Enrico Reiß
The physically motivated quantization procedure generates an operator (the so-called pseudo-differential operator, PDO) from a function on the phase space $$\mathbb{R}^n \times \mathbb{R}^n$$ (the so-called symbol). In the semiclassical limit, where the Planck's constant is supposed to be a small parameter, one finds conditions allowing to approximate the number of eigenvalues of an elliptic PDO by the phase space volume of the corresponding symbol (known under Weyl asymptotics).
A family of difference operators for functions on an $\epsilon$-scaled lattice can be interpreted as discrete versions of PDOs if the corresponding symbols are supposed to be periodic with respect to the momentum component. Adapting the techniques from the continuous setting to our discrete case, we can derive similar Weyl asymptotics for difference operators.
The discrete theory is motivated by statistical mechanics: Consider for example the Curie-Weiss model where N particles interact. The magnetization of the system takes values in a scaled lattice with lattice spacing 1/N. Time evolution in this model can be described by a reversible Markov chain in discrete time, with transition matrix P. Then 1-P is analog to a self-adjoint infinitesimal generator, or in physical language to the Hamilton operator of the system. After h-transform with respect to the reversible measure one has transformed to a space with counting measure. Hilbert space theory on this space can be written down by use of Fourier transform, which allows to realize these Hamilton operators as discrete pseudo-differential operators.
