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Within a well-known decay model describing a particle confined initially within a spherical potential shell, we consider the situation when the initial state has an unusual energy distribution decaying slowly as k; the simplest example corresponds to a wavefunction constant within the shell. We show that the non-decay probability as a function of time then behaves in a highly irregular, most likely fractal way.

Let T be be a zero-product preserving bounded linear map between C*-algebras A and B. Here neither A nor B is necessarily unital.

The problem of the existence of infinitely many prime values of a number-theoretic function f (x) has been one of the most important topics in Number Theory. Note that if f (x) represents infinitely many primes, then we can get this necessary condition: for any positive integer h , there exists a positive integer k such that ( f (k), h) =1and f (k) >1. Naturally, we are interested in the number-theoretic functions f (x) that satisfy the aforementioned necessary condition. Thus, there must exist the least positive integer n such that (f(n),h)-1andf(k)>1. Denote this least positive integer n byF_f(h). In this paper, we mainly focus on three famous number-theoretic functions:&f(x)=2^{2^x}+1,m(x)=2^x-1 and l(x)=x^2+1& proving they satisfy the aforementioned necessary condition respectively. Furthermore, we approximately estimate the upper bound of F_f(x)(h),Fm(x)(h)，Fl(x)(h) respectively, and obtain some interesting results.