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Chernoff's product formula: Semigroup approximations with non-uniform time intervals

J. Z. Bernád, A. B. Frigyik

TL;DR

This work extends Chernoff's product formula to non-uniform time intervals by imposing a commutativity condition on a contraction family $V$ and a mild regularity on the time-partition arrays $a_{n,i}$. The main result shows that, under these assumptions, the semigroup $T(t)$ generated by $A$ is recovered as $T(t)x=\lim_{n\to\infty}\prod_{i=1}^n V(a_{n,i}t)x$, uniformly for $t$ in compact sets, combining Chernoff's original framework with a Second Trotter–Kato approach. This yields nonuniform Lie–Trotter type product formulas and has concrete implications for quantum measurement models and diffusion-type limits in probability, including a central limit theorem via nonuniform time stepping. The paper also presents two illustrative examples: a quantum-mechanics-inspired Gaussian averaging leading to a specific generator on trace-class operators, and a CLT setting on $C_0(\mathbb{R})$ where the nonuniform product converges to a diffusion semigroup even when the spectral condition is not fulfilled by the naïve generator.

Abstract

Often, when we consider the time evolution of a system, we resort to approximation: Instead of calculating the exact orbit, we divide the time interval in question into uniform segments. Chernoff's results in this direction provide us with a general approximation scheme. There are situations when we need to break the interval into uneven pieces. In this paper, we explore alternative conditions to the one found by Smolyanov et al. such that Chernoff's original result can be extended to unevenly distributed time intervals. Two applications concerning the foundations of quantum mechanics and the central limit theorem are presented.

Chernoff's product formula: Semigroup approximations with non-uniform time intervals

TL;DR

This work extends Chernoff's product formula to non-uniform time intervals by imposing a commutativity condition on a contraction family and a mild regularity on the time-partition arrays . The main result shows that, under these assumptions, the semigroup generated by is recovered as , uniformly for in compact sets, combining Chernoff's original framework with a Second Trotter–Kato approach. This yields nonuniform Lie–Trotter type product formulas and has concrete implications for quantum measurement models and diffusion-type limits in probability, including a central limit theorem via nonuniform time stepping. The paper also presents two illustrative examples: a quantum-mechanics-inspired Gaussian averaging leading to a specific generator on trace-class operators, and a CLT setting on where the nonuniform product converges to a diffusion semigroup even when the spectral condition is not fulfilled by the naïve generator.

Abstract

Often, when we consider the time evolution of a system, we resort to approximation: Instead of calculating the exact orbit, we divide the time interval in question into uniform segments. Chernoff's results in this direction provide us with a general approximation scheme. There are situations when we need to break the interval into uneven pieces. In this paper, we explore alternative conditions to the one found by Smolyanov et al. such that Chernoff's original result can be extended to unevenly distributed time intervals. Two applications concerning the foundations of quantum mechanics and the central limit theorem are presented.
Paper Structure (4 sections, 7 theorems, 49 equations)

This paper contains 4 sections, 7 theorems, 49 equations.

Table of Contents

  1. Introduction
  2. Approximation with non-uniform time intervals
  3. Main result
  4. Examples

Key Result

Theorem 1

Suppose $X$ is a Banach space and there is a function from the non-negative reals into the bounded linear operators on $X$. Function $V$ satisfies the following conditions: it is power bounded, i.e. there is an $M \geq 1$ such that $\left\|[V(t)]^m\right\|\leq M$ for all $m\in \mathbb{N}$ and for each $t \geq 0$, and $V(0)=I$, where $I$ is the identity exists for all $x\in D(A)\subset X$, where $

Theorems & Definitions (13)

  • Theorem 1
  • Proposition 2
  • Proposition 3
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Theorem 6
  • proof
  • Corollary 7
  • ...and 3 more