Mathematics - Spectral Theory Publications (50)

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Mathematics - Spectral Theory Publications

This work establishes the Anderson localization in both the spectral exponential and the strong dynamical localization for the multi-particle Anderson tight-binding model with correlated but strongly mixing random external potential. The results are obtained near the lower edge of the spectrum of the multi-particle Hamiltonian. In particular, the exponential decay of the eigenfunctions is proved in the max-norm and the dynamical localization in the Hilbert-Schmidt norm. Read More


A measure $\mu$ on the unit circle $\mathbb{T}$ belongs to Steklov class $\mathcal{S}$ if its density $w$ with respect to the Lebesgue measure on $\mathbb{T}$ is strictly positive: $\inf_{\mathbb{T}} w > 0$. Let $\mu$, $\mu_{-1}$ be measures on the unit circle $\mathbb{T}$ with real recurrence coefficients $\{\alpha_k\}$, $\{-\alpha_k\}$, correspondingly. If $\mu \in \mathcal{S}$ and $\mu_{-1} \in \mathcal{S}$, then partial sums $s_k=\alpha_0+ \ldots + \alpha_k$ satisfy the discrete Muckenhoupt condition $\sup_{n > \ell\ge 0} \bigl(\frac{1}{n - \ell}\sum_{k=\ell}^{n-1} e^{2s_k}\bigr)\bigl(\frac{1}{n - \ell}\sum_{k=\ell}^{n-1} e^{-2s_k}\bigr) < \infty$. Read More


In statistical relational learning, knowledge graph completion deals with automatically understanding the structure of large knowledge graphs---labeled directed graphs---and predicting missing relationships---labeled edges. State-of-the-art embedding models propose different trade-offs between modeling expressiveness, and time and space complexity. We reconcile both expressiveness and complexity through the use of complex-valued embeddings and explore the link between such complex-valued embeddings and unitary diagonalization. Read More


We consider plasmon resonances and cloaking for the elastostatic system in $\mathbb{R}^3$ via the spectral theory of Neumann-Poincar\'e operator. We first derive the full spectral properties of the Neumann-Poincar\'e operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [8]. Read More


The Bessis-Moussa-Villani conjecture states that the trace of $exp(A-tB)$ is, as a function of the real variable t, the Laplace transform of a positive measure, where A and B are respectively a hermitian and positive semi-definite matrix. The long standing conjecture was recently proved by Stahl and streamlined by Eremenko. We report on a more concise yet self-contained version of the proof. Read More


We consider the energy-critical half-wave maps equation $$\partial_t \mathbf{u} + \mathbf{u} \wedge |\nabla| \mathbf{u} = 0$$ for $\mathbf{u} : [0,T) \times \mathbb{R} \to \mathbb{S}^2$. We give a complete classification of all traveling solitary waves with finite energy. The proof is based on a geometric characterization of these solutions as minimal surfaces with (not necessarily free) boundary on $\mathbb{S}^2$. Read More


We study the Strichartz estimates for Schr\"odinger operator $\mathcal{L}_V$ on metric cone $X$, where the metric cone $X=C(Y)=(0,\infty)_r\times Y$ and the cross section $Y$ is a compact $(n-1)$-dimensional Riemannian manifold $(Y,h)$. The equipped metric on $X$ is given by $g=dr^2+r^2h$, and let $\Delta_g$ be the Friedrich extension positive Laplacian on $X$ and $V=V_0 r^{-2}$ where $V_0\in\mathcal{C}^\infty(Y)$ such that $\Delta_h+V_0+(n-2)^2/4$ is a strictly positive operator on $L^2(Y)$. We establish the full range of the global-in-time Strichartz estimate without loss for the Sch\"odinger equation associated with the operator $\mathcal{L}_V=\Delta_g+V_0 r^{-2}$ except for the endpoint inhomogeneous inequality. Read More


For the analysis of the Schr\"odinger and related equations it is of central importance whether a unique continuation principle (UCP) holds or not. In continuum Euclidean space quantitative forms of unique continuation imply Wegner estimates and regularity properties of the integrated density of states (IDS) of Schr\"odinger operators with random potentials. For discrete Schr\"odinger equations on the lattice only a weak analog of the UCP holds, but it is sufficient to guarantee the continuity of the IDS. Read More


We derive a number of upper and lower bounds for the first nontrivial eigenvalue of a finite quantum graph in terms of the edge connectivity of the graph, i.e., the minimal number of edges which need to be removed to make the graph disconnected. Read More


