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Approximation of fractals by manifolds and other graph-like spacesFeb 08 2018We define a distance between energy forms on a graph-like metric measure space and on a discrete weighted graph using the concept of quasi-unitary equivalence. We apply this result to metric graphs and graph-like manifolds (e.g. a small neighbourhood ... More
Stability of Ricci de Turck flow on Singular SpacesFeb 08 2018In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical ... More
Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wellsFeb 08 2018In this article, we consider the semiclassical Schr\"odinger operator $P = - h^{2} \Delta + V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $\lambda_{k} ( P )$ as $h \to 0$. First, we ... More
Sharp operator-norm asymptotics for linearised elastic plates with rapidly oscillating periodic propertiesFeb 07 2018We analyse a system of partial differential equations describing the behaviour of an elastic plate with periodic moduli in the two planar directions. We assume that the displacement gradients of the points of the plate are small enough for the equations ... More
On recovering Sturm-Liouville differential operators with deviating argumentFeb 07 2018We consider second-order functional differential operators with a constant delay. Properties of their spectral characteristics are obtained and a nonlinear inverse problem is studied, which consists in recovering the operators from their spectra. We establish ... More
Age-Minimal Online Policies for Energy Harvesting Sensors with Incremental Battery RechargesFeb 06 2018A sensor node that is sending measurement updates regarding some physical phenomenon to a destination is considered. The sensor relies on energy harvested from nature to transmit its updates, and is equipped with a finite $B$-sized battery to save its ... More
KdV hierarchy via Abelian coverings and operator identitiesJan 31 2018We establish precise spectral criteria for potential functions $V$ of reflectionless Schr\"odinger operators $L_V = -\partial_x^2 + V$ to admit solutions to the Korteweg de-Vries (KdV) hierarchy with $V$ as an initial value. More generally, our methods ... More
The Maxwell operator with periodic coefficients in a cylinderJan 31 2018In the paper we consider the Maxwell operator in a three-dimensional cylinder with coefficients periodic along the axis of a cylinder. It is proved that for cylinders with circular and rectangular cross-section the spectrum of the Maxwell operator is ... More
Compressed Anomaly Detection with Multiple Mixed ObservationsJan 31 2018We consider a collection of independent random variables that are identically distributed, except for a small subset which follows a different, anomalous distribution. We study the problem of detecting which random variables in the collection are governed ... More
Preserving of the unconditional basis property under non-self-adjoint perturbations of self-adjoint operatorsJan 29 2018Let $T$ be a self-adjoint operator in a Hilbert space $H$ with domain $\mathcal D(T)$. Assume that the spectrum of $T$ is confined in the union of disjoint intervals $\Delta_k =[\alpha_{2k-1},\alpha_{2k}]$, $k\in \mathbb{Z}$, and $$ \alpha_{2k+1}-\alpha_{2k} ... More
Rellich-Kondrakov embedding of the Laplacian resolvent on the torusJan 27 2018This paper proves that the domain of the Laplacian, $\DEL,$ on a closed Riemannian manifold, $(M,g),$ is compactly embedded in $L^{2} (M) .$ Particularly, the resolvent of the Laplacian, $(\DEL + 1)^{-1},$ is shown to be compactly embedded on the torus. ... More
Fast binary embeddings, and quantized compressed sensing with structured matricesJan 26 2018This paper deals with two related problems, namely distance-preserving binary embeddings and quantization for compressed sensing . First, we propose fast methods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$, associated with the Euclidean ... More
Limit Operators, Compactness and Essential Spectra on Bounded Symmetric DomainsJan 25 2018This paper is a follow-up to a recent article about the essential spectrum of Toeplitz operators acting on the Bergman space over the unit ball. As mentioned in the said article, some of the arguments can be carried over to the case of bounded symmetric ... More
