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

Continuum limit for Laplace and Elliptic operators on lattices

Continuum limits of Laplace operators on general lattices are considered, and it is shown that these operators converge to elliptic operators on the Euclidean space in the sense of the generalized norm resolvent convergence. We then study operators on the hexagonal lattice, which does not apply the above general theory, but we can show its Laplace operator converges to the continuous Laplace operator in the continuum limit. We also study discrete operators on the square lattice corresponding to second order strictly elliptic operators with variable coefficients, and prove the generalized norm resolvent convergence in the continuum limit

Non-smoothness of the fundamental solutions for Schr\"{o}dinger equations with super-quadratic and spherically symmetric potential

We study non-smoothness of the fundamental solution for the Schr\"{o}dinger equation with a spherically symmetric and super-quadratic potential in the sence that $V(x)\geq C|x|^{2+\varepsilon}$ at infinity with constants $C>0$ and $\varepsilon>0$. More precisely, we show the fundamental solution $E(t,x,y)$ does not belong to $C^{1}$ as a function of $(t,x,y)$.Comment: 11 page

Construction of Isozaki-Kitada modifiers for discrete Schr\"odinger operators on general lattices

We consider a scattering theory for convolution operators on $\mathcal{H}=\ell^2(\mathbb{Z}^d; \mathbb{C}^n)$ perturbed with a long-range potential $V:\mathbb{Z}^d\to\mathbb{R}^n$. One of the motivating examples is discrete Schr\"odinger operators on $\mathbb{Z}^d$-periodic graphs. We construct time-independent modifiers, so-called Isozaki-Kitada modifiers, and we prove that the modified wave operators with the above-mentioned Isozaki-Kitada modifiers exist and that they are complete.Comment: 20 page

The First Beam Recirculation and Beam Tuning in the Compact ERL at KEK

Superconducting(SC)-linac-based light sources, which can produce ultra-brilliant photon beams in CW operation, are attracting worldwide attention. In KEK, we have been conducting R&D; efforts towards the energy-recovery-linac(ERL)-based light source* since 2006. To demonstrate the key technologies for the ERL, we constructed the Compact ERL (cERL)** from 2009 to 2013. In the cERL, high-brightness CW electron beams are produced using a 500-kV photocathode DC gun. The beams are accelerated using SC cavities, transported through a recirculation loop, decelerated in the SC cavities, and dumped. In the February of 2014, we succeeded in accelerating and recirculating the CW beams of 4.5 micro-amperes in the cERL; the beams were successfully transported from the gun to the beam dump under energy recovery operation in the main linac. Then, precise tuning of beam optics and diagnostics of beam properties are under way. We report our experience on the beam commissioning, as well as the results of initial measurements of beam properties