We consider the quantum graph Hamiltonian on square lattice in Euclidean space, and we show that spectrum of converges to corresponding Schr\"odinger operator space continuum limit, eigenfunctions eigenprojections also converge some sense. employ discrete as intermediate operator, use a recent result by second third author limit operator.

We report the results of a very high statistics Monte Carlo study of the continuum limit of the two dimensional O(3) non-linear σ model. We find a significant discrepancy between the continuum extrapolation of our data and the form factor prediction of Balog and Niedermaier, inspired by the Zamolodchikovs’ S-matrix ansatz. On the other hand our results for the O(3) and the dodecahedron model ar...

We study the classical continuum limit of the overlap topological charge Q (m) lat . For m = r = 1 , and with either periodic or anti-periodic boundary conditions on the spinor fields, we prove that Qlat reduces to the continuum topological charge Q when the continuum gauge field is in a singular gauge, vanishes outside a bounded region, and is pure gauge in a neighbourhood of the singularity. ...

We consider a general class of discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ with mesh size h > 0. In the continuum limit when h → 0, we prove that the limiting dynamics are given by a nonlinear Schrödinger equation (NLS) on R with the fractional Laplacian (− ) as dispersive symbol. In particular, we obtain that fractional powers 1 2 < α < 1 arise from long-range lattice int...