Effective Hamiltonians for Complexes of Unstable Particles

TL;DR

This paper explores the properties of effective Hamiltonians for unstable particle complexes derived via the KR equation, analyzing their long-time behavior and implications for decay and energy estimates.

Contribution

It provides a detailed analysis of time-dependent effective Hamiltonians for unstable systems using the KR equation, highlighting their asymptotic properties.

Findings

Real part of the effective Hamiltonian tends to the minimal energy as time approaches infinity.

Imaginary part of the effective Hamiltonian tends to zero at long times.

Effective Hamiltonians depend on time and influence decay dynamics.

Abstract

Effective Hamiltonians governing the time evolution in a subspace of unstable states can be found using more or less accurate approximations. A convenient tool for deriving them is the evolution equation for a subspace of state space sometime called the Krolikowski-Rzewuski (KR) equation. KR equation results from the Schr\"{o}dinger equation for the total system under considerations. We will discuss properties of approximate effective Hamiltonians derived using KR equation for $n$--particle, two particle and for one particle subspaces. In a general case these affective Hamiltonians depend on time $t$. We show that at times much longer than times at which the exponential decay take place the real part of the exact effective Hamiltonian for the one particle subsystem (that is the instantaneous energy) tends to the minimal energy of the total system when $t \rightarrow \infty$ whereas the imaginary part of this effective Hamiltonian tends to the zero as $t\rightarrow \infty$.