Time-decay of solutions to dissipative Schrdinger equations Xue - - PowerPoint PPT Presentation

Introduction Real resonances Time-decay of solutions

Time-decay of solutions to dissipative Schrödinger equations

Xue Ping WANG

Laboratoire de Mathématiques Jean Leray Université de Nantes, France

Introduction Real resonances Time-decay of solutions

Outline

1

Introduction Dissipative Schrödinger operators An overview

Introduction Real resonances Time-decay of solutions

Outline

1

Introduction Dissipative Schrödinger operators An overview

2

Real resonances Absence of low energy resonances Eigenvalues near zero

Introduction Real resonances Time-decay of solutions

Outline

1

Introduction Dissipative Schrödinger operators An overview

2

Real resonances Absence of low energy resonances Eigenvalues near zero

3

Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative Schrödinger operators

This talk is concerned with the Schrödinger operator with a complex-valued potential P = −∆ + V(x) V(x) = V1(x) − iV2(x), x ∈ Rn (1) with Vj real, V2(x) ≥ 0, V2 = 0 (i.e., dissipative) and |V(x)| ≤ Cx−ρ, x ∈ Rn, ρ > 1. (2)

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative Schrödinger operators

This talk is concerned with the Schrödinger operator with a complex-valued potential P = −∆ + V(x) V(x) = V1(x) − iV2(x), x ∈ Rn (1) with Vj real, V2(x) ≥ 0, V2 = 0 (i.e., dissipative) and |V(x)| ≤ Cx−ρ, x ∈ Rn, ρ > 1. (2) Physical backgrounds: Optical model of nuclear scattering, propagation of waves in media with variable absorption index, ...

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative Schrödinger operators

We are interested in time-decay of solutions to the time-dependent Schrödinger equation

• i ∂

∂t u(t)

= Pu(t), on Rn, u(0) = u0, (3) Dissipation: u(t)L2 is decreasing.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Selfadjoint case

When V(x) is real, P is selfadjoint and there are many results on time-decay estimates (local energy, Strichartz, dispersive). For example, when n = 3, ρ > 3 and s > 5/2 appropriately large, one has e−itP =

• λ∈σp(P)

e−iλtπλ + C tµ + O( 1 tµ+ǫ ), as operators L2,s → L2,−s. Here πλ is the spectral projection associated with the eigenvalue λ, L2,s = L2(x2sdx).

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Selfadjoint case

µ > 0 depends on threshold spectral properties of P: µ =

• 3

2,

if 0 is neither eigenvalue nor resonance of P,

1 2,

if 0 is eigenvalue or resonance of P. 0 is called a resonance if Pu = 0 admits a solution u ∈ L2,−s \ L2 for any s > 1

2.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Non-selfadjoint case

There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we

• nly mention:

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Non-selfadjoint case

There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we

• nly mention:
• G. Lebau (1994), damped waves in compact manifolds.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Non-selfadjoint case

There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we

• nly mention:
• G. Lebau (1994), damped waves in compact manifolds.
• L. Aloui, M. Khenissi (2002-2010), wave and Schrödinger

equations in exterior domains.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Non-selfadjoint case

There are several similarities between the dissipative Schrödinger equation and the damped wave equation. Time-decay for non-selfadjoint problems are studied by several authors. Here we

• nly mention:
• G. Lebau (1994), damped waves in compact manifolds.
• L. Aloui, M. Khenissi (2002-2010), wave and Schrödinger

equations in exterior domains.

