The fact the decay can be slowed or even prevented by frequently repeated measurements, noticed first by Beskow and Nilsson, and called quantum Zeno effect by Misra and Sudarshan, attracted recently a lot of attention since it seems to be nowadays experimentally accessible. Mathematically the issue was addressed, in particular, long time ago by Friedman and Chernoff, however, the question about existence of the "Zeno limit" remained open.
Let H and P be a non-negative self-adjoint operator acting in a separable Hilbert space H and an orthogonal projection on H, respectively. Suppose that the operator HP := (H1/2P)*(H1/2P) is densely defined, then we claim that the said limit exists and we find its value,
limn→∞ [P exp(-iεtH/n)P]n = exp(-iεtHP) P
for ε=±1. In fact, the proof given in [1] yields a stronger result with P on the left-hand side replaced by values of a projection-valued family such that P(t)→P as t→0 and P(t)P = P(t), as well as non-symmetric versions of such formulae. The method we use employs a modification of a Kato's result on Trotter product formula in combination with analyic continuation.
As an example consider H=-Δ in L2(Rd) and P projecting onto an open Ω⊂Rd with a smooth boundary. Then the limits exists and HP is equal to the Dirichlet Laplacian in L2(Ω); this provided a rigorous proof of the formal claim made recently in [2] using the method of stationary phase.
Thus Zeit RULES
KING BM, PhD, etc CONCURS
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