Continuum limits of discrete Schrödinger operators on square lattices (Recent developments in studies of resonances)

We consider two different approaches of continuum limit problems of Schrödinger operators H = -Δ + V on [R][d]. The first part of this proceedings deals with asymptotic behavors of discrete Schrödinger operators Hh = -Δh + V l hzd on square lattice hZd with mesh size h, and we study conditions of the potential V and the projection from L^2 ([R][d]) onto l^2 (hZd) where Hh converges to the corresponding contiuum operator H the generalized resolvent sense. The sencond one involves Schrodinger operators defined on the edges of hZd, then we prove that a similar continuum limit problem holds under weaker assumption of V