# Unitary dressing transformations and exponential decay below threshold for quantum spin systems. Parts I and II

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## Abstract

We consider a class of quantum spin systems defined on connected graphs of which the following Heisenberg We treat first the case in which

*XY*-model with a variable magnetic field gives an example:$$H_\lambda = \sum\limits_{x \in \mathbb{Z}^d } {h_x \sigma _x^{(3)} + \lambda } \sum\limits_{< x,y > \subset \mathbb{Z}^d } {(\sigma _x^{(1)} \sigma _y^{(1)} + \sigma _x^{(2)} \sigma _y^{(2)} )} .$$

*h*_{ x }=±1 for all sites*x*and we introduce a unitary dressing transformation to control the spectrum for λ small. Then, we consider a situation in which |*h*_{ x }| can be less than one for*x*in a finite set*L*and prove exponential decay away from*L*of dressed eigenfunctions with energy below the one-quasiparticle threshold. If the ground state is separated by a finite gap from the rest of the spectrum, this result can be strengthened and one can compute a second unitary transformation that makes the ground state of compact support. Finally, a case in which the singular set*L*is of finite density, is considered. The main technical tools we use are decay estimates on dressed Green's functions and variational inequalities.## Keywords

Magnetic Field Neural Network Statistical Physic Complex System Nonlinear Dynamics
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer-Verlag 1990