General and Refined Montgomery LemmataJan 23 2018Montgomery's Lemma on the torus $\mathbb{T}^d$ states that a sum of $N$ Dirac masses cannot be orthogonal to many low-frequency trigonometric functions in a quantified way. We provide an extension to general manifolds that also allows for positive weights: ... More
Code-Frequency Block Group Coding for Anti-Spoofing Pilot Authentication in Multi-Antenna OFDM SystemsJan 23 2018A pilot spoofer can paralyze the channel estimation in multi-user orthogonal frequency-division multiplexing (OFD- M) systems by using the same publicly-known pilot tones as legitimate nodes. This causes the problem of pilot authentication (PA). To solve ... More
Weighted local Weyl laws for elliptic operatorsJan 22 2018Let $A$ be an elliptic pseudo-differential operator of order $m$ on a closed manifold $\mathcal{X}$ of dimension $n>0$, formally positive self-adjoint with respect to some positive smooth density $d\mu_\mathcal{X}$. Then, the spectrum of $A$ is made up ... More
A Kotel'nikov Representation for WaveletsJan 17 2018This paper presents a wavelet representation using baseband signals, by exploiting Kotel'nikov results. Details of how to obtain the processes of envelope and phase at low frequency are shown. The archetypal interpretation of wavelets as an analysis with ... More
An Elementary Dyadic Riemann HypothesisJan 15 2018The connection zeta function of a finite abstract simplicial complex G is defined as zeta_L(s)=sum_x 1/lambda_x^s, where lambda_x are the eigenvalues of the connection Laplacian L defined by L(x,y)=1 if x and y intersect and 0 else. (I) As a consequence ... More
On Partly Overloaded Spreading Sequences with Variable Spreading FactorJan 12 2018Future wireless communications systems are expected to support multi-service operation, i.e. especially multi-rate as well as multi-level quality of service (QoS) requirements. This evolution is mainly driven by the success of the Internet of Things (IoT) ... More
Localised modes due to defects in high contrast periodic media via two-scale homogenizationJan 10 2018The spectral problem for an infinite periodic medium perturbed by a compact defect is considered. For a high contrast small $\ve$-size periodicity and a finite size defect we consider the critical $\ve^2$-scaling for the contrast. We employ (high contrast) ... More
On the Szegő formulas for truncated Wiener-Hopf operatorsJan 08 2018We consider functions of multi-dimensional versions of truncated Wiener--Hopf operators with smooth symbols, and study the scaling asymptotics of their traces. The obtained results extend the asymptotic formulas obtained by H. Widom in the 1980's to non-smooth ... More
On Absolutely Norm attaining OperatorsJan 08 2018We give necessary and sufficient conditions for a bounded operator defined between complex Hilbert spaces to be absolutely norm attaining. We discuss structure of such operators in the case of self-adjoint and normal operators separately. Finally, we ... More
Magnetic oscillations in a model of grapheneJan 05 2018We consider a quantum graph as a model of graphene in constant magnetic field and describe the density of states in terms of relativistic Landau levels satisfying a Bohr--Sommerfeld quantization condition. That provides semiclassical corrections (with ... More
Spectral properties of 2D Pauli operators with almost periodic electromagnetic fieldsJan 04 2018We consider a 2D Pauli operator with almost periodic field $b$ and electric potential $V$. First, we study the ergodic properties of $H$ and show, in particular, that its discrete spectrum is empty if there exists a magnetic potential which generates ... More
A new representation of Hankel operators and its spectral consequencesJan 02 2018We describe a new representation of Hankel operators $H$ as pseudo-differential operators $A$ in the space of functions defined on the whole axis. The amplitudes of such operators $A$ have a very special structure: they are products of functions of a ... More
