self-adjoint operators with spectral singularities

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Consequence: Asymptotic completeness

Theorem [F., Fr¨

hlich]

Suppose (H1)–(H6). Then ∗ H has no spectral singularities in Λ = ⇒ W−(H, H0) is asymptotically complete i.e., Ran(W−(H, H0)) =

Hb(H∗) ⊕ Hp(H∗)

⊥

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Applications

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: the nuclear optical model (I)

Setting

On H = L2(R3), H0 = −∆, HV = H0 + V (x), H = HV − iW (x)

References

Spectral and scattering theories for (HV , H0): Kato [’66], Reed-Simon [’78],

Isozaki-Kitada [’85], Koch-Tataru [’06], Yafaev [’10]

Relative smoothness/Analysis at threshold: Kato [’66], Jensen-Kato [’79],

Constantin-Saut [’89], Ben Artzi-Klainerman [’92], Jensen-Nenciu [’01], Fournais-Skibsted [’04], Schlag [’07]

Mourre’s theory: Mourre [’81], Boutet de Monvel-Georgescu [’96]
Resonances theory: Sj¨
strand [’02], Dyatlov-Zworski [’18]
Dissipative framework: Simon [’79], Wang [’11,’12]

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: the nuclear optical model (II)

Theorem

Let V , W ∈ L∞(R3; R). Assume that

V ∈ C2(R3) s.t. for |α| ≤ 2, ∂α

x V (x) = O(x−ρ−|α|) with ρ > 3,

W (x) ≥ 0, W (x) > 0 on a non-trivial open set, W (x) = O(x−δ) with δ > 2,
0 is neither an eigenvalue nor a resonance of HV

Then, ∗ S(λ) ∈ L(L2(S2)) is invertible ⇐ ⇒ λ is not a spectral singularity of H ∗ S(H, H0) is invertible in L(L2(R3)) ⇐ ⇒ H has no spectral singularities in Λ ∗ If the previous equivalent conditions hold, then Ran(W−(H, H0)) = Hd(H∗)⊥

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: the nuclear optical model (III)

Theorem

Let V , W ∈ L∞(R3; R). Assume that

V and W are compactly supported,
W (x) ≥ 0, W (x) > 0 on a non-trivial open set,
0 is neither an eigenvalue nor a resonance of HV

Then ∗ Hd(H) = Hp(H) ∗ W−(H, H0) is asymptotically complete ⇐ ⇒ Ran(W−(H, H0)) = Hp(H∗)⊥ ⇐ ⇒ H has no spectral singularities ∗ If the previous equivalent conditions hold, then ∗ S(H, H0) is invertible in L(L2(R3)), ∗ The restriction of H to Hp(H∗)⊥ is similar to H0 ∗ There exist m1 > 0 and m2 > 0 such that, for all u ∈ Hp(H∗)⊥, m1u ≤

e−itHu
≤ m2u,

t ∈ R

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: the nuclear optical model (IV)

Remarks

Spectral singularity = real resonance
H has no real eigenvalues
[Wang ’11]: 0 cannot be a resonance of H
[Wang ’12]: For any λ > 0, one can construct smooth compactly supported

potentials V and W such that λ is a spectral singularity of H

Work in progress: prove that “generically”, there is no spectral singularity

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: scattering for Lindblad master equations (I)

References

Davies [’80]
Different approach: Alicki [’81], Alicki-Frigerio [’83]
Falconi-F.-Fr¨
hlich-Schubnel [’16]

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: scattering for Lindblad master equations (II)

Lindbladian and quantum dynamical semigroup

If one considers a quantum particle interacting with a dynamical target, takes

the trace over the degrees of freedom of the target and studies the reduced effective evolution of the particle, then, in the kinetic limit, the dynamics of the particle is given by a quantum dynamical semigroup {e−itL}t≥0 generated by a Lindbladian L

On J1(H) (space of trace-class operators), L is given by

L(ρ) = Hρ − ρH∗ + i

j∈N

WjρW ∗

j ,

H = HV − i 2

j∈N

W ∗

j Wj,

where, for all j ∈ N, Wj ∈ L(H), and

j∈N W ∗ j Wj ∈ L(H)

