Absence of positive eigenvalues for hard-core N-body systems

TL;DR

This paper proves that certain multi-particle quantum systems with hard-core interactions and convex obstacles do not have positive eigenvalues, using advanced resolvent estimates and Mourre theory, with applications to models like the Helium atom.

Contribution

It introduces a new scheme for proving the absence of positive eigenvalues in N-body hard-core Schrödinger operators, extending previous methods to more complex systems.

Findings

No positive eigenvalues for 2-body hard-core Schrödinger operators with convex obstacles.

A general scheme for N-body systems N ≥ 2 is developed.

Application to the Helium atom model with infinite mass nucleus.

Abstract

We show absence of positive eigenvalues for generalized 2-body hard- core Schroedinger operators under the condition of bounded strictly convex obstacles. A scheme for showing absence of positive eigenvalues for generalized N -body hard-core Schroedinger operators, N \geq 2, is presented. This scheme involves high energy resolvent estimates, and for N = 2 it is implemented by a Mourre commutator type method. A particular example is the Helium atom with the assumption of infinite mass and finite extent nucleus.