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math.SP

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24 paper(s) to start with

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preprint2017arXiv

A partial inverse problem for the Sturm-Liouville operator on a star-shaped graph

The Sturm-Liouville operator on a star-shaped graph is considered. We assume that the potential is known a priori on all the edges except one, and study the partial inverse problem, which consists in recovering the potential on the remaining edge from the part of the spectrum. A constructive method is developed for the solution of this problem, based on the Riesz-basicity of some sequence of vector functions. The local solvability of the inverse problem and the stability of its solution are proved.

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preprint2013arXiv

Generalized Lawson tori and Klein bottles

Using Takahashi theorem we propose an approach to extend known families of minimal tori in spheres. As an example, the well-known two-parametric family of Lawson tau-surfaces including tori and Klein bottles is extended to a three-parametric family of tori and Klein bottles minimally immersed in spheres. Extremal spectral properties of the metrics on these surfaces are investigated. These metrics include i) both metrics extremal for the first non-trivial eigenvalue on the torus, i.e. the metric on the Clifford torus and the metric on the equilateral torus and ii) the metric maximal for the first non-trivial eigenvalue on the Klein bottle.

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preprint2016arXiv

Nodal Domains of Eigenvectors for $1$-Laplacian on Graphs

The eigenvectors for graph $1$-Laplacian possess some sort of localization property: On one hand, any nodal domain of an eigenvector is again an eigenvector with the same eigenvalue; on the other hand, one can pack up an eigenvector for a new graph by several fundamental eigencomponents and modules with the same eigenvalue via few special techniques. The Courant nodal domain theorem for graphs is extended to graph $1$-Laplacian for strong nodal domains, but for weak nodal domains it is false. The notion of algebraic multiplicity is introduced in order to provide a more precise estimate of the number of independent eigenvectors. A positive answer is given to a question raised in [{\sl K.~C. Chang, Spectrum of the $1$-Laplacian and Cheeger constant on graphs, J. Graph Theor., DOI: 10.1002/jgt.21871}], to confirm that the critical values obtained by the minimax principle may not cover all eigenvalues of graph $1$-Laplacian.

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preprint2016arXiv

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on the real hyperbolic space $\mathbb{H}^{n}$. It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

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preprint2017arXiv

On the location of maximal of solutions of Schrödinger's equation

We prove an inequality with applications to solutions of the Schrödinger equation. There is a universal constant $c>0$, such that if $Ω\subset \mathbb{R}^2$ is simply connected, $u:Ω\rightarrow \mathbb{R}$ vanishes on the boundary $\partial Ω$, and $|u|$ assumes a maximum in $x_0 \in Ω$, then $$ \inf_{y \in \partial Ω}{ \| x_0 - y\|} \geq c \left\| \frac{Δu}{u} \right\|^{-1/2}_{L^{\infty}(Ω)}.$$ It was conjectured by Pólya \& Szegő (and proven, independently, by Makai and Hayman) that a membrane vibrating at frequency $λ$ contains a disk of size $\sim λ^{-1/2}$. Our inequality implies a refined result: the point on the membrane that achieves the maximal amplitude is at distance $\sim λ^{-1/2}$ from the boundary. We also give an extension to higher dimensions (generalizing results of Lieb and Georgiev \& Mukherjee): if $u$ solves $-Δu = Vu$ on $Ω\subset \mathbb{R}^n$ with Dirichlet boundary conditions, then the ball $B$ with radius $\sim \|V\|_{L^{\infty}(Ω)}^{-1/2}$ centered at the point in which $|u|$ assumes a maximum is almost fully contained in $Ω$ in the sense that $|B \cap Ω| \geq 0.99 |B|.$

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preprint2016arXiv

Log-majorization of the moduli of the eigenvalues of a matrix polynomial by tropical roots

We show that the sequence of moduli of the eigenvalues of a matrix polynomial is log-majorized, up to universal constants, by a sequence of "tropical roots" depending only on the norms of the matrix coefficients. These tropical roots are the non-differentiability points of an auxiliary tropical polynomial, or equivalently, the opposites of the slopes of its Newton polygon. This extends to the case of matrix polynomials some bounds obtained by Hadamard, Ostrowski and Pólya for the roots of scalar polynomials. We also obtain new bounds in the scalar case, which are accurate for "fewnomials" or when the tropical roots are well separated.

