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preprint2020arXiv

Convexity properties of the difference over the real axis between the Steklov zeta functions of a smooth planar domain with $2π$ perimeter and of the unit disk

We consider the zeta function $ζ_Ω$ for the Dirichlet-to-Neumann operator of a simply connected planar domain $Ω$ bounded by a smooth closed curve of perimeter $2π$. We prove that $ζ_Ω''(0)\ge ζ_{\mathbb{D}}''(0)$ with equality if and only if $Ω$ is a disk where $\mathbb{D}$ denotes the closed unit disk. We also provide an elementary proof that for a fixed real $s$ satisfying $s\le-1$ the estimate $ζ_Ω''(s)\ge ζ_{\mathbb{D}}''(s)$ holds with equality if and only if $Ω$ is a disk. We then bring examples of domains $Ω$ close to the unit disk where this estimate fails to be extended to the interval $(0,2)$. Other computations related to previous works are also detailed in the remaining part of the text.

preprint2020arXiv

Misha Shubin. 1944 -- 2020

The article describes the biography and manifold contributions to research in mathematics of Mikhail Aleksandrovich Shubin.

preprint2020arXiv

High order isometric liftings and dilations

We show that a Hilbert space bounded linear operator has an $m$-isometric lifting for some integer $m\ge 1$ if and only if the norms of its powers grow polynomially. In analogy with unitary dilations of contractions, we prove that such operators also have an invertible $m$-isometric dilation. We also study $2$-isometric liftings of convex operators and $3$-isometric liftings of Foguel-Hankel operators.

preprint2020arXiv

Eigenvalue estimates for the one-particle density matrix

It is shown that the eigenvalues $λ_k, k=1, 2, \dots,$ of the one-particle density matrix satisfy the bound $λ_k\le C k^{-8/3}$ with a positive constant $C$.

preprint2020arXiv

The determinant of one-dimensional polyharmonic operators of arbitrary order

We obtain an explicit expression for the regularised spectral determinant of the polyharmonic operator $P_{n}=(-1)^{n} (\partial_x)^{2n}$ on $(0,T)$ with Dirichlet boundary conditions and $n$ a positive integer, and show that it satisfies the asymptotics $\log{(\det P_{n})} = -n^2 \log{n} + \left[\frac{7ζ(3)}{2π^2}+ \frac{3}{2}+\log\left(\frac{T}{4}\right)\right] n^2 + {\rm O}(n)$ for large $n$. This is a consequence of sharp upper and lower bounds for $\log{(\det P_{n})}$ valid for all $n$ and which coincide in the terms up to order $n$. These results form the basis to analyse more general operators with nonconstant coefficients and show that the corresponding determinants have a similar asymptotic behaviour.

preprint2020arXiv

Eigenvalue bounds for some classes of matrices associated with graphs

For a given complex square matrix $A$ with constant row sum, we establish two new eigenvalue inclusion sets. Using these bounds, first we derive bounds for the second largest and smallest eigenvalues of adjacency matrices of $k$-regular graphs. Then, we establish some bounds for the second largest and the smallest eigenvalues of the normalized adjacency matrices of graphs and the second smallest eigenvalue and the largest eigenvalue of the Laplacian matrices of graphs. Sharpness of these bounds are verified by examples.

preprint2020arXiv

Spectra of elliptic potentials and supersymmetric gauge theories

We study a relation between asymptotic spectra of the quantum mechanics problem with a four components elliptic function potential, the Darboux-Treibich-Verdier (DTV) potential, and the Omega background deformed N=2 supersymmetric SU(2) QCD models with four massive flavors in the Nekrasov-Shatashvili limit. The weak coupling spectral solution of the DTV potential is related to the instanton partition function of supersymmetric QCD with surface operator. There are two strong coupling spectral solutions of the DTV potential, they are related to strong coupling expansions of gauge theory prepotential at the magnetic and dyonic points in the moduli space. A set of duality transformations relate the two strong coupling expansions for spectral solution, and for gauge theory prepotential.

preprint2020arXiv

Concentration of quantum integrable eigenfunctions on a convex surface of revolution