Ground-state energy of one-dimensional free Fermi gases in the thermodynamic limitDec 30 2017We study the ground-state energy of one-dimensional, non-interacting fermions subject to an external potential in the thermodynamic limit. To this end, we fix some (Fermi) energy $\nu>0$, confine fermions with total energy below $\nu$ inside the interval ... More
Maximizing Riesz means of anisotropic harmonic oscillatorsDec 29 2017We consider problems related to the asymptotic minimization of eigenvalues of anisotropic harmonic oscillators in the plane. In particular we study Riesz means of the eigenvalues and the trace of the corresponding heat kernels. The eigenvalue minimization ... More
Spectral properties of the 2+1 fermionic trimer with contact interactionsDec 29 2017Jan 08 2018We qualify the main features of the spectrum of the Hamiltonian of point interaction for a three-dimensional quantum system consisting of three point-like particles, two identical fermions, plus a third particle of different species, with two-body interaction ... More
Spectral curves for the rogue wavesDec 26 2017Here we find the spectral curves, corresponding to the known rational or quasi-rational solutions of AKNS hierarchy equations, ultimately connected with the modeling of the rogue waves events in the optical waveguides and in hydrodynamics. We also determine ... More
Inverse dynamic and spectral problems for the one-dimensional Dirac system on a finite treeDec 25 2017We consider inverse dynamic and spectral problems for the one dimensional Dirac system on a finite tree. Our aim will be to recover the topology of a tree (lengths and connectivity of edges) as well as matrix potentials on each edge. As inverse data we ... More
A Low-Cost Robust Distributed Linearly Constrained Beamformer for Wireless Acoustic Sensor Networks with Arbitrary TopologyDec 23 2017We propose a new robust distributed linearly constrained beamformer (BF) which utilizes a set of linear equality constraints to reduce the cross power spectral density matrix to a block-diagonal form. The proposed BF has a convenient objective function ... More
Wireless Energy Beamforming Using Signal Strength FeedbackDec 10 2017Multiple antenna techniques, that allow energy beamforming, have been looked upon as a possible candidate for increasing the efficiency of the transfer process between the energy transmitter (ET) and the energy receiver (ER) in wireless energy transfer. ... More
Sparse Randomized Kaczmarz for Support Recovery of Jointly Sparse Corrupted Multiple Measurement VectorsNov 07 2017Jan 03 2018While single measurement vector (SMV) models have been widely studied in signal processing, there is a surging interest in addressing the multiple measurement vectors (MMV) problem. In the MMV setting, more than one measurement vector is available and ... More
A note on Kuttler-Sigillito's inequalitiesSep 28 2017We provide several inequalities between eigenvalues of some classical eigenvalue problems on domains with $C^2$ boundary in complete Riemannian manifolds. A key tool in the proof is the generalized Rellich identity on a Riemannian manifold. Our results ... More
Shannon information storage in noisy phase-modulated fringes and fringe-data compression by phase-shifting algorithmsAug 22 2017Nov 08 2017Optical phase-modulated fringe-patterns are usually digitized with XxY pixels and 8 bits/pixel (or higher) gray-levels. The digitized 8 bits/pixel are raw-data bits, not Shannon information bits. Here we show that noisy fringe-patterns store much less ... More
Integrable systems, symmetries and quantizationApr 21 2017Oct 16 2017These notes correspond to a mini-course given at the Poisson 2016 conference in Geneva. Starting from classical integrable systems in the sense of Liouville, we explore the notion of non-degenerate singularity and expose recent research in connection ... More
Corrigendum to "A perturbation of the Dunkl harmonic oscillator on the line", arXiv:1412.4655Apr 16 2017Jun 26 2017We correct the second main theorem of the previous paper "A perturbation of the Dunkl harmonic oscillator on the line", by the first two authors. The corrections concern mainly certain estimates, which were also improved by adding more methods.