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preprint2016arXiv

Geometric and spectral consequences of curvature bounds on tessellations

This is a chapter of a forthcoming Lecture Notes in Mathematics "Modern Approaches to Discrete Curvature" edited by L. Najman and P. Romon. It provides a survey on geometric and spectral consequences of curvature bounds. The geometric setting are tessellations of surfaces with finite and vanishing genus. We consider a curvature arising as an angular defect. Several of the results presented here have analogues in Riemannian geometry. In some cases one can go even beyond the Riemannian results and there also striking differences which shall be highlighted.

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preprint2016arXiv

Small volume expansion of the splitting of multiple Neumann Laplacian eigenvalues due to a grounded inclusion in two dimensions

The first terms of the small volume asymptotic expansion for the splitting of Neumann boundary condition Laplacian eigenvalues due to a grounded inclusion of size ε are derived. An explicit formula to compute the first term from the eigenvalues and eigenfunctions of the unperturbed domain, the inclusion size and position is given. As a consequence, when an eigenvalue of double multiplicity splits in two distinct eigenvalues, one decays like O(1/log(ε)), the other like O(ε^2).

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preprint2016arXiv

Cloaking by anomalous localized resonance for linear elasticity on a coated structure

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.

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preprint2016arXiv

An explicit formula for the transversal indices of the lifted Dolbeault operators

M. F. Atiyah proved that the index of a transversally elliptic operator relative to a free action can be computed by using indices of elliptic operators on the orbit manifold. In this paper, we derive an explicit formula for the transversal indices on S^1-bundles over complex projective spaces. Using this explicit formula and the index map, we prove Lefschetz formula for the n-dimensional complex projective space with the canonical action of T^{n + 1}.

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preprint2016arXiv

On the spectrum of operator families on discrete groups over minimal dynamical systems

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

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preprint2016arXiv

A spectral isoperimetric inequality for cones

In this note we investigate three-dimensional Schrödinger operators with $δ$-interactions supported on $C^2$-smooth cones, both finite and infinite. Our main results concern a Faber-Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.

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preprint2016arXiv

The Steklov spectrum and coarse discretizations of manifolds with boundary

We consider the class of compact n-dimensional Riemannian manifolds with cylindrical boundary, Ricci curvature bounded below by a given constant and injectivity radius bounded below by a positive constant, away from the boundary. For a manifold M of this class, we introduce a notion of discretization, leading to a graph with boundary which is roughly isometric to M, with constants depending only on the dimension and bounds on curvature and injectivity radius. In this context, we prove a uniform spectral comparison inequality between the Steklov eigenvalues of the manifold M and those of its discretization. Some applications to the construction of sequences of surfaces with boundary of fixed length and with arbitrarily large Steklov spectral gap are given. In particular, we obtain such a sequence for surfaces with connected boundary. The applications are based on the construction of graph-like surfaces which are obtained from sequences of graphs with good expansion properties.

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preprint2015arXiv

Logarithmic lower bound on the number of nodal domains

We prove that the number of nodal domains of a density one subsequence of eigenfunctions grows at least logarithmically with the eigenvalue on negatively curved `real Riemann surfaces'. The geometric model is the same as in prior joint work with Junehyuk Jung (arXiv:1310.2919, to appear in J. Diff. Geom), where the number of nodal domains was shown to tend to infinity, but without a specified rate. The proof of the logarithmic rate uses the new logarithmic scale quantum ergodicity results of Hezari-Riviere (arXiv:1411.4078) and X. Han (arXiv:1410.3911).

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preprint2016arXiv

On eigenvalue bounds for a general class of Sturm-Liouville operators

We consider Sturm-Liouville operators with measure-valued weight and potential, and positive, bounded diffusion coefficient which is bounded away from zero. By means of a local periodicity condition, which can be seen as a quantitative Gordon condition, we prove a bound on eigenvalues for the corresponding operator in $L_p$, for $1\leq p<\infty$. We also explain the sharpness of our quantitative bound, and provide an example for quasiperiodic operators.

