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Yoshihisa Miyanishi

Publications and source records attributed to Yoshihisa Miyanishi.

At least 19 recordsLinked to original sources
arXiv2024

Circle Foliations Revisited: Periods of Flows whose Orbits are all Closed

Our purpose here is to adapt the results of Geodesic circle foliations for Reeb flows or Hamiltonian flows on contact manifolds. Consequently, all periods are exactly the same if the contact manifold is connected and all orbits on the contact manifold are closed. We also present concrete examples of periodic flows, all of whose orbits are closed, such as Harmonic oscillators, Lotka-Volterra systems, and others. Lotka-Volterra systems, Reeb flows, and some geodesic flows have non-trivial periods, whereas the periods of Harmonic oscillators and similar systems can be easily obtained through direct calculations. As an application to quantum mechanics, we examine the spectrum of semiclassical Shrödinger operators. Then we have one of the semiclassical analogies of the Helton-Guillemin theorem.

math-ph↗
arXiv2024

Carleman factorization of layer potentials on smooth domains

One of the unexplored benefits of studying layer potentials on smooth, closed hypersurfaces of Euclidean space is the factorization of the Neumann-Poincaré operator into a product of two self-adjoint transforms. Resurrecting some pertinent indications of Carleman and M. G. Krein, we exploit this grossly overlooked structure by confining the spectral analysis of the Neumann-Poincaré operator to the amenable $L^2$-space setting, rather than bouncing back and forth the computations between Sobolev spaces of negative or positive fractional order. An enhanced, fresh new look at symmetrizable linear transforms enters into the picture in the company of geometric-microlocal analysis techniques. The outcome is manyfold, complementing recent advances on the theory of layer potentials, in the smooth boundary setting.

math.AP↗
arXiv2023

Generic properties of the Neumann-Poincaré operator: simplicity of eigenvalues and cyclic vectors

Two generic properties of the Neumann--Poincaré operator are investigated. We prove that non-zero eigenvalues of the Neumann--Poincaré operator on smooth boundaries in three dimensions and higher are generically simple in the sense of Baire category. We also prove that the functions defined by the fundamental solutions to the Laplace operator located at points outside the surface are generically cyclic vectors in the sense that the collection of those points where the functions are not cyclic vectors is of measure zero.

math.SP↗
arXiv2023

Decay rate of the eigenvalues of the Neumann-Poincaré operator

If the boundary of a domain in three dimensions is smooth enough, then the decay rate of the eigenvalues of the Neumann-Poincaré operator is known and it is optimal. In this paper, we deal with domains with less regular boundaries and derive quantitative estimates for the decay rates of the Neumann-Poincaré eigenvalues in terms of the Hölder exponent of the boundary. Estimates in particular show that the less the regularity of the boundary is, the slower is the decay of the eigenvalues. We also prove that the similar estimates in two dimensions. The estimates are not only for less regular boundaries for which the decay rate was unknown, but also for regular ones for which the result of this paper makes a significant improvement over known results.

math.SP↗
arXiv2021

Spectral structure of the Neumann-Poincaré operator on thin ellipsoids and flat domains

We investigate the spectral structure of the Neumann-Poincaré operator on thin ellipsoids. Two types of thin ellipsoids are considered: long prolate ellipsoids and flat oblate ellipsoids. We show that the totality of eigenvalues of the Neumann-Poincaré operators on a sequence of the prolate spheroids is densely distributed in the interval [0,1/2] as their eccentricities tend to 1, namely, as they become longer. We then prove that eigenvalues of the Neumann-Poincaré operators on the oblate ellipsoids are densely distributed in the interval [-1/2, 1/2] as the ellipsoids become flatter. In particular, this shows that even if there are at most finitely many negative eigenvalues on the oblate ellipsoids, more and more negative eigenvalues appear as the ellipsoids become flatter. We also show a similar spectral property for flat three dimensional domains.