Asymptotic behaviour of cuboids optimising Laplacian eigenvaluesMar 29 2017Oct 10 2017We prove that in dimension $n \geq 2$, within the collection of unit measure cuboids in $\mathbb{R}^n$ (i.e. domains of the form $\prod_{i=1}^{n}(0, a_n)$), any sequence of minimising domains $R_k^\mathcal{D}$ for the Dirichlet eigenvalues $\lambda_k$ ... More
Existence of metrics maximizing the first eigenvalue on closed surfacesMar 03 2017May 16 2017We prove that for closed surfaces of fixed topological type, orientable or non-orientable, there exists a unit volume metric, smooth away from finitely many conical singularities, that maximizes the first eigenvalue of the Laplace operator among all unit ... More
Large Deviations and the Lukic ConjectureMar 02 2017Mar 17 2017We use the large deviation approach to sum rules pioneered by Gamboa, Nagel and Rouault to prove higher order sum rules for orthogonal polynomials on the unit circle. In particular, we prove one half of a conjectured sum rule of Lukic in the case of two ... More
Pair correlations and equidistributionDec 16 2016A deterministic sequence of real numbers in the unit interval is called \emph{equidistributed} if its empirical distribution converges to the uniform distribution. Furthermore, the limit distribution of the pair correlation statistics of a sequence is ... More
Gram Determinants of Real Binary TensorsDec 13 2016A binary tensor consists of $2^n$ entries arranged into hypercube format $2 \times 2 \times \cdots \times 2$. There are $n$ ways to flatten such a tensor into a matrix of size $2 \times 2^{n-1}$. For each flattening, $M$, we take the determinant of its ... More
Existence of guided waves due to a lineic perturbation of a 3D periodic mediumDec 08 2016In this note, we exhibit a three dimensional structure that permits to guide waves. This structure is obtained by a geometrical perturbation of a 3D periodic domain that consists of a three dimensional grating of equi-spaced thin pipes oriented along ... More
A general trace formula for the differential operator on a segment with the last coefficient perturbed by a finite signed measureDec 07 2016A first order trace formula is obtained for a regular differential operator perturbed by a finite signed measure multiplication operator.
On Dirac operators with electrostatic δ-shell interactions of critical strengthDec 07 2016In this paper we prove that the Dirac operator with an electrostatic $\delta$-shell interaction of critical strength $\eta = \pm 2$ supported on a $C^2$-smooth compact surface $\Sigma$ is self-adjoint in $L^2(\mathbb{R}^3;\mathbb{C}^4)$ and we describe ... More
Lower bounds for the number of nodal domains for sums of two distorted plane waves in non-positive curvatureDec 06 2016In this paper, we will consider generalised eigenfunctions of the Laplacian on some surfaces of infinite area. We will be interested in lower bounds on the number of nodal domains of such eigenfunctions which are included in a given bounded set. We will ... More
Discrete spectrum of interactions concentrated near conical surfacesDec 06 2016We study the spectrum of two kinds of operators involving a conical geometry: the Dirichlet Laplacian in conical layers and Schr\"odinger operators with attractive $\delta$-interactions supported by infinite cones. Under the assumption that the cones ... More
Inequalities on the spectral radius, operator norm and numerical radius of Hadamard weighted geometric mean of positive kernel operatorsDec 06 2016Recently, several authors have proved inequalities on the spectral radius $\rho$, operator norm $\|\cdot\|$ and numerical radius of Hadamard products and ordinary products of non-negative matrices that define operators on sequence spaces, or of Hadamard ... More
Bounds on the joint and generalized spectral radius of Hadamard geometric mean of bounded sets of positive kernel operatorsDec 06 2016Let $\Psi _1, \ldots \Psi _m$ be bounded sets of positive kernel operators on a Banach function space $L$. We prove that for the generalized spectral radius $\rho$ and the joint spectral radius $\hat{\rho}$ the inequalities $$\rho \left(\Psi _1 ^{\left( ... More
Logarithmic convexity of fixed points of stochastic kernel operatorsDec 06 2016In this article we prove results on logaritmic convexity of fixed points of stochastic kernel operators. These results are expected to play a key role in the economic application to strategic market games.