We disprove a conjecture of Breuer-Last-Simon concerning the absolutely continuous spectrum of Jacobi matrices with coefficients that obey an $\ell^2$ bounded variation condition with step $q$. We prove existence of a.c. Read More


We consider Dirac operators with a Coulomb-type potential $V(x)\sim -\nu/|x|$, in the case where the Coulomb singularity is strong, that is $\sqrt3/2\leq\nu\leq1$ in units such that $mc^2=1$. This operator is not essentially self-adjoint but has a distinguished self-adjoint extension. In a first part we obtain new results on the domain of the distinguished self-adjoint extension, complementing previous works of Esteban and Loss. Read More


We study continuum Schr\"odinger operators on the real line whose potentials are comprised of two compactly supported square-integrable functions concatenated according to an element of the Fibonacci substitution subshift over two letters. We show that the Hausdorff dimension of the spectrum tends to one in the small-coupling and high-energy regimes, regardless of the shape of the potential pieces. Read More


We introduce a notion being a $k$-fold Lebesgue function for measure preserving transformations, where any $2$-fold Lebesgue function is just ordinary Lebesgue. We discuss how this new metrical isomorphisms invariant of dynamical systems is related to others classical notions in ergodic theory, mostly focusing on its spectral aspects. In particular, for transformations with sufficiently many multiple Lebesgue functions we treat the corresponding multiple analogs of very well-known problems of Banach and Rokhlin. Read More


We study inversion of the spherical Radon transform with centers on a sphere (the data acquisition set). Such inversions are essential in various image reconstruction problems arising in medical, radar and sonar imaging. In the case of radially incomplete data, we show that the spherical Radon transform can be uniquely inverted recovering the image function in spherical shells. Read More


We study the multi-particle Anderson model in the continuum and show that under some mild assumptions on the random external potential and the inter-particle interaction, for any finite number of particles, the multi-particle lower edges of the spectrum are almost surely constant in absence of ergodicity. We stress that this result is not quite obvious and has to be handled carefully. In addition, we prove the spectral exponential and the strong dynamical localization of the continuous multi-particle Anderson model at low energy. Read More


In 2002 Freiberg and Z\"ahle introduced and developed a harmonic calculus for measure-geometric Laplacians associated to continuous distributions. We show that their theory can be extended to encompass distributions with finite support and give a matrix representation for the resulting operators. In the case of a uniform discrete distribution we make use of this matrix representation to explicitly determine the eigenvalues and the eigenfunctions of the associated Laplacian. Read More


We show a fractal uncertainty principle with exponent $1/2-\delta+\epsilon$, $\epsilon>0$, for Ahflors-David regular subsets of $\mathbb R$ of dimension $\delta\in (0,1)$. This improves over the volume bound $1/2-\delta$, and $\epsilon$ is estimated explicitly in terms of the regularity constant of the set. The proof uses a version of techniques originating in the works of Dolgopyat, Naud, and Stoyanov on spectral radii of transfer operators. Read More


Following a number of recent studies of resolvent and spectral convergence of non-uniformly elliptic families of differential operators describing the behaviour of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a "defect": an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\l_0$ located in a gap of the limit spectrum for the periodic case. Read More


We prove that the elastic Neumann--Poincar\'e 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-Poincar\'e operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lam\'e parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators and the spectral mapping theorem. Read More


We introduce the notion of order-preserving multi-homogeneous mapping which allows to study Perron-Frobenius type theorems and nonnegative tensors in unified fashion. We prove a weak and strong Perron-Frobenius theorem for these maps and provide a Collatz-Wielandt principle for the maximal eigenvalue. Additionally, we propose a generalization of the power method for the computation of the maximal eigenvector and analyse its convergence. Read More


We address the computational spectral theory of Jacobi operators that are trace- class perturbations of the free Jacobi operator via the asymptotic properties of a connection coefficient matrix. In particular, we show that the computation of the spectrum can be reduced to a polynomial root finding problem, from a polynomial that is derived from the entries of a connection coefficient matrix. An explicit formula for the spectral measure of the operator is also derived from these entries. Read More


In this paper, we consider a backward in time problem for Ginzburg-Landau equation in multidimensional domain associated with some random data. The problem is ill-posed in the sense of Hadamard. To regularize the instable solution, we develop a new regularized method combined with statistical approach to solve this problem. Read More


