Pointwise behavpour of solutions to Schrödinger equations

Abstract

Suppose that H is a self-adjoint (possibly unbounded, but with dense domain) operator on the Hilbert space H. Take φ in H and let ω(t) be given by the formula

$$\psi (t) = \exp (itH)\phi t \in \mathbb{R}.$$

Then lim ω(t)=φ.

Suppose $\mathop {that}\limits^{t \to o}$ H is the space L²(X), for some measure space X. It is reasonable to ask when ω(t) converges to φ pointwise almost everywhere. We show that if |H|^αφ is in L²(X) for some α in (1/2,+∞), then pointwise convergence is verified.

To motivate our work, consider the following examples. If H=L²(ℝ), and

$$H\phi (x) = x\phi (x)x \in \mathbb{R},\phi \in Dom(H),$$

then

$$\psi (t,x) = exp(itx)\phi (x)x \in \mathbb{R},$$

and

$$\psi (t,x) \to \phi (x)a.e.x$$

as t→o for any φ in H. On the other hand, if

$${{H\phi (x) = id\phi (x)} \mathord{\left/{\vphantom {{H\phi (x) = id\phi (x)} {dxx \in \mathbb{R},\phi \in Dom(H),}}} \right.\kern-\nulldelimiterspace} {dxx \in \mathbb{R},\phi \in Dom(H),}}$$

then

$$\psi (t,x) = \phi (x - t)x \in \mathbb{R},$$

and for general φ in L²(ℝ),

$$\psi (t,x)\not \to \phi (x)a.e.x$$

as t→o. If however we assume that |H|^αφ є L²(ℝ) for some α in (1/2,+∞), then this forces φ to be continuous, and so pointwise convergence is obvious.

More recent examples arise in work of L. Carleson [1] and B.E.J. Dahlberg and C.E. Kenig [3], in which the case where

$$H\phi (x) = {{d^2 \phi (x)} \mathord{\left/{\vphantom {{d^2 \phi (x)} {dx^2 x \in \mathbb{R}}}} \right.\kern-\nulldelimiterspace} {dx^2 x \in \mathbb{R}}}$$

is treated. These authors show that |H|^αφ in L²(ℝ) is sufficient to guarantee pointwise convergence if and only if α⩾1/4.

Our approach to this problem is abstract. It is based on the ideas we present in fuller detail in [2]. In particular, we assume only that H is self-adjoint, and further, our results hold for any realisation of the Hilbert space H as L²(X).