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Time-decay of semigroups generated by dissipative
Schrödinger operators
Xue Ping Wang
To cite this version:
Xue Ping Wang. Time-decay of semigroups generated by dissipative Schrödinger operators. Journal of Differential Equations, Elsevier, 2012, 253 (12), pp.3523-3542. �hal-01005809�
SCHR ¨ODINGER OPERATORS
XUE PING WANG
Abstract. We establish a representation formula for semigroup of contraction in terms of global limiting absorption principle from the upper-half complex plane. As applications, we prove time-decay estimates of the semigroup of contractions generated by −iH where H is a dissipative Schr¨odinger operator.
1. Introduction
Let H = −∆ + V (x) be the Schr¨odinger operator with a complex-valued potential V satisfying V = V1 − iV2, where V1 and V2 are real functions satisfying V2(x) ≥ 0 and
V2(x) > 0 on some non-trivial open set. Suppose that
|Vj(x)| ≤ Chxi−ρ0, x ∈ Rn, (1.1) ass1
for some ρ0 > 1. Here hxi = (1 + |x|2)1/2. Mild local singularities can be included
with little additional effort. Denote H0 = −∆ and H1 = −∆ + V1. H defined
on D(−∆) is maximally dissipative and the numerical range of H is contained in {z; ℜz ≥ −R, −R ≤ ℑz ≤ 0} for some R > 0.
2. Some abstract results
Let H1 and V2 be selfadjoint operators on some Hilbert space H, with H1
semi-bounded from below, V2 ≥ 0 and relatively compact with respect to H1. H = H1− iV2
is maximally dissipative on H. Let S(t) := e−itH, t ≥ 0, be the strongly continuous
semigroup generated by −iH. In this Section, we give two results on S(t) based on the existence of a limiting absorption principle for H on the whole real axis. There results are to be applied in the next Section to a class of dissipative Schr¨odinger operators on Rn, n ≥ 2.
Since V2 is H1 compact, one sees that ∀δ0 > 0, ∃R1 > 0 such that the numerical range
of H is contained in the sector
{z ∈ C; ℜz > −R1, −δ0 ≤ arg(z + R1) ≤ 0}.
2000 Mathematics Subject Classification. 35J10, 35P15, 47A55.
Key words and phrases. Time-decay, semigroup of contractions, non-selfadjoint Schr¨odinger operators.
Research supported in part by the French National Research Project NONAa, No. ANR-08-BLAN-0228-01, on Spectral and microlocal analysis of non-selfadjoint operators.
2 XUE PING WANG
Let δ0 < π be fixed. For ǫ0 > 0 small enough and for all ǫ ∈]0, ǫ0], the set {λeiǫ; λ ∈
R, |λ| > R1} is contained in the resolvent set of H. Assume that there exists a dense subset D ⊂ H such that {f ∈ D ∩ D(H); Hf ∈ D} is dense in H and
• For any λ ∈ R, the limit
hR(λ + i0)f, gi = lim
ǫ→0+
hR(λ + iǫ)f, gi (2.1) ass1
exists for any f, g ∈ D and is continuous in λ ∈ R.
• There exist R > 1, k ∈ N∗, ρ, σ > 0 with ρ + kσ > 1 such that for any f, g in D,
λ → hR(λ + i0)f, gi is Ck for |λ| > R and
| d
j
dλjhR(λ + i0)f, gi| ≤ Cf,ghλi
−ρ−jσ, (2.2)
ass2
for j = 0, 1, . . . , k and |λ| > R.
Theorem 2.1. Under the conditions (2.1) and (ass1 2.2), one hasass2 he−itHf, gi = 1
2πi Z
R
e−itλhR(λ + i0)f, gidλ, t > 0, (2.3)
for f, g ∈ D.