H is a dissipative operator on H
On a suitable domain, L is the generator of a quantum dynamical semigroup

{e−itL}t≥0 (strongly continuous semigroup on J1(H) such that, for all t ≥ 0, e−itL preserves the trace and is a completely positive operator)

Free dynamics: group of isometries {e−itL0}t∈R,

L0(ρ) = H0ρ − ρH0

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Application: scattering for Lindblad master equations (III)

Modified wave operator ([Davies ’80], [Alicki ’81])

Π⊥

pp : H → H : orthogonal projection onto (Hb(H) ⊕ Hp(H))⊥

Modified wave operator:

˜ Ω+(L0, L) := s-lim

t→+∞ eitL0

Π⊥

ppe−itL(·)Π⊥ pp

Theorem

Suppose that Hypotheses (H1)–(H6) hold and that H has no spectral singularities in Λ. Then ˜ Ω+(L0, L) exists on J1(H)

Interpretation

For all ρ ∈ J1(H) with ρ ≥ 0 and tr(ρ) = 1, the number tr(˜ Ω+(L0, L)ρ) ∈ [0, 1] is interpreted as the probability that the particle, initially in the state ρ, eventually escapes from the target

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Thank you !

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Essential spectrum and absolutely continuous spectrum

The essential spectrum of H

The essential spectrum of the closed operator H is defined by σess(H) := C \ ρess(H), where ρess(H) :=

z ∈ C

, Ran(H − z) is closed, dim Ker(H − z) < ∞ or codim Ran(H − z) < ∞

.

The absolutely continuous spectral subspace for H

The absolutely continuous subspace for the dissipative operator H is defined by Hac(H) := M(H), where M(H) :=

u ∈ H, ∃cu > 0, ∀v ∈ H,

∞

e−itHu, v
2dt ≤ cuv2

.

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Dissipative quantum systems: scattering theory and spectral singularities J´ er´ emy Faupin Introduction: The nuclear

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model The abstract framework

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dissipative scattering theory Spectral singularities Asymptotic completeness Applications

Hypothesis (H1)

(H1) Spectrum of H0

The spectrum of H0 is purely absolutely continuous and has a constant multiplicity (which may be infinite) By the spectral theorem, there exists a unitary mapping F0 : H → ⊕

Λ

H(λ) dλ. such that and F0H0F∗

0 acts as multiplication by λ on H(λ). Since the spectrum of

H0 has a constant multiplicity, H(λ) ≃ M , dim M = k ⇒ ⊕

Λ

H(λ) dλ ≃ L2(Λ; M).

(H1) for the nuclear optical model

H0 = −∆ on L2(R3), then σ(H0) = σac(H0) = [0, ∞). In this case, M = L2(S2) and F0 is related to the usual Fourier transform F: F0f (λ)(ω) = λ

1 4 Ff (

√ λω), λ > 0, ω ∈ S2

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Hypothesis (H2)

(H2) Spectrum of HV

HV has finitely many eigenvalues counting multiplicity, no embedded eigenvalues, and σ(HV ) = ∅

(H2) for the nuclear optical model

HV = −∆ + V (x) acting on L2(R3). If V (x) = O(x−ρ)) with ρ > 1,

1 HV has no singular spectrum and no embedded eigenvalues except, perhaps, at 2 V− := sup(−V , 0) ∈ L3/2(R3) ⇒ HV has at most finitely many eigenvalues

counting multiplicities (Cwikel-Lieb-Rosembljum bound)

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Strong smoothness of V w.r.t. H0

Factorization of V

There exist an auxiliary Hilbert space G and operators G : H → G and K : G → G such that V = G ∗KG, where G(H

1 2

0 + 1)−1 ∈ L(H; G) and K ∈ L(G). Moreover, for all z ∈ C, Im(z) = 0,

GR0(z)G ∗ is compact

Regularity of G with respect to H0

The operator G is locally strongly H0-smooth with exponent s0 ∈ ( 1

2 , 1), i.e.

