Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Almost periodic potentials in higher dimensions

Author: Vassilis G. Papanicolaou
Journal: Trans. Amer. Math. Soc. 329 (1992), 679-696
MSC: Primary 35J10; Secondary 35P05, 47F05
MathSciNet review: 1042290
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Abstract: This work was motivated by the paper of R. Johnson and J. Moser (see [J-M] in the references) on the one-dimensional almost periodic potentials. Here we study the operator $ L = - \Delta /2 - q$, where $ q$ is an almost periodic function in $ {R^d}$. It is shown that some of the results of [J-M] extend to the multidimensional case (our approach includes the one-dimensional case as well).

We start with the kernel $ k(t,x,y)$ of the semigroup $ {e^{ - tL}}$. For fixed $ t > 0$ and $ u \in {R^d}$, it is known (we review the proof) that $ k(t,x,x + u)$ is almost periodic in $ x$ with frequency module not bigger than the one of $ q$. We show that $ k(t,x,y)$ is, also, uniformly continuous on $ [a,b] \times {R^d} \times {R^d}$. These results imply that, if we set $ y = x + u$ in the kernel $ {G^m}(x,y;z)$ of $ {(L - z)^{ - m}}$ it becomes almost periodic in $ x$ (for the case $ u = 0$ we must assume that $ m > d/2$), which is a generalization of an old one-dimensional result of Scharf (see [S.G]). After this, we are able to define $ {w_m}(z) = {M_x}[{G^m}(x,x;z)]$, and, by integrating this $ m$ times, an analog of the complex rotation number $ w(z)$ of [J-M]. We also show that, if $ e(x,y;\lambda )$ is the kernel of the projection operator $ {E_\lambda }$ associated to $ L$, then the mean value $ \alpha (\lambda ) = {M_x}[e(x,x;\lambda )]$ exists. In one dimension, this (times $ \pi $) is the rotation number. In higher dimensions ($ d = 1$ included), we show that $ d\alpha (\lambda )$ is the density of states measure of [A-S] and it is related to $ {w_m}(z)$ in a nice way. Finally, we derive a formula for the functional derivative of $ {w_m}(z;q)$ with respect to $ q$, which extends a result of [J-M].

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Keywords: Almost periodic functions of several variables, Schrödinger semigroup, resolvent, (complex) rotation number, (integrated) density of states
Article copyright: © Copyright 1992 American Mathematical Society