math.SP↗
arXiv2020

Surface localization of plasmons in three dimensions and convexity

The Neumann--Poincaré operator defined on a smooth surface has a sequence of eigenvalues converging to zero, and the single layer potentials of the corresponding eigenfunctions, called plasmons, decay to zero, i.e., are localized on the surface, as the index of the sequence $j$ tends to infinity. We investigate quantitatively the surface localization of the plasmons in three dimensions. The results are threefold. We first prove that on smooth bounded domains of general shape the sequence of plasmons converges to zero off the boundary surface almost surely at the rate of $j^{-1/2}$. We then prove that if the domain is strictly convex, then the convergence rate becomes $j^{-\infty}$, namely, it is faster than $j^{-N}$ for any integer $N$. As a consequence, we prove that cloaking by anomalous localized resonance does not occur on three-dimensional strictly convex smooth domains. We then look into the surface localization of the plasmons on the Clifford torus by numerical computations. The Clifford torus is taken as an example of non-convex surfaces. The computational results show that the torus exhibits the spectral property completely different from strictly convex domains. In particular, they suggest that there is a subsequence of plasmons on the torus which has much slower decay than other entries of the sequence.

math.AP↗
arXiv2020

Spectral structure of the Neumann--Poincaré operator on thin domains in two dimensions

We consider the spectral structure of the Neumann--Poincaré operators defined on the boundaries of thin domains of rectangle shape in two dimensions. We prove that as the aspect ratio of the domains tends to $\infty$, or equivalently, as the domains get thinner, the spectra of the Neumann--Poincaré operators are densely distributed in the interval $[-1/2,1/2]$.

math.SP↗
arXiv2020

Spectral analysis of Neumann-Poincaré operator

This is a survey of accumulated spectral analysis observations spanning more than a century, referring to the double layer potential integral equation, also known as Neumann-Poincaré operator. The very notion of spectral analysis has evolved along this path. Indeed, the quest for solving this specific singular integral equation, originally aimed at elucidating classical potential theory problems, has inspired and shaped the development of theoretical spectral analysis of linear transforms in XX-th century. We briefly touch some marking discoveries into the subject, with ample bibliographical references to both old, sometimes forgotten, texts and new contributions. It is remarkable that applications of the spectral analysis of the Neumann-Poincaré operator are still uncovered nowadays, with spectacular impacts on applied science. A few modern ramifications along these lines are depicted in our survey.

math.SP↗
arXiv2019

Eigenvalues of the Neumann-Poincare operator in dimension 3: Weyl's law and geometry

We consider the asymptotic properties of the eigenvalues of the Neumann-Poincare (NP) operator in three dimensions. The region $Ω\subset \mathbb{R}^3$ is bounded by a compact surface $Γ=\partial Ω$, with certain smoothness conditions imposed. The NP operator is defined by $ \mathcal{K}_Γ[ψ](\mathbf{x}):=\frac{1}{4π}\int_Γ\frac{\langle \mathbf{y}-\mathbf{x},\mathbf{n}(\mathbf{y})\rangle}{|\mathbf{x}-\mathbf{y}|^3}ψ(\mathbf{y})dS_{\mathbf{y}},$ where $dS_{\mathbf{y}}$ is the surface element and $\mathbf{n}(\mathbf{y})$ is the outer unit normal vector on $Γ$. The first-named author established earlier that the singular numbers $s_j(\mathcal{K}_Γ)$ of $\mathcal{K}_Γ$ and the ordered moduli of its eigenvalues $λ_j(\mathcal{K}_Γ)$ satisfy the Weyl law with coefficient expressed in geometric terms. Our main purpose here is to investigate the asymptotic behavior of positive and negative eigenvalues separately under the condition of infinite smoothness of the boundary $Γ$. These formulas are used, in particular, to obtain certain answers to the long-standing problem of the existence or finiteness of negative eigenvalues of $\mathcal{K}_Γ$.

math.FA↗
arXiv2019

Spectral properties of the Neumann-Poincaré operator in 3D elasticity

We consider the double layer potential (Neumann-Poincaré) operator appearing in 3-dimensional elasticity. We show that the recent result about the polynomial compactness of this operator for the case of a homogeneous media follows without additional calculations from previous considerations by Agranovich et.al., based upon pseudodifferential operators. Further on, we define the NP operator for the case of a nonhomogeneous isotropic media and show that its properties depend crucially on the character of non-homogeneity. If the Lamé parameters are constant along the boundary, the NP operator is still polynomially compact. On the other hand, if these parameters are not constant, two or more intervals of continuous spectrum may appear, so the NP operator ceases to be polynomially compact. However, after a certain modification, it becomes polynomially compact again. Finally, we evaluate the rate of convergence of discrete eigenvalues of the NP operator to the tips of the essential spectrum.