On the Bonsall cone spectral radius and the approximate point spectrumDec 06 2016We study the Bonsall cone spectral radius and the approximate point spectrum of (in general non-linear) positively homogeneous, bounded and supremum preserving maps, defined on a max-cone in a given normed vector lattice. We prove that the Bonsall cone ... More
Formulas of Szegő type for the periodic Schrödinger operatorDec 06 2016We prove asymptotic formulas of Szeg\H{o} type for the periodic Schr\"odinger operator $H=-\frac{d^2}{dx^2}+V$ in dimension one. Admitting fairly general functions $h$ with $h(0)=0$, we study the trace of the operator $h(\chi_{(-\alpha,\alpha)}\chi_{(-\infty,\mu)}(H)\chi_{(-\alpha,\alpha)})$ ... More
A family of degenerate elliptic operators: maximum principle and its consequencesDec 06 2016In this paper we investigate the validity and the consequences of the maximum principle for degenerate elliptic operators whose higher order term is the sum of "k" eigenvalues of the Hessian. In particular we shed some light on some very unusual phenomena ... More
Resolvent expansions for the Schrödinger operator on the discrete half-lineDec 05 2016Simplified models of transport in mesoscopic systems are often based on a small sample connected to a finite number of leads. The leads are often modelled using the Laplacian on the discrete half-line $\mathbb N$. Detailed studies of the transport near ... More
The First Eigenvalue for the Bi-Beltrami-Laplacian on Minimal Isoparametric Hypersurfaces of $\mathbb{S}^{n+1}(1)$Dec 05 2016In this paper, we investigate the first eigenvalues of two closed eigenvalue problems of the bi-Beltrami-Laplacian on minimal embedded isoparametric hypersurface in the unit sphere $\mathbb{S}^{n+1}(1)$. Although many mathematicians want to derive the ... More
Nikishin systems on star-like sets: ratio asymptotics of the associated multiple orthogonal polynomialsDec 04 2016We investigate the ratio asymptotic behavior of the sequence $(Q_{n})_{n=0}^{\infty}$ of multiple orthogonal polynomials associated with a Nikishin system of $p\geq 1$ measures that are compactly supported on the star-like set of $p+1$ rays $S_{+}=\{z\in\mathbb{C}: ... More
Level repulsion for Schroedinger operators with singular continuous spectrumDec 04 2016We describe a family of half-line continuum Schroedinger operators with purely singular continuous essential spectrum, exhibiting asymptotic strong level repulsion (known as clock behavior). This follows from the convergence of the renormalized continuum ... More
Peculiar spectral statistics of ensembles of branched polymersDec 03 2016The spectral statistics of ensembles of exponentially weighted full binary trees and $p$-branching star graphs is investigated. It is shown that spectral densities demonstrate peculiar ultrametric structure typical for sparse graphs. In particular, the ... More
The discrete Pompeiu problem on the planeDec 01 2016We say that a finite subset $E$ of the Euclidean plane $\mathbb{R}^2$ has the discrete Pompeiu property with respect to isometries (similarities), if, whenever $f:\mathbb{R}^2\to \mathbb{C}$ is such that the sum of the values of $f$ on any congruent (similar) ... More
Anderson localization for weakly interacting multi-particle models in the continuumNov 30 2016For the weakly interacting one-dimensional multi-particle Anderson model in the continuum space of configurations, we prove the spectral exponential and the strong dynamical localization. The results require the interaction amplitude to be sufficiently ... More
Distribution of the nodal sets of eigenfunctions on analytic manifoldsNov 30 2016The nodal set of the Laplacian eigenfunction has co-dimension one and has finite hypersurface measure on a compact Riemannian manifold. In this paper, we investigate the distribution of the nodal sets of eigenfunctions, when the metric on the manifold ... More
