Time decay for solutions of Schrödinger equations with rough and time-dependent potentials

By Igor Rodnianski, Wilhelm Schlag. 2001

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In this paper we establish dispersive estimates for solutions to the linear Schrödinger equation in three dimensions 0.1$$\frac{1}{i}\partial_t \psi - \triangle \psi + V\psi = 0,\qquad \psi(s)=f$$ where V(t,x) is a time-dependent potential that satisfies the conditions $$\sup_{t}\|V(t,\cdot)\|_{L^{\frac{3}{2}}(\mathbb{R}^3)} + \sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \int_{-\infty}^\infty\frac{|V(\hat{\tau},x)|}{|x-y|}\,d\tau\,dy < c_0.$$ Here c0 is some small constant and $V(\hat{\tau},x$) denotes the Fourier transform with respect to the first variable. We show that under these conditions (0.1) admits solutions ψ(·)∈L∞t(L2x(ℝ3))∩L2t(L6x(ℝ3)) for any f∈L2(ℝ3) satisfying the dispersive inequality 0.2$$\|\psi(t)\|_{\infty} \le C|t-s|^{-\frac32}\,\|f\|_1 \text{\ \ for all times $t,s$.}$$ For the case of time independent potentials V(x), (0.2) remains true if $$\int_{\mathbb{R}^6} \frac{|V(x)|\;|V(y)|}{|x-y|^2} \, dxdy <(4\pi)^2\text{\ \ \ and\ \ \ }\|V\|_{\mathcal{K}}:=\sup_{x\in\mathbb{R}^3}\int_{\mathbb{R}^3} \frac{|V(y)|}{|x-y|}\,dy<4\pi.$$ We also establish the dispersive estimate with an ε-loss for large energies provided $\|V\|_{\mathcal{K}}+\|V\|_2<\infty$.Finally, we prove Strichartz estimates for the Schrödinger equations with potentials that decay like |x|-2-ε in dimensions n≥3, thus solving an open problem posed by Journé, Soffer, and Sogge.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/058d0ffd5e65d54106b2d60e63589dd93abe462b

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Citations in the corpus

Dispersive Estimates for Higher Dimensional Schrödinger Operators with Threshold Eigenvalues I: The Odd Dimensional Case (2014, background): “Estimates of type (1) are proved in [21, 28, 25, 13, 26, 4, 6, 3, 15] in various dimensions, and with different characterizations of the potential V (x).”
Dispersive Estimates for Four Dimensional Schrödinger and Wave Equations with Obstructions at Zero Energy (2013, background): “There is a substantial body of work concerning the validity of dispersive estimates for a Schrödinger operator of the form H = −∆ + V , where V is a real-valued potential on Rn decaying at spatial infinity with the assumption that zero is a regular point of the spectrum of H, see for example [31, 40”
Dispersive Estimates for Schrödinger Equations with Threshold Resonance and Eigenvalue (2005, background): “3, following basically the argument of [ 25 ] and [8] but more concisely.”
Controllability of the discrete-spectrum Schrödinger equation driven by an external field (2008, background): “Let us mention that, in some relevant cases in which the spectrum of A has a nontrivial continuous component the solution can be defined as in [30, 34] by means of Strichartz estimates.”
Selection of the ground state for nonlinear schrödinger equations (2003, background): “The L(1) → L and, more generally, L → L estimates, are known in the self-adjoint case [25, 61, 37].”
Scattering theory for nonlinear Schrödinger equations with inverse-square potential (2013, background): “And also it is known that for any potential V (x) ∼ |x|−2−ǫ, the Strichartz estimates are satisfied, without any further assumption on the monotonicity or regularity of V (x); see RodnianskiSchlag [27].”
On the Solvability Conditions for a Linearized Cahn-Hilliard Equation (2011, background): “3 of [17], under our Assumption 1 above on the potential function, the operator −∆ − V (x) − a is self-adjoint and unitarily equivalent to −∆ − a on L(2)(R(3)) via the wave operators (see [10, 14]) Ω := s − limt→∓∞e e with the limit understood in the strong L(2) sense (see, e.”
Restriction estimates, sharp spectral multipliers and endpoint estimates for Bochner-Riesz means (2012, background): “We also refer to Rodnianski and Schlag [49] for more references on dispersive estimates for Schrödinger operators.”
Strichartz and Smoothing Estimates for Dispersive Equations with Magnetic Potentials (2007, methodology): “The method we shall follow is inspired by [29], [6] and consists in mixing Strichartz and smoothing estimates for the free operator with smoothing estimates for the perturbed operator.”
Solvability in the sense of sequences to some non-Fredholm operators (2013, neutral): “3 of [16], under Assumption 3 above on the potential function, the operator−∆+V (x)−a on L(2)(R(3)) is self-adjoint and unitarily equivalent to −∆− a via the wave operators (see [10], [13]) Ω± := s− limt→∓∞eit(−∆+V e, where the limit is understood in the strong L(2) sense (see e.”
On the solvability conditions for the diffusion equation with convection terms (2011, neutral): “3 of [13], under our Assumption 2 on the potential function, the operator Ha is self-adjoint and unitarily equivalent to −∆ − a on L(2)(R(3)) via the wave operators (see [6], [9]) Ω± := s− limt→∓∞eit(−∆+W0)eit∆”
Counterexamples of Strichartz Inequalities for Schrodinger Equations with Repulsive Potentials (2006, background): “There is a huge literature on the subject, let us recall in particular [1], [2], [3], [6], [7], [8], [11], [16], [17], [18], [19], [21], even if the list is very far from being complete.”
Solvability conditions for some non-Fredholm operators (2009, neutral): “The left-hand side of the inequality above is usually referred to as the Rollnik norm (see, for example, [15]) and the upper bound we obtained on it is the sufficient condition for the operator H0 = −∆ + V (x) on L2(R3) to be self-adjoint and unitarily equivalent to −∆ via the wave operators (see, f”