math.SP↗
arXiv2019

Convergence rate for eigenvalues of the elastic Neumann--Poincaré operator on smooth and real analytic boundaries in two dimensions

The elastic Neumann--Poincaré operator is a boundary integral operator associated with the Lamé system of linear elasticity. It is known that if the boundary of a planar domain is smooth enough, it has eigenvalues converging to two different points determined by Lamé parameters. We show that eigenvalues converge at a polynomial rate on smooth boundaries and the convergence rate is determined by smoothness of the boundary. We also show that they converge at an exponential rate if the boundary of the domain is real analytic.

math.SP↗
arXiv2018

Spectral structure of the Neumann--Poincaré operator on tori

We address the question whether there is a three-dimensional bounded domain such that the Neumann--Poincaré operator defined on its boundary has infinitely many negative eigenvalues. It is proved in this paper that tori have such a property. It is done by decomposing the Neumann--Poincaré operator on tori into infinitely many self-adjoint compact operators on a Hilbert space defined on the circle using the toroidal coordinate system and the Fourier basis, and then by proving that the numerical range of infinitely many operators in the decomposition has both positive and negative values.

math.SP↗
arXiv2018

Weyl's law for the eigenvalues of the Neumann--Poincaré operators in three dimensions: Willmore energy and surface geometry

We deduce eigenvalue asymptotics of the Neumann--Poincaré operators in three dimensions. The region $Ω$ is $C^{2, α}$ ($α>0$) bounded in ${\mathbf R}^3$ and the Neumann--Poincaré operator ${\mathcal K}_{\partialΩ} : L^2(\partial Ω) \rightarrow L^2(\partial Ω) $ is defined by $$ {\mathcal K}_{\partialΩ}[ψ]({\bf x}) := \frac{1}{4π} \int_{\partial Ω} \frac{\langle {\bf y}-{\bf x}, {\bf n}({\bf y}) \rangle}{|{\bf x}-{\bf y}|^3} ψ({\bf y})\; dS_{\bf y} $$ where $dS_{\bf y}$ is the surface element and ${\bf n}({\bf y})$ is the outer normal vector on $\partial Ω$. Then the ordering eigenvalues of the Neumann--Poincaré operator $λ_j ({\mathcal K}_{\partial Ω})$ satisfy $$ |λ_j({\mathcal K}_{\partial Ω})| \sim \Big\{\frac{3W(\partial Ω) - 2πχ(\partial Ω)}{128 π} \Big\}^{1/2} j^{-1/2}\quad \text{as}\ j \rightarrow \infty. $$ Here $W(\partial Ω)$ and $χ(\partial Ω)$ denote, respectively, the Willmore energy and the Euler charateristic of the boundary surface $\partialΩ$. This formula is the so-called Weyl's law for eigenvalue problems of Neumann--Poincaré operators.

math.SP↗
arXiv2018

The first Hadamard variation of Neumann-Poincaré eigenvalues on the sphere

The Neumann-Poincaré operator on the sphere has $\frac{1}{2(2k+1)}$, $k=0,1,2,\ldots$, as its eigenvalues and the corresponding multiplicity is $2k+1$. We consider the bifurcation of eigenvalues under deformation of domains, and show that Frechét derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincaré operator.

math.SP↗
arXiv2017

Elastic Neumann-Poincaré operators on three dimensional smooth domains: Polynomial compactness and spectral structure

We prove that the elastic Neumann--Poincaré operator defined on the smooth boundary of a bounded domain in three dimensions, which is known to be non-compact, is in fact polynomially compact. As a consequence, we prove that the spectrum of the elastic Neumann-Poincaré operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators and the spectral mapping theorem.

math.SP↗
arXiv2015

Notes on Feynman path integral-like methods of quantization on Riemannian manifolds

We propose an alternative method for Feynman path integrals on compact Riemannian manifolds. Our method employs action integrals along the shortest paths. In the case of rank 1 locally symmetric Riemannian manifolds, we prove the strong convergence of time slicing products of oscillatory integrals for low energy functions. Moreover, the strong limit includes Dewitt curvature $R/6$, where $R$ denotes the scalar curvature of a Riemannian manifold.

math-ph↗