• M. Goldberg(2010), M. Beceanu and M. Goldberg(2012),

Schrödinger equation with complex-valued potentials, under the assumption of the absence of real resonances in R+.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

• Definition. Let L2,s = L2(Rn; x2sdx). We call λ ∈ R+ a (real)

resonance of P = −∆ + V(x) (V(x) complex-valued) if (P − λ)u = 0 admits a solution u ∈ L2,−s \ L2,

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

• Definition. Let L2,s = L2(Rn; x2sdx). We call λ ∈ R+ a (real)

resonance of P = −∆ + V(x) (V(x) complex-valued) if (P − λ)u = 0 admits a solution u ∈ L2,−s \ L2, provided that ρ > 2 and s > 1 if λ = 0; ρ > 1 and s > 1/2 if λ > 0 and u satisfies one of the Sommerfeld’s radiation conditions: u(x) = e±i

√ λ|x|

|x|n/2−1 (a(θ) + o(1)), θ = x/|x| for some a ∈ L2(Sn−1) and a = 0.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

One knows very few about real resonances. Real resonances form a compact set of measure zero.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic".

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic".

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Real resonances

One knows very few about real resonances. Real resonances form a compact set of measure zero. It is unknown if the complex potentials without any real resonances are “generic". The goal of this talk is to show some time-decay estimates for the dissipative Schrödinger equation without assumptions on real resonances.

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative quantum scattering

Since this is a Scattering Session, we end this Introduction by a word

• n dissipative quantum scattering.

Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958).

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative quantum scattering

Since this is a Scattering Session, we end this Introduction by a word

• n dissipative quantum scattering.

Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980).

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative quantum scattering

Since this is a Scattering Session, we end this Introduction by a word

• n dissipative quantum scattering.

Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980).

Introduction Real resonances Time-decay of solutions Dissipative Schrödinger operators An overview

Dissipative quantum scattering

Since this is a Scattering Session, we end this Introduction by a word

• n dissipative quantum scattering.

Physical models: H. Feshbach, Unified theory of nuclear reactions, Ann. Phys., (1958). Mathematical analysis: Ph. Martin (1976), E.B. Davies (1979-1982), B. Simon (1980). Open question. Asymptotic completeness: Is the dissipative scattering operator bijective?

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

An example of positive resonance

For any λ > 0, one can show that there exits C∞

0 potentials V(x)

such that λ is a resonance of the dissipative Schrödinger operator P = −∆ + V(x).

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

An example of positive resonance

For any λ > 0, one can show that there exits C∞

0 potentials V(x)

such that λ is a resonance of the dissipative Schrödinger operator P = −∆ + V(x). In fact, let f ∈ C∞

0 (R3) with f ≥ 0, and f = 0 and

supp f ⊂ {x; |x| ≤

π 4 √ λ}. Put

u = R0(λ − i0)f = 1 4π

• R3 e−i

√ λ|x−y| f(y)

|x − y| dy. One has ℜu(x) > 0, ℑu(x) < 0 for x ∈ supp f. Take V(x) = − f(x) u(x). Then V ∈ C∞

0 (R3), ℜV ≤ 0 and ℑV ≤ 0 and

Pu = λu.

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

Let P = −∆ + V1(x) − iV2(x), V2 ≥ and V2 = 0, and P1 = −∆ + V1(x). Lemma 1 Let s ∈ [0, 1[, n ≥ 1. Suppose that the condition (2) is satisfied with ρ ≥ 2s. Then u ∈ L2,−s and Pu = 0 imply u = 0.

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

Let P = −∆ + V1(x) − iV2(x), V2 ≥ and V2 = 0, and P1 = −∆ + V1(x). Lemma 1 Let s ∈ [0, 1[, n ≥ 1. Suppose that the condition (2) is satisfied with ρ ≥ 2s. Then u ∈ L2,−s and Pu = 0 imply u = 0.

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

Let P = −∆ + V1(x) − iV2(x), V2 ≥ and V2 = 0, and P1 = −∆ + V1(x). Lemma 1 Let s ∈ [0, 1[, n ≥ 1. Suppose that the condition (2) is satisfied with ρ ≥ 2s. Then u ∈ L2,−s and Pu = 0 imply u = 0.

