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Yukihide Tadano

Publications and source records attributed to Yukihide Tadano.

9 recordsLinked to original sources
arXiv2025

On the dispersive estimates for the discrete Schrödinger equation on a honeycomb lattice

The discrete Schrödinger equation on a two-dimensional honeycomb lattice is a fundamental tight-binding approximation model that describes the propagation of waves on graphene. For free evolution, we first show that the degenerate frequencies of the dispersion relation are completely characterized by three symmetric periodic curves (Theorem 2.1), and that the three curves meet at Dirac points where conical singularities appear (see Figure 2.1). Based on this observation, we prove the $L^1\to L^\infty$ dispersion estimates for the linear flow depending on the frequency localization (Theorem 2.3). Collecting all, we obtain the dispersion estimate with $O(|t|^{-2/3})$ decay as well as Strichartz estimates. As an application, we prove small data scattering for a nonlinear model (Theorem 2.10). The proof of the key dispersion estimates is based on the associated oscillatory integral estimates with degenerate phases and conical singularities at Dirac points. Our proof is direct and uses only elementary methods.

math.AP↗
arXiv2024

Non-smoothness of the fundamental solutions for Schrödinger equations with super-quadratic and spherically symmetric potential

We study non-smoothness of the fundamental solution for the Schrödinger 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 $(t,x,y)$.

math.AP↗
arXiv2023

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.

math-ph↗
arXiv2023

Construction of Isozaki-Kitada modifiers for discrete Schrödinger 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 Schrödinger 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.

math-ph↗
arXiv2022

Continuum limit of the lattice quantum graph Hamiltonian

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

math-ph↗
arXiv2019

Uniform bounds of discrete Birman-Schwinger operators

In this note, uniform bounds of the Birman-Schwinger operators in the discrete setting are studied. For uniformly decaying potentials, we obtain the same bound as in the continuous setting. However, for non-uniformly decaying potential, our results are weaker than in the continuous setting. As an application, we obtain unitary equivalence between the discrete Laplacian and the weakly coupled systems.

math-ph↗
arXiv2018

Long-range scattering theory for discrete Schrödinger operators on graphene

We consider a long-range scattering theory for discrete Schrödinger operators on the hexagonal lattice, which describe tight-binding Hamiltonians on the graphene sheet. We construct Isozaki-Kitada modifiers for a pair of the difference Laplacian on the hexagonal lattice and perturbed operators with long-range potentials. We prove that these modified wave operators exist and that they are asymptotically complete.

math-ph↗
arXiv2018

Long-range scattering for discrete Schrödinger operators

In this paper, we define time-independent modifiers to construct a long-range scattering theory for discrete schrödinger operators on the square lattice $\mathbb{Z}^N$. We prove the existence and completeness of modified wave operators in terms of the above mentioned time-independent modifiers.

math-ph↗