A Variation on the Donsker-Varadhan Inequality for the Principial EigenvalueNov 28 2016The purpose of this short note is to give a variation on the classical Donsker-Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain $\Omega$ by the largest mean first exit time of the associated ... More
Three-body problem in 3D space: ground state, (quasi)-exact-solvabilityNov 24 2016We study aspects of the quantum and classical dynamics of a $3$-body system in 3D space with interaction depending only on mutual distances. The study is restricted to solutions in the space of relative motion which are functions of mutual distances only. ... More
Local Decay for Weak Interactions with Massless ParticlesNov 23 2016We consider a mathematical model for the weak decay of the intermediate boson $Z^0$ into neutrinos and antineutrinos. We prove that the total Hamiltonian has a unique ground state in Fock space and we establish a limiting absorption principle, local decay ... More
Limits of orbifold $0$-spectra under collapsingNov 23 2016We consider the behavior of the eigenvalue spectrum of the Laplacian of a connected sum of two Riemannian orbifolds as one of the orbifolds in the pair is collapsed to a point. We show that the limit of the eigenvalue spectrum of the connected sum equals ... More
Sharp lower bounds on the spectral radius of uniform hypergraphs concerning degreesNov 22 2016Let $\mathcal{A}(H)$ and $\mathcal{Q}(H)$ be the adjacency tensor and signless Laplacian tensor of an $r$-uniform hypergraph $H$. Denote by $\rho(H)$ and $\rho(\mathcal{Q}(H))$ the spectral radii of $\mathcal{A}(H)$ and $\mathcal{Q}(H)$, respectively. ... More
A matricial view of the Karpelevi{č} TheoremNov 21 2016The question of the exact region in the complex plane of the possible single eigenvalues of all n-by-n stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevic in 1951 after a partial result by Dmitriev and Dynkin in 1946. The Karpelevic ... More
A matricial view of the Karpelevič TheoremNov 21 2016Dec 01 2016The question of the exact region in the complex plane of the possible single eigenvalues of all $n$-by-$n$ stochastic matrices was raised by Kolmogorov in 1937 and settled by Karpelevi\v{c} in 1951 after a partial result by Dmitriev and Dynkin in 1946. ... More
A spectral characterization of geodesic balls in non-compact rank one symmetric spacesNov 18 2016In constant curvatures spaces, there are a lot of characterizations of geodesic balls as optimal domain for shape optimization problems. Although it is natural to expect similar characterizations in rank one symmetric spaces, very few is known in this ... More
Moler Tensors and Their PropertiesNov 18 2016Moler matrices are an important class of test matrices in MATLAB. They are symmetric and positive definite, yet each Moler matrix has a very small positive eigenvalue. In this paper, we extend Moler matrices to Moler tensors and Moler-like tensors, and ... More
Moler Tensors and Their PropertiesNov 18 2016Nov 26 2016Moler matrices are named after the founder of MATLAB -- Cleve Moler, and form a class of test matrices in MATLAB. They are symmetric and positive definite, yet each Moler matrix has a very small positive eigenvalue. In this paper, we extend Moler matrices ... More
Equidistribution of random waves on small ballsNov 18 2016Following \cite{Ha2} by the first-named author, we continue our investigation of the equidistribution, at small scale, of random Laplacian eigenfunctions on a compact manifold $\mathbb{M}$. First we generalise the small scale expectation and variance ... More
Singular Values of the Attenuated Photoacoustic Imaging OperatorNov 17 2016We analyse the ill-posedness of the photoacoustic imaging problem in the case of an attenuating medium. To this end, we introduce an attenuated photoacoustic operator and determine the asymptotic behaviour of its singular values. Dividing the known attenuation ... More
