arXiv subjects
Publications and source records attributed to Tomio Umeda.
We prove that the discrete Dirac operators in three dimensions converge to the continuum Dirac operators in the strong resolvent sense, but not in the norm resolvent sense.
We discuss the continuum limit of discrete Dirac operators on the square lattice in $\mathbb R^2$ as the mesh size tends to zero. To this end, we propose the most natural and simplest embedding of $\ell^2(\mathbb Z_h^d)$ into $L^2(\mathbb R^d)$, which enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space $L^2(\mathbb R^2)^2$. In particular, we prove that the discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on $\mathbb R^2$ and allowed to be complex matrix-valued. We also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense. This is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis.
Let $$L_0=\suml_{j=1}^nM_j^0D_j+M_0^0,\,\,\,\,D_j=\frac{1}{i}\frac{\pa}{\paxj}, \quad x\in\Rn,$$ be a constant coefficient first-order partial differential system, where the matrices $M_j^0$ are Hermitian. It is assumed that the homogeneous part is strongly propagative. In the nonhomegeneous case it is assumed that the operator is isotropic . The spectral theory of such systems and their potential perturbations is expounded, and a Limiting Absorption Principle is obtained up to thresholds. Special attention is given to a detailed study of the Dirac and Maxwell operators. The estimates of the spectral derivative near the thresholds are based on detailed trace estimates on the slowness surfaces. Two applications of these estimates are presented: \begin{itemize} \item Global spacetime estimates of the associated evolution unitary groups, that are also commonly viewed as decay estimates. In particular the Dirac and Maxwell systems are explicitly treated. \item The finiteness of the eigenvalues (in the spectral gap) of the perturbed Dirac operator is studied, under suitable decay assumptions on the potential perturbation. \end{itemize}
The spectra of massless Dirac operators are of essential interest e.g. for the electronic properties of graphene, but fundamental questions such as the existence of spectral gaps remain open. We show that the eigenvalues of massless Dirac operators with suitable real-valued potentials lie inside small sets easily characterised in terms of properties of the potentials, and we prove a Schnol'-type theorem relating spectral points to polynomial boundedness of solutions of the Dirac equation. Moreover, we show that, under minimal hypotheses which leave the potential essentially unrestrained in large parts of space, the spectrum of the massless Dirac operator covers the whole real line; in particular, this will be the case if the potential is nearly constant in a sequence of regions.
Discussed are $\pm m$ modes and $\pm m$ resonances of Dirac operators with vector potentials $H_{\!A}= α\cdot (D - A(x)) + m β$. Asymptotic limits of $\pm m$ modes at infinity are derived when $|A(x)| \le C ^{-ρ}$, $ρ> 1$, provided that $H_A$ has $\pm m$ modes. In wider classes of vector potentials, sparseness of the vector potentials which give rise to the $\pm m$ modes of $H_A$ are established. It is proved that no $H_A$ has $\pm m$ resonances if $|A(x)|\le C ^{-ρ}$, $ρ>3/2$.
Generalized eigenfunctions of the two-dimensional relativistic Schrödinger operator $H=\sqrt{-Δ}+V(x)$ with $|V(x)|\leq C< x>^{-σ}$, $σ>3/2$, are considered. We compute the integral kernels of the boundary values $R_0^\pm(λ)=(\sqrt{-Δ}-(λ\pm i0))^{-1}$, and prove that the generalized eigenfunctions $ϕ^\pm(x,k)$ are bounded on $R_x^2\times\{k | a\leq |k|\leq b\}$, where $[a,b]\subset(0,\infty)\backslashσ_p(H)$, and $σ_p(H)$ is the set of eigenvalues of $H$. With this fact and the completeness of the wave operators, we establish the eigenfunction expansion for the absolutely continuous subspace for $H$. Finally, we show that each generalized eigenfunction is asymptotically equal to a sum of a plane wave and a spherical wave under the assumption that $σ>2$.
We show that the eigenspaces of the Dirac operator $H=α\cdot (D - A(x)) + m β$ at the threshold energies $\pm m$ are coincide with the direct sum of the zero space and the kernel of the Weyl-Dirac operator $σ\cdot (D - A(x))$. Based on this result, we describe the asymptotic limits of the eigenfunctions of the Dirac operator corresponding to these threshold energies. Also, we discuss the set of vector potentials for which the kernels of $H\mp m$ are non-trivial, i.e. ${Ker}(H\mp m) \not = \{0 \}$.
Asymptotic behaviors of zero modes of the massless Dirac operator $H=α\cdot D + Q(x)$ are discussed, where $α= (α_1, α_2, α_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-ρ} $, $ρ>1$. We shall show that for every zero mode $f$, the asymptotic limit of $|x|^2f(x)$ as $|x| \to +\infty$ exists. The limit is expressed in terms of an integral of $Q(x)f(x)$.
The zero modes and zero resonances of the Dirac operator $H=α\cdot D + Q(x)$ are discussed, where $α= (α_1, α_2, α_3)$ is the triple of $4 \times 4$ Dirac matrices, $ D=\frac{1}{i} \nabla_x$, and $Q(x)=\big(q_{jk} (x) \big)$ is a $4\times 4$ Hermitian matrix-valued function with $| q_{jk}(x) | \le C < x >^{-ρ} $, $ρ>1$. We shall show that every zero mode $f(x)$ is continuous on ${\mathbb R}^3$ and decays at infinity with the decay rate $|x|^{-2}$. Also, we shall show that $H$ has no zero resonance if $ρ> 3/2$.
Generalized eigenfunctions of the 3-dimensional relativistic Schrödinger operator $\sqrtΔ + V(x)$ with $|V(x)|\le C < x >^{-σ}$, $σ> 1$, are considered. We show that the generalized eigenfunctions can be expressed as the sum of plane waves and solutions to the time-independent relativistic Schrödinger equation with the radiation condition. If $σ>3$, then we can give pointwise estimates of the differences between the sums and the solutions.
We shall investigate the asymptotic behavior of the extended resolvent R(s) of the Dirac operator as |s| increases to infinity, where s is a real parameter. It will be shown that the norm of R(s), as a bounded operator between two weighted Hilbert spaces of square integrable functions on the 3-dimensional Euclidean space, stays bounded. Also we shall show that R(s) converges 0 strongly as |s| increases to infinity. This result and a result of Yamada [15] are combined to indicate that the extended resolvent of the Dirac operator decays much more slowly than those of Schroedinger operators.