Absence of Negative Energy Spectrum for N-Particle Hamiltonians

Abstract

We consider the spectrum of the quantum Hamiltonian H for a system of N one-dimensional particles. H is given by \(H = \sum\nolimits_{i = 1}^n { - \frac{1}{{2m_i }}\frac{{\partial ^2 }}{{\partial x_i^2 }}} + \sum {_{1 \leqslant i 0, and a function Vij(x), |x|>a, with \(\smallint _a^\infty \left| {x - a} \right|\left| {V_{ij} \left( x \right)} \right|dx < \infty \). We give conditions on V−ij(x), the negative part of Vij(x), which imply that H has no negative energy spectrum for all N. For example, this is the case if V−ij(x) has finite range 2a and $$2m_i \smallint _a^{2a} \left| {x - a} \right|\left| {V_{ij}^ - \left( x \right)} \right|dx < 1.$$ If V−ij is not necessarily small we also obtain a thermodynamic stability bound inf σ(H)≥−cN, where 0<c<∞, is an N-independent constant.