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Using rearrangements of matrix-valued sequences, we prove that with certain boundary conditions the solution of the one-dimensional Schrödinger equation increases or decreases under monotone rearrangements of its potential.

No abstract uploaded!

It is well known that every Hölder continuous function on the unit circle is the sum of two functions such that one of these functions extends holomorphically into the unit disc and the other extends holomorphically into the complement of the unit disc. We prove that an analogue of this holds for Hölder continuous functions on an annulus A which have zero averages on all circles contained in A which surround the hole.

Set $$\theta (s/t): = (s/t - 1)(t/s)^{\frac{{s/t}}{{s/t - 1}}} = (s - t)\frac{{t^{t/(s - t)} }}{{s^{s/(s - t)} }}$$ if 0<$t$<$s$. The key result of the paper shows that if ($T (t))$_{$t$>0}is a nontrivial strongly continuous quasinilpotent semigroup of bounded operators on a Banach space then there exists δ>0 such that ║$T(t)-T(s)$║>θ(s/t) for 0<$t$<$s$≤δ. Also if ($T(t)$)_{$t$>0}is a strongly continuous semigroup of bounded operators on a Banach space, and if there exists η>0 and a continuous function$t$→$s(t)$on [0, ν], satisfying$s$(0)=0, and such that 0<$t$<$s(t)$and ║$T(t)-T(s(t))║<θ(s/t)$for$t$∈(o, η], then the infinitesimal generator of the semigroup is bounded. Various examples show that these results are sharp.