• Proof. The equation Pu = 0 gives

u, P1u = iu, V2u. (4) P1u, u is a real number, although u is not in the domain of P1. (This intuition is wrong if Pu = Eu with E > 0!) Since V2 ≥ 0 and V2 = 0, one has V2u = 0 and u(x) = 0 for x in a non trivial open set Ω. The equation P1u = 0 implies u = 0 on Rn.

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

For n ≥ 3 and ρ > 2, Pu = 0 and u ∈ L2,−s for any s > 1 imply that u ∈ L2,−s for any s > 1/2. It follows from Lemma 1 that Corollary 1 Under the assumption on V with ρ > 2 and n ≥ 3, there is some c0 > 0 such that there are no low-energy real resonances lying in [0, c0].

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

For n ≥ 3 and ρ > 2, Pu = 0 and u ∈ L2,−s for any s > 1 imply that u ∈ L2,−s for any s > 1/2. It follows from Lemma 1 that Corollary 1 Under the assumption on V with ρ > 2 and n ≥ 3, there is some c0 > 0 such that there are no low-energy real resonances lying in [0, c0].

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Absence of low-energy resonances

For n ≥ 3 and ρ > 2, Pu = 0 and u ∈ L2,−s for any s > 1 imply that u ∈ L2,−s for any s > 1/2. It follows from Lemma 1 that Corollary 1 Under the assumption on V with ρ > 2 and n ≥ 3, there is some c0 > 0 such that there are no low-energy real resonances lying in [0, c0].

• Remark. In the case n = 2 and ρ > 2, Lemma 1 only implies that if 0

is a resonance, then it is geometrically simple. Recall that in the selfadjoint case, zero resonance in dimension two can be of multiplicity at most three.

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Non-accumulation of eigenvalues towards zero

Let R(z) = (P − z)−1, R0(z) = (−∆ − z)−1. Theorem 1 Assume that n ≥ 3 and ρ > 2 so that Corollary 1 holds. Then, zero is not an accumulation point of the complex eigenvalues of P and there exists c0 > 0 such that the limits R(λ ± i0) = lim

ǫ→0+ R(λ ± iǫ)

(5) exist as bounded operators from L2,s to L2,−s for any s > 1, uniformly in λ ∈ [−c0, c0].

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Non-accumulation of eigenvalues towards zero

Let R(z) = (P − z)−1, R0(z) = (−∆ − z)−1. Theorem 1 Assume that n ≥ 3 and ρ > 2 so that Corollary 1 holds. Then, zero is not an accumulation point of the complex eigenvalues of P and there exists c0 > 0 such that the limits R(λ ± i0) = lim

ǫ→0+ R(λ ± iǫ)

(5) exist as bounded operators from L2,s to L2,−s for any s > 1, uniformly in λ ∈ [−c0, c0].

Introduction Real resonances Time-decay of solutions Absence of low energy resonances Eigenvalues near zero

Non-accumulation of eigenvalues towards zero

Let R(z) = (P − z)−1, R0(z) = (−∆ − z)−1. Theorem 1 Assume that n ≥ 3 and ρ > 2 so that Corollary 1 holds. Then, zero is not an accumulation point of the complex eigenvalues of P and there exists c0 > 0 such that the limits R(λ ± i0) = lim

ǫ→0+ R(λ ± iǫ)

(5) exist as bounded operators from L2,s to L2,−s for any s > 1, uniformly in λ ∈ [−c0, c0]. Consequence. If in addition V is dilation-analytic, then P has only a finite number of complex eigenvalues.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

A global limiting absorption principle

Combining Theorem 1 with a result of J. Royer (C.P .D.E.,2010) for positive energies, we obtain a global limiting absorption principle: Assume ρ > 2, n ≥ 3 and s > 1. One has x−sR(λ + i0)x−s ≤ Csλ−1/2, λ ∈ R. (6) Under some additional assumptions on V, one can show that R(λ + i0) is differentiable for λ away from zero.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

