International audienceWe discuss the uniform time-decay of semi-groups generated by dissipative Schrödinger operators in the semiclassical regime

Wang, Xue Ping

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Uniform time-decay of semigroups of contractionXue Ping WangTo cite this version:Xue Ping Wang. Uniform time-decay of semigroups of contraction. Integral Equations andOperator Theory, Springer Verlag, 2012, 73 (1), pp.3- 4. <hal-01006429>HAL Id: hal-01006429https://hal.archives-ouvertes.fr/hal-01006429Submitted on 16 Jun 2014HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Uniform time-decay of semigroupsof contractionsXue Ping WangAbstract. We discuss the uniform time-decay of semi-groups generatedby dissipative Schro¨dinger operators in the semiclassical regime.Mathematics Subject Classification (2010). 35J10, 35P15, 47A55.Keywords. Dissipative Schro¨dinger operators, semiclassical analysis.High frequency analysis of propagation of waves in media with variable ab-sorption index leads to the following dissipative Schro¨dinger equation:{ih ∂∂tuh(x, t) = P (h)uh(x, t),uh(x, 0) = uh0 (x),where P (h) = −h2∆+V1(x)−ihV2(x), x ∈ Rn, h ∈]0, h0] is a small parameterproportional to wave length and Vj , j = 1, 2, are real functions with V2 ≥ 0and V2 ̸= 0. Assume that Vj is smooth, satisfying for some ρ > 0|∂αxVj(x)| ≤ Cα⟨x⟩−ρ−|α|, j = 1, 2;here ⟨x⟩ = (1 + |x|2)1/2. Let Sh(t) = e−itP (h)/h, t ≥ 0, be the associatedsemigroup of contractions in L2(Rn). Then ∥Sh(t)∥ ≤ 1 for all t ≥ 0 and h ∈]0, h0]. The interplay between propagation along the flow of the Hamiltonianp1(x, ξ) = ξ2 + V1(x) and the dissipation governs the long-time behavior ofsolutions. A natural question in this connection is the followingQuestion. Can one establish a uniform time-decay estimate for the semigroupSh(t) in the form∥⟨x⟩−sSh(t)⟨x⟩−s∥ ≤ w(t), t > 0, (1)uniformly in h ∈ ]0, h0]? Here s > 0 and w(t) is independent of h and suchthat w(t)→ 0 as t→∞.When n = 3 and ρ > 2, one can show that for each fixed h > 0, one cantake w(t) = Ch⟨t⟩−r for any r < s and r ∈ [0, 3/2]. In the regime h→ 0, long-time behaviors of the quantum evolution are closely related to the classicaldynamics. Let (x(t; y, η), ξ(t; y, η)) denote the classical Hamiltonian flow of2 Xue Ping Wangp1(x, ξ) with initial data (y, η). Making use of Egorov’s theorem, one candeduce that a necessary condition for (1) to be true is that|⟨x(t; y, η)⟩−se−2∫t0V2(x(τ ;y,η))dτ ⟨y⟩−s| ≤ w(t) (2)for all (y, η) ∈ R2n. Since w(t) tends to zero, estimate (2) implies that eachbounded classical trajectory should pass through the open set {x;V2(x) > 0}.In addition, one derives from (2) that w(t) ≥ C⟨t⟩−sσ with σ = min{1, τ0},where τ0 is the divergence rate in t of non-trapping trajectories with energy0. Is the condition (2) sufficient for a uniform time-decay estimate of the form(1)? If yes, can one take w(t) = C⟨t⟩−min{n2,sσ}? The restriction on the decayrate by n2 comes from the threshold behavior of the semigroup for fixed h.Recall that in the selfadjoint case (V2 = 0) and with a localization inenergies away from R−, the result is true with w(t) = Cs⟨t⟩−s for any s > 0.More precisely, let U(t, h) = e−itP1(h)/h, P1(h) = −h2∆+ V1, I =]a, b[ witha > 0, χ ∈ C∞0 (I). Then the estimate∥⟨x⟩−sχ(P1(h))U(t, h)⟨x⟩−s∥ ≤ Cs⟨t⟩−ϵ, t ∈ R, (3)holds for some s, ϵ > 0 and uniformly in h ∈]0, h0] if and only if every energyE in supp χ is non-trapping. If the latter is satisfied, (3) holds with ϵ = sand for any s > 0; see [2].For non-selfadjoint operators, one can no longer use compactly sup-ported cut-off. The main difficulty to prove a result like (1) is the semiclas-sical analysis near the threshold zero for dissipative Schro¨dinger operators.A closely related problem is a global limiting absorption principle on thewhole real axis from the the upper half-complex plane and a nice resolventestimate in h > 0. For energies away from zero and under the condition (2),this was recently obtained by J. Royer (see [1]). The question is open nearthe threshold zero.References[1] J. Royer, Limiting absorption principle for the dissipative Helmholtz equation,Comm. Part. Diff. Equ. 35(2010), no.8, 1458-1489.[2] X. P. Wang, Time-decay of scattering solutions and classical trajectories, Ann.I.H.P., Sect. A. 47(1987), no.1, 25-37.Xue Ping WangDe´partement de Mathe´matiquesUniversite´ de Nantes44322 Nantes Cedex 3, Francee-mail: xue-ping.wang@univ-nantes.fr

Uniform time-decay of semigroups of contraction

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Uniform time-decay of semigroups of contraction

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