Proof. Denote Hǫ = e−iǫH, ǫ > 0. Then the numerical range of Hǫ is contained in
Nǫ := {e−iǫz; ℜz > −R1, −δ0 ≤ arg(z + R1) ≤ 0}
for some R1 > 1. Therefore for each ǫ > 0 small enough, Hǫ − iR1ǫ is maximally
dissipative and strictly m-sectorial and −iHǫ generates a semigroup e−itHǫ, t ≥ 0, which
can be represented in a usual way (cf. [12], pp 489-491). For Rk 0 > ǫR1, let Γǫ,R be a
contour in ρ(Hǫ) composed of the segment {ℑz = R0, ℜz ∈ [−R1− 1, R1+ 1]} and the
two rays (−R1− 1 + iR0) + ei(π+ǫ/2)R+ and (R1+ 1 + iR0) + e−iǫ/2R+ (running from the
infinity with arg z = −π + ǫ2 to the infinity with arg z = −2ǫ). Then one has e−itHǫ = 1 2πi Z Γǫ,R0 e−itz(H ǫ− z)−1dz := Fǫ(t), (2.4) Uepsilon
for t > 0. In addition, one has the estimate
ke−itHǫk ≤ etR1ǫ, t ≥ 0. (2.5) norm1
Under the condition (2.2), by an argument of perturbation, one can deduce that forass2 j = 0, 1, · · · , k, | d j dλjhR(λe iη)f, gi| ≤ C f,ghλi−ρ−jσ, (2.6)
uniformly in λ ≥ 1 and η > 0 small enough. Making use of techniques of oscillatory integrals, we deduce from (ass12.1) and (ass22.2) that the integral
hF (t)f, gi := 1 2πi
Z
R
e−itλhR(λ + i0)f, gidλ, (2.7)
converges for f, g ∈ D. By the same method, one can show that the integral defining Kǫ(t) converges uniformly in ǫ > 0 and one has
hF (t)f, gi = lim
R,ǫ→0+
for any f, g ∈ D. The bound on kFǫ(t)k shows that |hF (t)f, gi| ≤ kf kkgk. Therefore
F (t) can be extended to a contraction on H, still denoted by F (t) and limǫ→0+Fǫ(t) =
F (t) weakly for t > 0.
For f ∈ D ∩ D(H) with Hf ∈ D and g ∈ D, one has Z
Γǫ,R0
d dthe
−itz(H
ǫ− z)−1f, gidz = −ie−iǫ
Z
Γǫ,R0
he−itz(H
ǫ− z)−1Hf, gidz
converges uniformly in ǫ, R0 > 0 small. It follows that hF (t)f, gi is differentiable in
t > 0 and
hdF (t)
dt f, gi = h−iF (t)Hf, gi = h−iHF (t)f, gi.
By an argument of density, we deduce that for any f ∈ D(H), t → F (t)f is weakly differentiable and
dF (t)
dt f = −iF (t)Hf = −iHF (t)f. (2.8)
To show that F (t) = e−itH, we prove that F (t) → 1 weakly as t → 0
+. For each
t ∈]0, 1], take R0 = t−1 ≥ 1. Making a change of variables, we obtain that
Fǫ(t) = 1 2πi Z Γǫ,1 e−iζζ−1(Hǫ− ζ t) −1dζ t , t ∈]0, 1]. Noticing that 1 2πi Z Γǫ,1 e−iζζ−1dζ = −1 for every ǫ > 0, one deduces
h(Fǫ(t) − 1)f, gi = 1 2πi Z Γǫ,1 e−iζζ−1h(H −ζe iǫ t ) −1Hf, gidζ (2.9)
for f ∈ D(H). For ζ ∈ Γǫ,1, one has |ζ| ≥ 12 and |e−iζ| ≤ C uniformly in ǫ > 0 small.
By the condition (2.2),ass2
|h(H −ζe iǫ t ) −1Hf, gi| ≤ C f,g( t |ζ|) ρ, t ∈]0, 1],
for any f, g ∈ vD with Hf ∈ D, uniformly in ǫ > 0. It follows that
|h(Fǫ(t) − 1)f, gi| ≤ C′tρ, t ∈]0, 1], (2.10)
uniformly in ǫ. Taking the limit ǫ → 0 in the above inequality, we obtain
|h(F (t) − 1)f, gi| ≤ C′tρ. (2.11)
This proves that limt→0+h(F (t) − 1)f, gi = 0 for any g ∈ D and f ∈ D ∩ D(H) with
Hf ∈ D. Since kF (t) − 1k ≤ 2, an argument of density shows that F (t)f → f weakly for any f ∈ H as t → 0+.
Now for any f ∈ D(H), let f (t) = F (t)f and u(t) = e−itHf . Then f (t) ∈ D(H) for
any t > 0 and one has d
dshF (s)e
4 XUE PING WANG
for any g ∈ H. Integrating the above equation gives that hf (t), gi − hu(t), gi = c for some constant c. Since both f (t) and u(t) converge weakly to f as t → 0+, one has
c = 0, hence hf (t), gi = hu(t), gi for any g ∈ H. This proves that F (t)f = e−itHf for
any f ∈ D(H). Since D(H) is dense in H, F (t) coincides with the semigroup generated
by −iH. ¤
Another consequence of global limiting absorption principle is the Kato’s smoothness estimate for semigroup of contractions which is useful for dissipative quantum scattering. We give below a simple proof, using the theory of selfadjoint dilation. See also roy[16] in some special case.