F0G ∗

X : G → Cs0(X; M) is continuous,

where GX := GE0(X), E0(X) is the spectral projection associated to H0 on X ⋐ Λ, and Cs0(X; M) denotes the set of H¨

lder continuous M-valued functions of order s0

Remarks

1 We suppose that s0 > 1/2 in order to insure that the only possible “embedded

singularities” of HV are eigenvalues, (which are absent by (H2))

2 Under this assumption, the wave operators W±(HV , H0) exist and are complete

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Strong smoothness of V w.r.t. H0 in the nuclear optical model

Factorization of V

Here H = G = L2(R3). If V (x) = O(x−2ρ) with ρ > 0, we have many possible choices for G and K:

1 G =

|V | and K = sgn(V ).

2 G = x−α, with 0 < α ≤ ρ, and K = x2αV .

Regularity of G with respect to H0

If V (x) = O(x−2α), with α > 1

2 , taking G = x−α, and K = x2αV , then G is

strongly H0-smooth with exponent belonging to ( 1

2 , 1).

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Strong smoothness of C w.r.t. HV

Regularity of C with respect to HV

1 The operator C is locally strongly HV -smooth with exponent s ∈ (0, 1) on any

compact set X ⋐ Λ, i.e. F±C ∗

X : H → Cs(Λ; H) is continuous,

where F± := F0W ∗

±(HV , H0), CX := CEV (X) and EV (X) is the spectral

projection for HV on X ⋐ Λ.

2 CRV (z)C ∗ is compact for all z ∈ C, Im(z) = 0. 3 The map

˚ Λ ∈ λ → C

RV (λ + i0) − RV (λ − i0)
C ∗ ∈ L(H)

is bounded

Strong smoothness implies smoothness

Under the previous assumption, C is HV -smooth, i.e., there exists a constant cV > 0, such that

R
Ce−itHV u
2

Hdt ≤ c2 V u2 H,

for all u ∈ Hac(HV ).

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Strong smoothness of C w.r.t. HV in the nuclear optical model

Regularity of C with respect to HV in the nuclear optical model

For V ∈ C2(R3) with ∂α

x V (x) = O(x−ρ−|α|), |α| ≤ 2, for some ρ > 0, one has for

s > 1

2 and z, z′ ∈ {z ∈ C, ±Im(z) ≥ 0} (limiting absorption principle):

x−s(RV (z) − RV (z′))x−s ≤ C |z − z′|s−1/2. Previous hypothesis is satisfied if W (x) = O(x−δ) with δ > 1, taking C = √ W . Moreover, √ W RV (z) √ W is clearly compact for all z ∈ C, Im(z) = 0

Condition 3

We have to verify that CRV (λ ± i0)C ∗ is bounded for λ near 0 and ∞.

1 Near λ = ∞: if V (x) and W (x) are equal to O(x−ρ) with ρ > 1, then

√

W RV (λ ± i0) √ W = O(λ− 1

2 ),

λ → ∞.

2 Near λ = 0: if V (x) = O(x−ρ) with ρ > 3, and that 0 is neither an eigenvalue

nor a resonance of HV , then

√

W RV (λ ± i0) √ W = O(1), λ → 0, provided that W (x) = O(x−δ) with δ > 1

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Γ0(λ) and Γ±(λ)

Γ0(λ)

Recall F0 : H → ⊕

Λ

M dλ = L2(Λ; M), dim M = k. For f , g ∈ F∗

0 (C∞ 0 (Λ, M)),

f , gH =

Λ
Γ0(λ)f , Γ0(λ)g
M dλ,

Γ0(λ)f := F0f (λ), λ ∈ Λ Formally, Γ0(λ)∗Γ0(λ) = 1 2iπ (R0(λ + i0) − R0(λ − i0)),

Γ±(λ)

For f ∈ F∗

0 (C∞ 0 (˚

Λ, M)) and a.e. λ ∈ Λ, let Γ±(λ)f := Γ0(λ)(I − VRV (λ ± i0))f . Then the operator F± : F∗

0 (C∞ 0 (˚

Λ, M)) → L2(Λ, M) defined by F±f (λ) = Γ±(λ)f extends to a unitary map from H to L2(Λ, M) satisfying W±(HV , H0) = F∗

±F0.