Asymptotic shape optimization for Riesz means of the Dirichlet Laplacian over convex domainsNov 17 2016For $\Omega \subset \mathbb{R}^n$, a convex and bounded domain, we study the spectrum of $-\Delta_\Omega$ the Dirichlet Laplacian on $\Omega$. For any $\Lambda>0$ and $\gamma \geq 0$ let $\Omega_{\Lambda, \gamma}(\mathcal{A})$ denote any extremal set ... More
Very weak solutions of wave equation for Landau Hamiltonian with irregular electromagnetic fieldNov 17 2016In this paper we study the Cauchy problem for the Landau Hamiltonian wave equation, with time dependent irregular (distributional) electromagnetic field and similarly irregular velocity. For such equations, we describe the notion of a `very weak solution' ... More
Pointwise Bounds for Steklov EigenfunctionsNov 16 2016Dec 01 2016Let $(\Omega,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial \Omega.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions ... More
Pointwise Bounds for Steklov EgenfunctionsNov 16 2016Let $(\Omega,g)$ be a compact, real-analytic Riemannian manifold with real-analytic boundary $\partial \Omega.$ The harmonic extensions of the boundary Dirchlet-to-Neumann eigenfunctions are called Steklov eigenfunctions. We show that the Steklov eigenfuntions ... More
Infinite and finite dimensional generalized Hilbert tensorsNov 16 2016In this paper, we introduce the concept of an $m$-order $n$-dimensional generalized Hilbert tensor $\mathcal{H}_{n}=(\mathcal{H}_{i_{1}i_{2}\cdots i_{m}})$, $$ \mathcal{H}_{i_{1}i_{2}\cdots i_{m}}=\frac{1}{i_{1}+i_{2}+\cdots i_{m}-m+a},\ a\in \mathbb{R}\setminus\mathbb{Z}^-;\ ... More
Dispersion Estimates for Spherical Schrödinger Equations with Critical Angular MomentumNov 16 2016We derive a dispersion estimate for one-dimensional perturbed radial Schr\"odinger operators where the angular momentum takes the critical value $l=-\frac{1}{2}$. We also derive several new estimates for solutions of the underlying differential equation ... More
Meromorphic continuation approach to noncommutative geometryNov 15 2016Following an idea of Nigel Higson, we develop a method for proving the existence of a meromor-phic continuation for some spectral zeta functions. The method is based on algebras of generalized differential operators. The main theorem states, under some ... More
Does Levinson's theorem count complex eigenvalues ?Nov 15 2016Yes it does ! Indeed an extended version of Levinson's theorem is proposed for a system involving complex eigenvalues. The perturbed system corresponds to a realization of the Schroedinger operator with inverse square potential on the half-line, while ... More
Cantor spectra of magnetic chain graphsNov 14 2016We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the ... More
Cantor spectra of magnetic chain graphsNov 14 2016Nov 21 2016We demonstrate a one-dimensional magnetic system can exhibit a Cantor-type spectrum using an example of a chain graph with $\delta$ coupling at the vertices exposed to a magnetic field perpendicular to the graph plane and varying along the chain. If the ... More
Passage through a potential barrier and multiple wellsNov 13 2016Consider the semiclassical limit, as the Planck constant $\hbar\ri 0$, of bound states of a one-dimensional quantum particle in multiple potential wells separated by barriers. We show that, for each eigenvalue of the Schr\"odinger operator, the Bohr-Sommerfeld ... More
Convolution generated by Riesz basesNov 11 2016In this note we discuss notions of convolutions generated by biorthogonal systems of elements of a Hilbert space. We describe the associated biorthogonal Fourier analysis, discuss properties of convolutions and give a number of examples.
