arXiv subjects
Publications and source records attributed to Kazunori Ando.
We prove a Rellich-Vekua type theorem for Schrödinger operators with exponentially decreasing potentials on a class of lattices including square, triangular, hexagonal lattices and their ladders. We also discuss the unique continuation theorem and the non-existence of eigenvalues embedded in the continuous spectrum.
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.
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.
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.
We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported and symmetric, we show that the S-matrix for all energies in any given open set in the continuous spectrum determines the potentials.
We consider the inverse scattering on the quantum graph associated with the hexagonal lattice. Assuming that the potentials on the edges are compactly supported, we show that the S-matrix for all energies in any open set in the continuous spectrum determines the potentials.
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.
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 $[-1/2,1/2]$.
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.
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.
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.
We study the inverse scattering for Schr{ö}dinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a boundary value problem on a finite part of the graph, and reconstruct scalar potentials as well as the graph structure from the knowledge of the S-matrix. In particular, we give a procedure for probing defects in hexagonal lattices (graphene).
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.
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.
We investigate anomalous localized resonance on the circular coated structure and cloaking related to it in the context of elasto-static systems. The structure consists of the circular core with constant Lamé parameters and the circular shell of negative Lamé parameters proportional to those of the core. We show that the eigenvalues of the Neumann-Poincaré operator corresponding to the structure converges to certain non-zero numbers determined by Lamé parameters and derive precise asymptotics of the convergence. We then show with estimates that cloaking by anomalous localized resonance takes place if and only if the dipole type source lies inside critical radii determined by the radii of the core and the shell.
We show that the eigenvalues of the Neumann-Poincaré operator on analytic boundaries of simply connected bounded planar domains tend to zero exponentially fast, and the exponential convergence rate is determined by the maximal Grauert radius of the boundary. We present a few examples of boundaries to show that the estimate is optimal.
We first investigate spectral properties of the Neumann-Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, its spectrum consists of eigenvalues which accumulates to two numbers determined by Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on disks and ellipses. We then investigate resonance occurring at eigenvalues and anomalous localized resonance at accumulation points of eigenvalues. We show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.
We study resonance for the Helmholz equation with a finite frequency in a plasmonic material of negative dielectric constant in two and three dimensions. We show that the quasi-static approximation is valid for diametrically small inclusions. In fact, we quantitatively prove that if the diameter of a inclusion is small compared to the loss parameter, then resonance occurs exactly at eigenvalues of the Neumann-Poincaré operator associated with the inclusion.