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Regularity at the boundary Λ \ ˚ Λ

Definition

Let λ ∈ Λ \ ˚ Λ. We say that λ is a regular spectral point of H if there exists a compact interval Kλ ⊂ R whose interior contains λ, such that all µ ∈ Kλ ∩ ˚ Λ are regular, and such that the map Kλ ∩ ˚ Λ ∋ µ → CR(µ − i0)C ∗ ∈ L(H) is bounded

Definition

If Λ is right-unbounded, we say that +∞ is regular if there exists m > 0 such that all µ ∈ [m, ∞) ∩ ˚ Λ are regular, and such that the map [m, ∞) ∩ ˚ Λ ∋ µ → CR(µ − i0)C ∗ ∈ L(H) is bounded

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Regularity at the boundary Λ \ ˚ Λ in the nuclear optical model

Regularity at the endpoints

We have to verify that CR(λ ± i0)C ∗ is bounded for λ near 0 and ∞.

1 Near λ = ∞: if V (x) = O(x−ρ) and W (x) = O(x−δ) near ∞, with

ρ, δ > 1, one can use a Neumann series argument together with the fact that x−γ(−∆ − (λ − i0))−1x−γ = O(λ− 1

2 ) , γ > 1/2,

2 Near λ = 0: Assuming that V (x) = O(x−ρ) and W (x) = O(x−δ) near ∞,

with ρ, δ > 2, W > 0 on a nontrivial set, the result was proven in [Wang ’12]

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Resonances

Compact perturbations of −∆

If V and W are supposed to be compactly supported, a resonance may be defined as a pole of the map C ∋ z → (H − z2)−1 : L2

c(R3) → L2 loc(R3),

(given as the meromorphic extension of the meromorphic map originally defined on {z ∈ C, Im(z) > 0}), where L2

c (R3) = {u ∈ L2(R3), u is compactly supported} and

L2

loc(R3) = {u : R3 → C, u ∈ L2(K) for all compact set K ⊂ R3}

Short range potentials

If V and W are supposed to satisfy V (x) = O(x−ρ) and W (x) = O(x−δ) near ∞, with ρ, δ > 1, then ±λ1/2 (with λ > 0) may be called a resonance of H if the equation (H − λ)u = 0 admits a solution u ∈ H2

loc(R3) \ L2(R3) satisfying the

Sommerfeld radiation condition u(x) = |x|−1e±iλ

1 2 |x|

a( x |x| ) + o(1)

,

|x| → ∞, with a ∈ L2(S2), a = 0.

Remark

Spectral singularity at λ > 0 = resonance at −λ1/2

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Ideas of the proofs

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Step 1 : the range of the wave operator and the inverse semigroup

Lemma

Suppose that Hypotheses (H1), (H3), (H4) hold. Then Ran(W−(H, H0)) = S(H) ∩ Hb(H)⊥ where S(H) =

u ∈ H, sup

t≥0

eitHu
< ∞
Elements of the proof
[Davies ’80] : Hb(H)⊥ = Hac(H), where Hac(H) is the closure of

M(H) =

u ∈ H, ∃cu > 0, ∀v ∈ H,

∞

e−itHu, v
2dt ≤ cuv2
Easy to verify that Ran(W−(H, H0)) ⊂ S(H) ∩ Hb(H)⊥
Converse inclusion : uses that for all u ∈ S(H) and v ∈ Hac(H),
v, eitHu
→ 0,

t → ∞

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Step 2 : spectral projections

Spectral projections for non-self-adjoint operators

Let I ⊂ [0, ∞) be a closed interval and

EH(I) = w-lim

ε↓0

1 2iπ

I
(H − (λ + iε))−1 − (H − (λ − iε))−1

dλ Then EH(I) is a well-defined projection if H does not have spectral singularities in I, EH(I)EH(J) = EH(I ∩ J)

Considered in [Dunford ’52, ’58], [J. Schwartz ’60]
Studied in relation with stationary scattering theory : [Mochizuki ’67,’68],

[Goldstein ’70,’71], [Huige ’71]

