Zeno Product Formula Revisited

Abstract

We introduce a new product formula which combines an orthogonal projection with a complex function of a non-negative operator. Under certain assumptions on the complex function the strong convergence of the product formula is shown. Under more restrictive assumptions even operator-norm convergence is verified. The mentioned formula can be used to describe Zeno dynamics in the situation when the usual non-decay measurement is replaced by a particular generalized observables in the sense of Davies.

Remarks on the Trotter–Kato Product Formula for Unitary Groups

“…This allows us to extend the L 2 -convergence of the imaginary-time Trotter product formula to holomorphic Kato functions. Using the concept of admissible functions introduced in [6] we prove this result also for the Trotter-Kato product formula.…”

“…In [10] Ichinose proposed a modified Trotter-type product formula. He proved in that paper that 6) where E A (·) and E B (·) denote the spectral measures of the operators A and B, respectively, and a ≥ 0, b ≥ 0, 0 < δ < π/2. If one introduces the functions…”

“…holds for any h ∈ h and T > 0, where P is the orthogonal projection from H onto h. Subsequently, an attempt was made in [6] to replace the strong L 2 -topology of [5] by the usual strong topology of H. For admissible functions φ satisfying ℑm (φ(x)) ≤ 0, x ∈ R + , it was shown in [6] that…”

Sufficient conditions for the anti-Zeno effect

“…For the first time it was formulated explicitly in this context by Beskow and Nilsson [3] and soon after a mathematical analysis [5,14] revealed sufficient conditions under which it exists; it became truly popular after the authors of [20] coined its present name. Recently the effect attracted a new wave of mathematical [8,9,19,22] and physical [11,10,12,13,16,18] interest; in the mentioned papers one can find a more complete bibliography.Although the opposite situation, in which a frequent measurement can on the contrary speed up the decay, or ideally to lead to an immediate disappearance of the unstable system, was also noticed early [6], it attracted attention only recently -see, e.g., [1,2,17,21] and also [22] and references therein. As in the case of the Zeno effect, the problem can be tackled from two points of view.…”

“…For the first time it was formulated explicitly in this context by Beskow and Nilsson [3] and soon after a mathematical analysis [5,14] revealed sufficient conditions under which it exists; it became truly popular after the authors of [20] coined its present name. Recently the effect attracted a new wave of mathematical [8,9,19,22] and physical [11,10,12,13,16,18] interest; in the mentioned papers one can find a more complete bibliography.…”