On the Lie-Trotter theorems inL ^p-spaces

Abstract

In this paper we consider operatorsH ₀ andV possessing the following properties:

• (1) H ₀ is a positive self-adjoint operator acting inL ²(M, γ) with γ a probability measure, so that exp(−tH ₀) is a contraction onL ¹(M, γ) for eacht>0.

• (2) V is a semibounded multiplicative operator acting inL ²(M, γ) {fx379-1}

Under these assumptions theorems of Lie-Trotter type are derived for the operatorsH, H ₀, V, whereH is a self-adjoint extension of the algebraic sumH ₀+V, and is built by the form method. Under the additional assumption thatV(·)∈L ²(M, γ) we prove an essential self-adjointness ofH ₀+V. The results obtained are applicable to non-relativistic quantum mechanics.