On time dependent Schrödinger equations: Global well-posedness and growth of Sobolev norms

Abstract

In this paper we consider time dependent Schrödinger linear PDEs of the form i∂tψ = L(t)ψ, where L(t) is a continuous family of self-adjoint operators. We give conditions for well-posedness and polynomial growth for the evolution in abstract Sobolev spaces.is a perturbation smooth in time and H is a self-adjoint positive operator whose spectrum can be enclosed in spectral clusters whose distance is increasing, we prove that the Sobolev norms of the solution grow at most as t ǫ when t → ∞, for any ǫ > 0. If V (t) is analytic in time we improve the bound to (log t) γ , for some γ > 0. The proof follows the strategy, due to Howland, Joye and Nenciu, of the adiabatic approximation of the flow. We recover most of known results and obtain new estimates for several models including 1-degree of freedom Schrödinger operators on R and Schrödinger operators on Zoll manifolds.

Growth of Sobolev norms in time dependent semiclassical anharmonic oscillators

“…Remark that, in the case of equation (1.1), assumption (H cl ) is easily checked, while assumption (H qu ) follows by Theorem A.1, which is a semiclassical version of the abstract theorem of growth proved in [31].…”

“…Therefore condition (A.3) holds with τ = 1 2 and Theorem A.1 implies that sup ∈(0,1]U (t, s) L(H r ) ≤ C ′ r t − s r ;in particular condition(3.3) holds with µ = 1. Thus, by Theorem 3.2, Lemma 4.5 (with ǫ = 1/2) and using also (4.1),(4.20), one finds constants K 1 , K 2 , K 3 > 0 such thatU (t, 0)ϕ z0 r ≥ U 2 (t, 0)ϕ z0 r − U (t, 0)ϕ z0 − U 2 (t, 0)ϕ z0 r ≥ K 1 [log(2 + t)] r for all times 2 ≤ t ≤ K 2 log K A Asemiclassical abstract theorem on growth of Sobolev normsWe prove here a semiclassical version of Thereom 1.5 of[31]. Thus consider an Hilbert space H 0 and a positive, invertible, selfadjoint operator K (possibly -dependent) acting on it.…”

“…If the map t → β(t) is real analytic in time, the subpolynomial bound (1.13) can be improved into a logarithmic one [31]:…”

A theoretical investigation of time-dependent Kohn–Sham equations: new proofs

“…The classical approach based on semi-group theory, has been extended to the case of time-dependent potential. We refer to [13,Chapters 6 and 7] and, e.g., to the very recent work [21]. However, to use these results the potentials must be continuous functions in time; see, e.g., Hypothesis (3) of Theorem 6.2.5 in [13] and Hypothesis H0 in [21].…”

“…We refer to [13,Chapters 6 and 7] and, e.g., to the very recent work [21]. However, to use these results the potentials must be continuous functions in time; see, e.g., Hypothesis (3) of Theorem 6.2.5 in [13] and Hypothesis H0 in [21]. This assumption is in general not suitable for concrete applications and in optimal control theory, where the time-dependency of the potentials is due to timedependent control functions that are, in general, much less regular than continuously differentiable functions.…”