# 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
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© Springer-Verlag 1990