Let H be maximal dissipative on a Hilbert space H. −iH is generator of a semigroup of contractions T (s) = e−isH, t ≥ 0. According to the theory of Foia¸s-Sz. Nagy
(Masson,1967, Ch.III, § 9), ∃ a Hilbert space G ⊃ H and a unitary group U (t) = e−itG
on G such that
Π0U (s)|H = T (s), s ≥ 0, (2.12)
where Π0 : G → H is the projection. G is called a selfadjoint dilation of H.
th2.2 Theorem 2.2. Assume that there exits A : H → H continuous such that sup λ∈R,δ∈]0,1] kA(H − (λ + iδ))−1A∗k ≤ γ. (2.13) Then Z ∞ 0 (kAT (s)f k2+ kAT (s)∗f k2)ds ≤ 2γkf k2, f ∈ H. (2.14)
Proof. Let G be a selfadjoint dilation of H. Then
Π0(G − z)−1|H = (H − z)−1, Π0(G − z)−1|H = (H∗− z)−1,
for ℑz > 0. Therefore
k(AΠ0)(G − z)−1(AΠ0)∗k ≤ γ, 0 < |ℑz| ≤ 1.
By Kato’s smoothness estimate for selfadjoint operators,
Z ∞ −∞ kAΠ0U (s)gk2 ds ≤ Ckgk2, g ∈ G, with C = sup 0<ℑz≤1 k(AΠ0)[(G − z)−1− (G − z)−1)(AΠ0)∗k ≤ 2γ.
For g = f ∈ H, one has
Z ∞
0
(kAT (s)f k2+ kAT (s)∗f k2) ds ≤ 2γkf k2, f ∈ H.
3. Dissipative Schr¨odinger operators
Let R(z) = (H − z)−1, z 6∈ σ(H). Let R
j(z) = (Hj − z)−1. Denote L2,s =
L2(Rn; hxisdx) and kf k
s = kf kL2,s. It is well know that if n ≥ 3, the limit
F0 = lim
z→0,z6∈R+
R0(z) : L2,s→ L2,−s
exists if s > 1.
prop1 Proposition 3.1. Assume that n ≥ 3 and ρ0 > 2. Then one has
(a). 1 + F0V is invertible on L2,−s for any s ∈]1, ρ0/2[ and there exists c0 > 0 such
that the limit
R(λ + i0) = lim
ǫ→0+
R(λ + iǫ) : L2,s→ L2,−s exists for s > 1 and λ ∈ [−c0, c0].
(b). Zero is not an accumulating point of the eigenvalues of H and there exists δ0 > 0
such that
σ(H) ⊂ {z; −π + δ0 ≤ arg z ≤ 0}. (3.1)
This result is proved by X.P. Wang 2009.
th1 Theorem 3.2. Assume that n ≥ 3 and ρ0 > 2. Then
(a). For any s > 1
sup
ǫ∈]0,1],λ∈R
hλi12khxi−sR(λ + iǫ)hxi−sk < ∞ (3.2) global-resolv
The limit
R(λ + i0) = lim
ǫ→0+
R(λ + iǫ) : L2,s→ L2,−s
exists for s > 1 and is continuous on R.
(b) Let k ∈ N. Assume that |(x · ∇)jV (x)| ≤ Chxi−ρ0, j = 0, 1, · · · , k, ρ
0 > 2. Then for j = 0, 1, · · · , k + 1 and s > j + 1 2, one has khxi−s dj dλjR(λ + i0)hxi −sk ≤ C shλi− j+1 2 , (3.3) for λ > 1.
Ideas. For λ > 0, one uses the equation R(z) = (1 + R0(z)V )−1R0(z) by the argument
of S. Agmon (1975) (ρ0 > 1). For more general situations (an abstract Mourre’s theory
for dissipative operators), see J. Royer (Commun. in PDE, to appear). For λ near 0, we apply Proposition . n ≥ 3 and ρ0 > 2 are needed.
prop1
4. Optimal time-decay rate
As an application of the representation of the semigroup, we can show the following dispersive estimate for dissipative Schr¨odinger operators.
dispersive Theorem 4.1. Assume n = 3 and |V (x)| + |x · ∇V (x)| ≤ Chxi−ρ0, ρ
0 > 2. Then one has ke−itHf k L∞ ≤ Ct− 3 2kf kL1, ∀ f ∈ L1(R3), t > 0. (4.1) dispative-disp
6 XUE PING WANG
To prove the Theorem, it suffices to prove |hU (t)u, vi| ≤ C|t|−3/2kuk
L1kvkL1, u, v ∈ C0∞.
