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
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
then
and
as t→o for any φ in H. On the other hand, if
then
and for general φ in L2(ℝ),
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
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).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
L. Carleson, Some analytic problems related to statistical mechanics, in Euclidean Harmonic Analysis, Lecture Notes in Math. 779 (1979), 5–45.
M. Cowling, Harmonic analysis on semigroups, to appear, Annals of Math..
B.E.J. Dahlberg and C.E. Kenig, A note on the almost everywhere behaviour of solutions to the Schrödinger equation, in Harmonic Analysis, Lecture Notes in Math. 908 (1982), 205–209.
E.C. Titchmarsh, The Theory of Functions. Oxford Univ. Press, Oxford, etc., 1978.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1983 Springer-Verlag
About this paper
Cite this paper
Cowling, M.G. (1983). Pointwise behavpour of solutions to Schrödinger equations. In: Mauceri, G., Ricci, F., Weiss, G. (eds) Harmonic Analysis. Lecture Notes in Mathematics, vol 992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0069152
Download citation
DOI: https://doi.org/10.1007/BFb0069152
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-12299-9
Online ISBN: 978-3-540-39885-1
eBook Packages: Springer Book Archive