arXiv subjects

Yoshimi Saito

Publications and source records attributed to Yoshimi Saito.

7 recordsLinked to original sources

arXiv2010

Eigenfunctions at the threshold energies of magnetic Dirac operators

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 $|A(x)|\le C ^{-ρ}$, $ρ>3/2$.

math.SP↗

arXiv2007

The asymptotic limits of zero modes of massless Dirac operators

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 $Q(x)f(x)$.

math.SP↗

arXiv2007

The zero modes and zero resonances of massless Dirac operators

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 $ρ> 3/2$.

math.SP↗

arXiv1999

Resolvent estimates of the Dirac operator

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.

math.SP↗

arXiv1999

On the spectrum of the reduced wave operator with cylindrical discontinuity

Consider the differential operator H = -(1/m(x))L, where L is the N-dimensional Laplacian, in the weighted Hilbert space of square integrable functions on N-dimensional Euclidean space with weight m(x)dx. Here m(x) is a positive step function with a surface S of discontinuity (the separation surface). So far the stratified media in which the separating surface S consists of paralell planes have been vigorously studied. Also the case where S has a cone shape has been discussed. In this work we shall deal with a new type of discontinuity which we call cylindrical discontinuity. Under this condition we shall use the limiting absorption method to prove that H is absolute continuous. Our method is based on a apriori estimates of radiation condition term.

math.SP↗

arXiv1999

The reduced wave equation in layered materials

Let H = -(1/m(x))L be the reduced wave operator defined on the N-dimensional Euclidean space, where \f L is the Laplacian. Here m(x) is a positive step function with possible countably infinte surfaces of discontinuity (separating surfaces) under the compatibilty condition (1.12) on each separating surface. These compatibily condition allows us to treat the cases, among others, the separating surfaces are cylinders. The case where the separating surface has only one connected component was discussed in [9]. Also the case where the separating surface is cone-shaped was considered by Eidus [6] and others ([10], [11]). We shall prove the limiting absorption principle for H. Also we shall discuss the case where m(x) is perturbed by a short-range or long-range function.

math.SP↗

arXiv1998

The uniqueness of the solution of the Schrodinger equation with discontinuous coefficients

Consider the Schroeodinger equation: - Du(x) - l(x)u + s(x)u = 0, where D is the Laplacian, l(x) > 0 and s(x) is dominated by l(x). We shall extend the celebrated Kato's result on the asymptotic behavior of the solution to the case where l(x) has unbounded discontinuity. The result will be used to establish the limiting absorption principle for a class of reduced wave operators with discontinuous coefficients.

math.SP↗