Lemma

Suppose that Hypotheses (H1), (H3) and (H4) hold. Let I ⊂ [0, ∞) be a closed interval containing no spectral singularities of H. Then Ran(EH(I)) ⊂ Ran(W−(H, H0))

Element of the proof

Use that Ran(W−(H, H0)) = S(H) ∩ Hb(H)⊥

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Step 3 : proof that Hd(H) = Hp(H) (I)

Use of spectral projections

Ran(EH∗(I)) ⊂ Ran(W+(H∗, H0))
Taking orthogonal complements

Ran(W+(H∗, H0))⊥ ⊂

I⊂[0,∞)

Ran(EH∗(I))⊥, i.e. Ker(W+(H0, H)) ⊂

I⊂[0,∞)

Ker(EH(I)), where the intersection runs over all closed intervals I ⊂ [0, ∞) with the property that I does not contain any spectral singularities of H

Since

Hb(H) ⊕ Hd(H) = Ker(W+(H0, H)), it suffices to prove that K :=

I⊂[0,∞)

Ker(EH(I)) ⊂ Hb(H) ⊕ Hp(H)

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Step 3 : proof that Hd(H) = Hp(H) (II)

Spectral mapping theorem and Riesz projections

Let R = (H − i)−1. Then

Id = Πpp(R) + 1 2iπ

Γε

(µ − R)−1dµ, (∗∗) where Πpp(R) = sum of Riesz projections corresponding to isolated eigenvalues

f R, and Γε = Γ1,ε ∪ Γ2,ε ∪ Γ3,ε ∪ Γ4,ε
Ran(Πpp(R)) = Hb(H) ⊕ Hp(H)
If H does not have spectral singularities, taking the weak limit ε ↓ 0 gives

Id = Πpp(R) + w-lim

ε↓0

1 2iπ ∞

H − (λ + iε)

−1 −

H − (λ − iε)

−1 dλ and therefore K =

I⊂[0,∞)

Ker(EH(I)) = Ker(EH([0, ∞))) ⊂ Hb(H) ⊕ Hp(H)

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Γ3,ε Γ1,ε Γ2,ε Γ4,ε

Figure: The spectrum of R = (H − i)−1 and the contour Γε.

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Step 3 : proof that Hd(H) = Hp(H) (III)

Regularizing the spectral singularities

If {λ1, . . . λn} ⊂ [0, ∞) are the spectral singularities of H, let µj = (λj − i)−1,

j = 1, . . . , n, be the corresponding “spectral singularities” of R = (H − i)−1

Composing (∗∗) by R4 n

j=1(R − µj)νj gives

R4

n

j=1

(R − µj)νj =R4

n

j=1

(R − µj)νj Πpp(R) − 1 2iπ

Γε

µ4

n

j=1

(µ − µj)νj (R − µ)−1dµ

Taking the weak limit ε ↓ 0 gives the modified spectral decomposition formula

n

j=1

(R − µj)νj =

n

j=1

(R − µj)νj Πpp(R) + w-lim

ε↓0

1 2iπ ∞

n

j=1
(λ − i)−1 − µj

νj

H − (λ + iε)

−1 −

H − (λ − iε)

−1 dλ

Using in particular Lebesgue’s dominated convergence theorem

K =

I⊂[0,∞)

Ker(EH(I)) ⊂ Hb(H) ⊕ Hp(H)

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Step 4 : asymptotic completeness

Parseval’s theorem

For all ε > 0 and u ∈ (Hb(H∗) ⊕ Hp(H∗))⊥, ∞ e−sε CeisHu

2ds = 1

2π

R
C(H − (λ − iε))−1u
2dλ

(∗ ∗ ∗)

Asymptotic completeness

Recall Ran(W−(H, H0)) = S(H) ∩ Hb(H)⊥
Observe that

S(H) =

u ∈ H,

∞

CeisHu
2ds < ∞
If H does not have spectral singularities, uniform bound in ε > 0 in the right

side of (∗ ∗ ∗) implies that u ∈ S(H) and hence to Ran(W−(H, H0))

If H does have a spectral singularity, one constructs a vector