Distinguish two regimes: λ ∈ [−R, R] and |λ| > R. Consider only the case λ ≥ 0. By the change of variable λ → λ2 and an integration by parts, we are led to prove
| Z
e−itλ2ρ(λ)hG′(λ)u, vi dλ| ≤ C|t|−1/2kuk1kvk1. (4.2) est1
Here ρ is an appropriate cut-off and G(λ) = r(λ2) which can be written as G(λ) =
G0(λ)(1 + V G0(λ))−1.
One can calculate
G′(λ) = (1 − G0(λ)V )G′0(λ)(1 + V G0(λ))−1 (4.3) G
and G′
0(λ) is the operator with integral kernel: ie
iλ|x−y|
4π .
Take ρ with support in ]λ0− δ, λ0+ δ[, δ > 0 to be adjusted. If one replaces λ by λ0
in (1 − G0(λ)V ) and (1 + V G0(λ))−1 in (
G
4.3), one sees that
| Z R3 x Z R3 y Z R
e−itλ2+iλ|x−y|ρ(λ)˜u(x)˜v(y)dλdxdy| ≤ C|t|−1/2kuk1kvk1.
Note that V G0(λ) : L1 → L1 is H¨older-continuous in λ. For λ ∈ supp ρ, one expands
(1 + V G0(λ))−1= ∞
X
k=0
(−1)k(S0D(λ))kS0
with S0 = 1 + V G0(λ0), D(λ) = V (G0(λ) − G0(λ0)). The integral kernel of D(λ) is
V(x)e
iλ|x−y|− eiλ0|x−y| 4π|x − y| .
One can prove that Z R kFλ→τρ(λ)(S0D(λ))kS0uk1dτ ≤ (CVδǫ)kkuk1. It follows that | Z R
hG′0(λ)e−itλ2ρ(λ)(S0D(λ))kS0u, A(λ)vidλ| ≤ |t|−1/2(CVδǫ)kkuk1kvk1.
Here A(λ) = 1 − V G0(λ0) or D(λ).
Taking δ s.t. CVδǫ < 1 and summing up in k, we obtain the desired estimate (
est1
4.2) for each fixed energy.
Little modification is needed when λ0 = 0.
The high energy estimate in the limiting absorption principle implies k(V G0(λ))kuk1 ≤ Ckλ−(k−2)kuk1
Lemma 4.2. Let fk : R → R be a sequence of continuous functions such that |fk(λ)| ≤ Ckhλi−k+2, Z R hτ iǫ0| ˆf k(τ )|dτ ≤ Ck.
Then for χ a cut-off for the interval [R, ∞[, one has Z
|dχfk(τ )|dτ ≤ C1khRi
−(k−2)ǫ0
2(1+ǫ0).
One uses the series for λ large
(1 + V G0(λ))−1 = ∞
X
k=0
(−1)k(V G0(λ))k
By the decay condition on V , one has Z
hτ iǫ0kF
λ→τρ(λ)(V G0(λ))kuk1dτ ≤ Ckkuk1,
uniformly in R. Here ρ is a cut-off for [R, ∞[. The Lemma gives Z
R
kFλ→τρ(V G0(λ))kuk1dτ ≤ CkR−
(k−2)ǫ0 2(1+ǫ0)kuk1.
For R > 1 large enough,
CkR−(k−2)ǫ02(1+ǫ0) < ǫk, ǫ < 1.
One can deduce that
| Z
e−itλ2ρ(λ)hG′0(λ)(V G0(λ))ku, A(λ)vidλ| ≤ |t|−1/2Cǫkkuk1kvk1.
Taking the summation in k, we obtain the desired high energy estimate. ¤
This result is to compare with the dispersive estimate for selfadjoint operator H1
(V2 = 0). Assume that n = 3 and |V1(x)| ≤ Chxi−ρ0, ρ0 > 2 and that 0 is not an
eigenvalue nor a resonance of H1. Then one has
ke−itH1P
acf kL∞ ≤ Ct− 3
2kf kL1, ∀ f ∈ L1(R3), t 6= 0, (4.4)
where Pac is the projection onto the absolutely continuous spectral subspace of H1.
As a consequence of Theorem 4.1, one hasdispersive
Corollary 4.3. Under the condition of Theorem 4.1, one has for any s ∈ [0, 3/2] anddispersive s′ > s,
khxi−s′e−itHhxi−s′kL(L2) ≤ Chti−s, t > 0. (4.5)
Proof. For s = 3/2 and s′ > 3/2, it follows from Theorem dispersive4.1 that the above estimate
8 XUE PING WANG
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D´epartement de Math´ematiques, UMR 6629 CNRS, Universit´e de Nantes, 44322 Nantes Cedex 3 France, E-mail: xue-ping.wang@univ-nantes.fr