Let $(S^2,g)$ be a convex surface of revolution and $H \subset S^2$ the unique rotationally invariant geodesic. Let $φ^\ell_m$ be the orthonormal basis of joint eigenfunctions of $Δ_g$ and $\partial_θ$, the generator of the rotation action. The main result is an explicit formula for the weak-* limit of the normalized empirical measures, $Σ_{m = -\ell}^\ell ||φ^\ell_m||^2_{L^2(H)} δ_{\frac{m}{\ell}}(c)$ on $[-1,1]$. The explicit formula shows that, asymptotically, the $L^2$ norms of restricted eigenfunctions are minimal for the zonal eigenfunction $m = 0$, maximal for Gaussian beams $m = \pm 1$, and exhibit a $(1 - c^2)^{-\frac{1}{2}}$ type singularity at the endpoints. For a pseudo-differential operator $B$ we also compute the limits of the normalized measures $\sum_{m = -\ell}^\ell \langle B φ^\ell_m , φ^\ell_m \rangle δ_{\frac{m}{\ell}}(c)$.

preprint2020arXiv

Maximizing the ratio of eigenvalues of non-homogeneous partially hinged plates

We study the spectrum of non-homogeneous partially hinged plates having structural engineering applications. A possible way to prevent instability phenomena is to maximize the ratio between the frequencies of certain oscillating modes with respect to the density function of the plate; we prove existence of optimal densities and we investigate their analytic expression. This analysis suggests where to locate reinforcing material within the plate; some numerical experiments give further information and support the theoretical results.

preprint2020arXiv

Lieb-Thirring constant on the sphere and on the torus

We prove on the 2D sphere and on the 2D torus the Lieb-Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.

preprint2020arXiv

Quantum trees which maximize higher eigenvalues are unbalanced

The isoperimetric problem of maximizing all eigenvalues of the Laplacian on a metric tree graph within the class of trees of a given average edge length is studied. It turns out that, up to rescaling, the unique maximizer of the $k$-th positive eigenvalue is the star graph with three edges of lengths $2 k - 1$, $1$ and $1$. This complements the previously known result that the first nonzero eigenvalue is maximized by all equilateral star graphs and indicates that optimizers of isoperimetric problems for higher eigenvalues may be less balanced in their shape -- an observation which is known from numerical results on the optimization of higher eigenvalues of Laplacians on Euclidean domains.

preprint2020arXiv

On the Robin spectrum for the hemisphere

We study the spectrum of the Laplacian on the hemisphere with Robin boundary conditions. It is found that the eigenvalues fall into small clusters around the Neumann spectrum, and satisfy a Szegő type limit theorem. Sharp upper and lower bounds for the gaps between the Robin and Neumann eigenvalues are derived, showing in particular that these are unbounded. Further, it is shown that except for a systematic double multiplicity, there are no multiplicities in the spectrum as soon as the Robin parameter is positive, unlike the Neumann case which is highly degenerate. Finally, the limiting spacing distribution of the desymmetrized spectrum is proved to be the delta function at the origin.

preprint2020arXiv

Lower bounds for fractal dimensions of spectral measures of the period doubling Schrödinger operator

It is shown that there exits a lower bound $α>0$ to the Hausdorff dimension of the spectral measures of the one-dimensional period doubling substitution Schrödinger operator, and, generically in the hull of such sequence, $α$ is also a lower bound to the upper packing dimension of spectral measures.

preprint2020arXiv

On Ambarzumyan-type Inverse Problems of Vibrating String Equations

We consider the inverse spectral theory of vibrating string equations. In this regard, first eigenvalue Ambarzumyan-type uniqueness theorems are stated and proved subject to separated, self-adjoint boundary conditions. More precisely, it is shown that there is a curve in the boundary parameters' domain on which no analog of it is possible. Necessary conditions of the $n$-th eigenvalue are identified, which allows to state the theorems. In addition, several properties of the first eigenvalue are examined. Lower and upper bounds are identified, and the areas are described in the boundary parameters' domain on which the sign of the first eigenvalue remains unchanged. This paper contributes to inverse spectral theory as well as to direct spectral theory.

preprint2020arXiv

A perturbation result of m-accretive linear operators in Hilbert spaces

A new sufficient condition is given for the sum of linear m-accretive operator and accretive operator one in a Hilbert space to be m-accretive. As an application, an extended result to the operator-norm error bound estimate for the exponential Trotter-Kato product formula is given.

preprint2020arXiv

The IDS and Asymptotic of the Largest Eigenvalue of Random Schrödinger Operators with Decaying Random Potential