Quantum heat tracesNov 11 2016We study new invariants of elliptic partial differential operators acting on sections of a vector bundle over a closed Riemannian manifold that we call the relativistic heat trace and the quantum heat traces. We obtain some reduction formulas expressing ... More
$L_\infty$-estimates for the torsion function and $L_\infty$-growth of semigroups satisfying Gaussian boundsNov 11 2016We investigate selfadjoint $C_0$-semigroups on Euclidean domains satisfying Gaussian upper bounds. Major examples are semigroups generated by second order uniformly elliptic operators with Kato potentials and magnetic fields. We study the long time behaviour ... More
Continuity of solutions of a class of fractional equationsNov 11 2016In practice many problems related to space/time fractional equations depend on fractional parameters. But these fractional parameters are not known a priori in modelling problems. Hence continuity of the solutions with respect to these parameters is important ... More
Dirac equation: the stationary and dynamical scattering problemsNov 10 2016We prove that for the radial Dirac equation with Coulomb-type potential the generalized dynamical scattering operator coincides with the corresponding generalized stationary scattering operator. This fact is a quantum mechanical analogue of ergodic results ... More
On a spectral theorem of WeylNov 10 2016We give a geometric proof of a theorem of Weyl on the continuous part of the spectrum of Sturm-Liouville operators on the half-line with asymptotically constant coefficients. Earlier proofs due to Weyl and Kodaira depend on special features of Green's ... More
Spectral and pseudospectral functions of various dimensions for symmetric systemsNov 10 2016The main object of the paper is a symmetric system $J y'-B(t)y=\l\D(t) y$ defined on an interval $\cI=[a,b) $ with the regular endpoint $a$. Let $\f(\cd,\l)$ be a matrix solution of this system of an arbitrary dimension and let $(Vf)(s)=\int\limits_\cI ... More
On the Diffusion Geometry of Graph Laplacians and ApplicationsNov 09 2016We study directed, weighted graphs $G=(V,E)$ and consider the (not necessarily symmetric) averaging operator $$ (\mathcal{L}u)(i) = -\sum_{j \sim_{} i}{p_{ij} (u(j) - u(i))},$$ where $p_{ij}$ are normalized edge weights. Given a vertex $i \in V$, we define ... More
Row Cones, Perron Similarities, and Nonnegative SpectraNov 08 2016In further pursuit of the diagonalizable \emph{real nonnegative inverse eigenvalue problem} (RNIEP), we study the relationship between the \emph{row cone} $\mathcal{C}_r(S)$ and the \emph{spectracone} $\mathcal{C}(S)$ of a Perron similarity $S$. In the ... More
Interrogating surface length spectra and quantifying isospectralityNov 07 2016This article is about inverse spectral problems for hyperbolic surfaces and in particular how length spectra relate to the geometry of the underlying surface. A quantitative answer is given to the following: how many questions do you need to ask a length ... More
Anomalies in local Weyl laws and applications to random topology at critical dimensionNov 07 2016Let $\mathcal{M}$ be a smooth manifold of positive dimension $n$ equipped with a smooth density $d\mu_{\mathcal{M}}$. Let $A$ be a polyhomogeneous elliptic pseudo-differential operator of positive order $m$ on $\mathcal{M}$ which is symmetric for the ... More
Final value problem for nonlinear space fractional diffusion equation with random noiseNov 07 2016This 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 use the trigonometric method in nonparametric regression associated ... More
Ultrarelativistic bound states in the shallow spherical wellNov 06 2016We determine approximate eigenvalues and eigenfunctions shapes for bound states in the $3D$ shallow spherical ultrarelativistic well. Existence thresholds for the ground state and first excited states are identified, both in the purely radial and orbitally ... More
Krein's trace formula for unitary operators and operator Lipschitz functions (English translation)Nov 05 2016The main result of this paper is a description of the space of functions on the unit circle, for which Krein's trace formula holds for arbitrary pairs of unitary operators with trace class difference. This space coincides with the space of operator Lipschitz ... More
Operator Lipschitz functions (English translation)Nov 05 2016The purpose of this survey is a comprehensive study of operator Lip\-schitz functions. A continuous function $f$ on the real line ${\Bbb R}$ os called operator Lipschitz if $\|f(A)-f(B)\|\le\operatorname{const}\|A-B\|$ for arbitrary self-adjoint operators ... More
L-Borderenergetic graphs and Normalized Laplacian EnergyNov 04 2016In this paper we present new L-borderenergetic graphs, this is, graphs which are L-noncospectral with Kn but have the same Laplacian energy. We also present some graphs which are noncospectral to respective normalized Laplacian energy and they have the ... More