A representation formula of the semigroup

A consequence of this global limiting absorption principle is a representation formula for the semigroup e−itP. Theorem 2 Assume that n ≥ 3 and |V(x)| + |x · ∇V(x)| ≤ Cx−ρ, ρ > 2. Then e−itPf, g = 1 2πi

• R

e−itλR(λ + i0)f, gdλ, t > 0, (7) for f, g ∈ L2,s = L2(Rn; x2sdx), s > 5/2.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Dispersive estimate in dimension three

Time-decay estimates can be deduced from Theorem 2 as in selfadjoint case. Theorem 3 Assume n = 3 and |V(x)| ≤ Cx−ρ, x ∈ R3, ρ > 2. (8) Then one has

e−itPu0L∞ ≤ Ct−3/2u0L1, t > 0. (9)

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Dispersive estimate in dimension three

Time-decay estimates can be deduced from Theorem 2 as in selfadjoint case. Theorem 3 Assume n = 3 and |V(x)| ≤ Cx−ρ, x ∈ R3, ρ > 2. (8) Then one has

e−itPu0L∞ ≤ Ct−3/2u0L1, t > 0. (9)

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Dispersive estimate in dimension three

Time-decay estimates can be deduced from Theorem 2 as in selfadjoint case. Theorem 3 Assume n = 3 and |V(x)| ≤ Cx−ρ, x ∈ R3, ρ > 2. (8) Then one has

e−itPu0L∞ ≤ Ct−3/2u0L1, t > 0. (9)

• Remark. One does not see in (9) contributions from complex

eigenvalues and real resonances of P as t → ∞.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Consequences

Optimal time-decay in weighted spaces Corollary 2 Let n = 3 and ρ > 2. One has for any s > 3/2, x−se−itPx−sL(L2) ≤ Cst−3/2, t > 0. (10)

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Consequences

Optimal time-decay in weighted spaces Corollary 2 Let n = 3 and ρ > 2. One has for any s > 3/2, x−se−itPx−sL(L2) ≤ Cst−3/2, t > 0. (10)

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Consequences

Optimal time-decay in weighted spaces Corollary 2 Let n = 3 and ρ > 2. One has for any s > 3/2, x−se−itPx−sL(L2) ≤ Cst−3/2, t > 0. (10) Rate of energy dispersion of eigenfunctions Corollary 3 There exists a universal constant C > such that for any z ∈ C− and u with Pu = zu, uL1 = 1, one has u∞ ≤ C|ℑz|3/2.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Long-time asymptotics of solutions

Theorem 4 Let n ≥ 3. Assume that there exists k ≥ n

2 and ρ > n such that

|(x · ∇)jV(x)| ≤ Cx−ρ, j = 0, 1, · · · , k. (11) Then one has S(t) = C0 tn/2 + O( 1 tn/2+ǫ ) : L2,s → L2,−s (12) for s > n+1

2 , t → +∞. Here C0 is an operator of rank one.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Remark on positive resonances

Let E > 0 be a resonance of P. Then under some assumptions, one can show that there exists some operator T of finite rank such that R(z) − T E − z = O(1) : L2,s → L2,−s, s > 5/2, for any z ∈ C− with |z − E| < ǫ0. This is to compare with the fact that λ → R(λ + i0) is continuous at λ = E.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Remark on positive resonances

Let E > 0 be a resonance of P. Then under some assumptions, one can show that there exists some operator T of finite rank such that R(z) − T E − z = O(1) : L2,s → L2,−s, s > 5/2, for any z ∈ C− with |z − E| < ǫ0. This is to compare with the fact that λ → R(λ + i0) is continuous at λ = E. The complex eigenvalues and real resonances of P will affect the long-time behavior of solutions as t → −∞, but not as t → +∞.

Introduction Real resonances Time-decay of solutions A representation formula Time-decay estimates Remark on positive resonances

Thanks!