Chaoticity for some linear operators in quantum mechanics
by
H. Emamirad
University of Poitiers

In [ Gulisashvili A. , MacCluer C. R. , Linear chaos in the unforced quantum harmonic oscillator, ASME J. Dyn. Syst. Meas. Control 118 (1996) 337-338] it is proved that the oscillator annihilation operator A = [ 1/squareroot2](x +[ d/dx]) is chaotic in the Fréchet space

F: = ì í î \psi in H     |     \psi = \infty å n=0  cn\psin with \infty å n=0  |cn|2(n+1)l < \infty     for alll ü ý þ .

Here, we prove the same result by replacing F by the Fréchet space E(A), where E(A): = { x in H  |   \sum_k=0^\infty |t|ⁿ ||Aⁿ x|| / n! < \infty, for all t in R} is the space of all entire vectors for A, defined in a complex Hilbert space H. Inspired by the above result we can show also that the limit case ( as b --> 0) of the Gribov operator H_{-1, b, 1} which can be expressed by H : = H_{-1, 0, 1} = z[(d²)/(dz²)]+ z²[ d/dz] is also chaotic in the H-valued Bargmann space F(H)₀:={f in F(H)  |   f(0)=0}, where F(H) is the space of entire functions with Hilbert space structure (f, g) = \int_C <f(z), g(z)>d\mu(z), with d\mu(z) = \pi^-1 e^-[`z].zdxdy,   (z=x+iy) the gaussian measure on C.

(T)

Date received: April 22, 2000