Asymptotics of information entropies of some Toda-like potentials

The spreading of the quantum probability density for the highly-excited states of a single-particle system with an exponential-type potential on the positive semiaxis is quantitatively determined in both position and momentum spaces by means of the Boltzmann-Shannon information entropy. This problem boils down to the calculation of the asymptotics of the entropy-like integrals of the modified Bessel function of the second kind (also called the Mcdonald function or Basset function). The dependence of the two physical entropies on the large quantum number n is given in detail. It is shown that the semiclassical (WKB) position-space entropy grows slower than the corresponding quantity of not only the harmonic oscillator but also the single-particle systems with any power-type potential of the form V(x) = x^2k, x ∈ ℤ and k ∈ℕ. The momentum-space entropy, calculated with a method based on the properties of the Mcdonald function, is rigorously found to have a behavior of the form - ln ln n, in strong contrast with the corresponding quantity of other one-dimensional systems known up to now (power-type potentials, infinite well).