9,804 research outputs found
Approximate closed-form formulas for the zeros of the Bessel Polynomials
We find approximate expressions x(k,n) and y(k,n) for the real and imaginary
parts of the kth zero z_k=x_k+i y_k of the Bessel polynomial y_n(x). To obtain
these closed-form formulas we use the fact that the points of well-defined
curves in the complex plane are limit points of the zeros of the normalized
Bessel polynomials. Thus, these zeros are first computed numerically through an
implementation of the electrostatic interpretation formulas and then, a fit to
the real and imaginary parts as functions of k and n is obtained. It is shown
that the resulting complex number x(k,n)+i y(k,n) is O(1/n^2)-convergent to z_k
for fixed kComment: 9 pages, 2 figure
Survival and Nonescape Probabilities for Resonant and Nonresonant Decay
In this paper we study the time evolution of the decay process for a particle
confined initially in a finite region of space, extending our analysis given
recently (Phys. Rev. Lett. 74, 337 (1995)). For this purpose, we solve exactly
the time-dependent Schroedinger equation for a finite-range potential. We
calculate and compare two quantities: (i) the survival probability S(t), i.e.,
the probability that the particle is in the initial state after a time t; and
(ii) the nonescape probability P(t), i.e., the probability that the particle
remains confined inside the potential region after a time t. We analyze in
detail the resonant and nonresonant decay. In the former case, after a very
short time, S(t) and P(t) decay exponentially, but for very long times they
decay as a power law, albeit with different exponents. For the nonresonant case
we obtain that both quantities differ initially. However, independently of the
resonant and nonresonant character of the initial state we always find a
transition to the ground state of the system which indicates a process of
``loss of memory'' in the decay.Comment: 26 pages, RevTex file, figures available upon request from
[email protected] (To be published in Annals of Physics
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