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preprint2016arXiv

Scaling of Harmonic Oscillator Eigenfunctions and Their Nodal Sets Around the Caustic

We study the scaling asymptotics of the eigenspace projection kernels $Π_{\hbar, E}(x,y)$ of the isotropic Harmonic Oscillator $- \hbar ^2 Δ+ |x|^2$ of eigenvalue $E = \hbar(N + \frac{d}{2})$ in the semi-classical limit $\hbar \to 0$. The principal result is an explicit formula for the scaling asymptotics of $Π_{\hbar, E}(x,y)$ for $x,y$ in a $\hbar^{2/3}$ neighborhood of the caustic $\mathcal C_E$ as $\hbar \to 0.$ The scaling asymptotics are applied to the distribution of nodal sets of Gaussian random eigenfunctions around the caustic as $\hbar \to 0$. In previous work we proved that the density of zeros of Gaussian random eigenfunctions of $\hat{H}_{\hbar}$ have different orders in the Planck constant $\hbar$ in the allowed and forbidden regions: In the allowed region the density is of order $\hbar^{-1}$ while it is $\hbar^{-1/2}$ in the forbidden region. Our main result on nodal sets is that the density of zeros is of order $\hbar^{-\frac{2}{3}}$ in an $\hbar^{\frac{2}{3}}$-tube around the caustic. This tube radius is the `critical radius'. For annuli of larger inner and outer radii $\hbar^α$ with $0< α< \frac{2}{3}$ we obtain density results which interpolate between this

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preprint2016arXiv

Heat kernel coefficients on Kahler manifolds

Polterovich proved a remarkable closed formula for heat kernel coefficients of the Laplace operator on compact Riemannian manifolds involving powers of Laplacians acting on the distance function. In the case of Kähler manifolds, we prove a combinatorial formula for powers of the complex Laplacian and use it to derive an explicit graph theoretic formula for the numerics in heat coefficients as a linear combination of metric jets based on Polterovich's formula.

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preprint2016arXiv

Schrödinger operator with non-zero accumulation points of complex eigenvalues

We study Schrödinger operators $H=-Δ+V$ in $L^2(Ω)$ where $Ω$ is $\mathbb R^d$ or the half-space $\mathbb R_+^d$, subject to (real) Robin boundary conditions in the latter case. For $p>d$ we construct a non-real potential $V\in L^p(Ω)\cap L^{\infty}(Ω)$ that decays at infinity so that $H$ has infinitely many non-real eigenvalues accumulating at every point of the essential spectrum $σ_{\rm ess}(H)=[0,\infty)$. This demonstrates that the Lieb-Thirring inequalities for selfadjoint Schrödinger operators are no longer true in the non-selfadjoint case.

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preprint2016arXiv

A special structure of the scattering operator and infrared divergences in quantum electrodynamics

We assume that the unperturbed operators $A_0$ are known. Then, the fact that the scattering operators $S$ and the unperturbed operators $A_0$ are pairwise permutable provides some important information about the structure of the scattering operators. Using this information and the ideas from the theory of generalized wave operators, we present a new approach to the divergence problems in quantum electrodynamics. We show that the so called infrared divergences appeared because the deviations of the initial and final waves from the free waves were not taken into account.

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preprint2016arXiv

Stark resonances in a quantum waveguide with analytic curvature

We investigate the influence of an electric field on trapped modes arising in a two-dimensional curved quantum waveguide ${\bf Ω}$ i.e. bound states of the corresponding Laplace operator $-Δ\_{\bf Ω}$. Here the curvature of the guide is supposed to satisfy some assumptions of analyticity, and decays as $O(|s|^{-\varepsilon}), \varepsilon > 3$ at infinity. We show that under conditions on the electric field $ \bf F$, ${\bf H}(F):= -Δ\_{\bf Ω} + {\bf F}. {\bf x} $ has resonances near the discrete eigenvalues of $-Δ\_{\bf Ω}$.

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preprint2016arXiv

Operator Lipschitz Functions

The purpose of this survey article is a comprehensive study of operator Lipschitz functions. A continuous function $f$ on the real line ${\Bbb R}$ is called operator Lipschitz if $\|f(A)-f(B)\|\le{\rm const}\|A-B\|$ for arbitrary self-adjoint operators $A$ and $B$. We give sufficient conditions and necessary conditions for operator Lipschitzness. We also study the class of operator differentiable functions on ${\Bbb R}$. Then we consider operator Lipschitz functions on closed subsets of the plane as well as commutator Lipschitz functions on such subsets. Am important role is played by double operator integrals and Schur multipliers.

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preprint2016arXiv

Approximate solutions of inverse problems for nonlinear space fractional diffusion equations with randomly perturbed data

This paper is concerned with backward problem for nonlinear space fractional diffusion with additive noise on the right-hand side and the final value. To regularize the instable solution, we develop some new regularized method for solving the problem. In the case of constant coefficients, we use the truncation methods. In the case of perturbed time dependent coefficients, we apply a new quasi-reversibility method. We also show the convergence rate between the regularized solution and the sought solution under some a priori assumption on the sought solution.

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