Exact Nilpotent Collapse of Born-Neumann Expansions in Finite Quantum Systems: A SON Formulation for Exact Algebraic Closures of Scattering Series

TL;DR

This paper demonstrates that for acyclic finite quantum systems, the Born series collapses into an exact finite sum due to the nilpotency of the transfer operator, enabling precise algebraic solutions and interference analysis.

Contribution

It establishes a rigorous link between acyclic graph structures and the nilpotency of the transfer operator, leading to exact algebraic solutions for quantum scattering series.

Findings

The Born series collapses exactly for acyclic systems with nilpotent transfer operators.

The transition amplitude for a four-level diamond-graph system is exactly expressed as a finite algebraic sum.

First-order Born approximation fails in all regimes, highlighting the need for exact methods.

Abstract

We identify a class of finite quantum systems, namely, acyclic systems whose transition graph is a directed acyclic graph (DAG), for which the Born series collapses into an exact algebraic identity with finitely many terms and strictly zero truncation error. The sufficient condition is the nilpotency of the transfer operator T = G_0(E)V. If T^{m+1} = 0, then the exact solution of the Lippmann-Schwinger equation is the finite sum |psi> = sum_{k=0}^{m} T^k |phi>, with no condition on ||T||. We prove that the acyclicity of the transition graph implies the nilpotency of T (Theorem 19), and that the nilpotency index coincides with the maximal path length of the graph (Proposition 21). The main result (Theorem 23) concerns the four-level quantum system with diamond-graph structure. In this case, the transition amplitude toward the final state is A_4 = t_{42}t_{21} + t_{43}t_{31}, an exact algebraic identity encoding constructive interference, exact destructive interference (dark state formation), and partial interference. The first-order Born approximation predicts identically zero amplitude in all regimes, thereby failing quantitatively in 100% of the cases. The Born-SON framework additionally provides the exact full resolvent, the exact T-matrix, explicit error control in the quasi-nilpotent regime, and a scalar structural metric, the Born-SON depth, quantifying the intrinsic complexity of an acyclic quantum system.