In this work we obtain the integrated density of states for the Schrödinger operators with decaying random potentials acting on $\ell^2(\mathbb{Z}^d)$. We also study the asymptotic of the largest and smallest eigenvalues of its finite volume approximation

preprint2020arXiv

Log-dimensional bounds for the spectral measure of the disordered Holstein model

We demonstrate Hausdorff log dimensional bounds on the continuity of the spectral measure of the strongly disordered Holstein operator.

preprint2020arXiv

Discrete Laplacian in a half-space with a periodic surface potential I: Resolvent expansions, scattering matrix, and wave operators

We present a detailed study of the scattering system given by the Neumann Laplacian on the discrete half-space perturbed by a periodic potential at the boundary. We derive asymptotic resolvent expansions at thresholds and eigenvalues, we prove the continuity of the scattering matrix, and we establish new formulas for the wave operators. Along the way, our analysis puts into evidence a surprising relation between some properties of the potential, like the parity of its period, and the behaviour of the integral kernel of the wave operators.

preprint2020arXiv

Ballistic transport for one-dimensional quasiperiodic Schrödinger operators

In this paper, we show that one-dimensional discrete multi-frequency quasiperiodic Schrödinger operators with smooth potentials demonstrate ballistic motion on the set of energies on which the corresponding Schrödinger cocycles are smoothly reducible to constant rotations. The proof is performed by establishing a local version of strong ballistic transport on an exhausting sequence of subsets on which reducibility can be achieved by a conjugation uniformly bounded in the $\mathrm{C}^{\ell}$-norm. We also establish global strong ballistic transport under an additional integral condition on the norms of conjugation matrices. The latter condition is quite mild and is satisfied in many known examples.

preprint2020arXiv

Spectral properties of spiral-shaped quantum waveguides

We investigate properties of a particle confined to a hard-wall spiral-shaped region. As a case study we analyze in detail the Archimedean spiral for which the spectrum above the continuum threshold is absolutely continuous away from the thresholds. The subtle difference between the radial and perpendicular width implies, however, that in contrast to `less curved' waveguides, the discrete spectrum is empty in this case. We also discuss modifications such a multi-arm Archimedean spirals and spiral waveguides with a central cavity; in the latter case bound state already exist if the cavity exceeds a critical size. For more general spiral regions the spectral nature depends on whether they are `expanding' or `shrinking'. The most interesting situation occurs in the asymptotically Archimedean case where the existence of bound states depends on the direction from which the asymptotics is reached.

preprint2020arXiv

Spectral data characterization for the Sturm-Liouville operator on the star-shaped graph

The inverse spectral problems are studied for the Sturm-Liouville operator on the star-shaped graph and for the matrix Sturm-Liouville operator with the boundary condition in the general self-adjoint form. We obtain necessary and sufficient conditions of solvability for these two inverse problems, and also prove their local solvability and stability.

preprint2020arXiv

Quantum ergodicity and localization of plasmon resonances

We are concerned with the geometric properties of the surface plasmon resonance (SPR). SPR is a non-radiative electromagnetic surface wave that propagates in a direction parallel to the negative permittivity/dielectric material interface. It is known that the SPR oscillation is very sensitive to the material interface. However, we show that the SPR oscillation asymptotically localizes at places with high magnitude of curvature in a certain sense. Our work leverages the Heisenberg picture of quantization and quantum ergodicity first derived by Shnirelman, Zelditch, Colin de Verdière and Helffer-Martinez-Robert, as well as certain novel and more general ergodic properties of the Neumann-Poincaré operator to analyze the SPR field, which are of independent interest to the spectral theory and the potential theory.

preprint2020arXiv

The inverse problem for a spectral asymmetry function of the Schrödinger operator on a finite interval

For the Schrödinger equation $-d^2 u/dx^2 + q(x)u = λu$ on a finite $x$-interval, there is defined an "asymmetry function" $a(λ;q)$, which is entire of order $1/2$ and type $1$ in $λ$. Our main result identifies the classes of square-integrable potentials $q(x)$ that possess a common asymmetry function. For any given $a(λ)$, there is one potential for each Dirichlet spectral sequence.

preprint2020arXiv

Uniform resolvent estimates for the discrete Schrödinger operator in dimension three

In this note, we prove the uniform resolvent estimate of the discrete Schrödinger operator with dimension three. To do this, we show a Fourier decay of the surface measure on the Fermi surface.