Quantum Hamiltonians with weak random abstract perturbation. I. Initial length scale estimate

TL;DR

This paper investigates the spectral properties of large random quantum Hamiltonians with weak perturbations, establishing bounds on eigenvalues and Green function decay, relevant for understanding quantum systems with disorder.

Contribution

It provides new deterministic and probabilistic estimates on the spectral minimum and Green function decay for large-scale random Hamiltonians with weak perturbations.

Findings

Derived bounds on the lowest eigenvalue of the Hamiltonian.

Established exponential decay of the Green function near the spectral bottom.

Analyzed the asymptotic behavior as system size tends to infinity.

Abstract

We study random Hamiltonians on finite-size cubes and waveguide segments of increasing diameter. The number of random parameters determining the operator is proportional to the volume of the cube. In the asymptotic regime where the cube size, and consequently the number of parameters as well, tends to infinity, we derive deterministic and probabilistic variational bounds on the lowest eigenvalue, i.e. the spectral minimum, as well as exponential off-diagonal decay of the Green function at energies above, but close to the overall spectral bottom.