We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in $\mathbb{Z}^d$. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. Read More


We say that a list of complex numbers is "realisable" if it is the spectrum of some (entrywise) nonnegative matrix. The Nonnegative Inverse Eigenvalue Problem (NIEP) is the problem of characterising all realisable lists. Although the NIEP remains unsolved, it has been solved in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. Read More


We consider the Dirichlet Pauli operator in bounded connected domains in the plane, with a semi-classical parameter. We show, in particular, that the ground state energy of this Pauli operator will be exponentially small as the semi-classical parameter tends to zero and estimate this decay rate. This extends our results, discussing the results of a recent paper by Ekholm--Kova\v{r}\'ik--Portmann, to include also non-simply connected domains. Read More


We study the spectral properties of curl, a linear differential operator of first order acting on differential forms of appropriate degree on an odd-dimensional closed oriented Riemannian manifold. In three dimensions its eigenvalues are the electromagnetic oscillation frequencies in vacuum without external sources. In general, the spectrum consists of the eigenvalue 0 with infinite multiplicity and further real discrete eigenvalues of finite multiplicity. Read More


We consider Laplace operators on periodic discrete graphs perturbed by guides, i.e., graphs which are periodic in some directions and finite in other ones. Read More


We consider Schr\"odinger operators with complex decaying potentials (in general, not from trace class) on the lattice. We determine trace formulae and estimate of eigenvalues and singular measure in terms of potentials. The proof is based on estimates of free resolvent and analysis of functions from Hardy space. Read More


Inside a fixed bounded domain $\Omega$ of the plane, we look for the best compact connected set $K$, of given perimeter, in order to maximize the first Dirichlet eigenvalue $\lambda_1(\Omega\setminus K)$. We discuss some of the qualitative properties of the maximizers, passing toward existence, regularity and geometry. Then we study the problem in specific domains: disks, rings, and, more generally, disks with several holes. Read More


We analyze a general class of difference operators $H_\varepsilon = T_\varepsilon + V_\varepsilon$ on $\ell^2(\varepsilon \mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well potential and $\varepsilon$ is a small parameter. We construct formal asymptotic expansions of WKB-type for eigenfunctions associated with the low lying eigenvalues of $H_\varepsilon$. These are obtained from eigenfunctions or quasimodes for the operator $H_\varepsilon$, acting on $L^2(\mathbb{R}^d)$, via restriction to the lattice $\varepsilon\mathbb{Z}^d$. Read More


We introduce and investigate a class of real semi-infinite banded Toeplitz matrices satisfying the condition that the spectra of their principal submatrices accumulate onto a real interval when the size of the submatrix grows to $\infty$. This class contains both Hermitian and non-Hermitian matrices. Using the symbol of a Toeplitz matrix we present necessary and sufficient conditions guaranteeing that a matrix belongs to this class. Read More


Inverse spectral problems are studied for the second order integro-differential operators on a finite interval. Properties of spectral characteristic are established, and the uniqueness theorem is proved for this class of inverse problems. Read More


Using the Kato-Rosenblum theorem, we describe the absolutely continuous spectrum of a class of weighted integral Hankel operators in $L^2(\mathbb R_+)$. These self-adjoint operators generalise the explicitly diagonalisable operator with the integral kernel $s^\alpha t^\alpha(s+t)^{-1-2\alpha}$, where $\alpha>-1/2$. Our analysis can be considered as an extension of J. Read More


We prove the theorem on the completeness of the root functions of the Schroedinger operator $L=-d^2/dx^2+p(x)$ on the semi-axis $\mathbb R_+$ with a complex--valued potential $p(x)$. It is assumed that the potential $p = q \pm ir$ is such that the real functions $q$ and $r$ are subject the conditions $$ q(x) \geqslant c r(x), \quad r(x) \geqslant c_0+ c_1 x^\alpha, \quad \alpha >0, $$ where the constants $c, \ c_0\in \mathbb R$, $c_1>0$ and $\arg(\pm i+c) < 2\alpha\pi/(2+\alpha)$. For the case of the Airy operator $L_c=-d^2/dx^2+cx$, $c=const$, this theorem imply the completeness of the system of the eigenfunctions of this operator if $|\arg c|<2\pi/3$. Read More


We prove Davies-Gaffney-Grigor'yan lemma for heat kernels of bounded discrete Hodge Laplacians on simplicial complexes. Read More


We give a complete description, provided with a mathematical proof, of the shape of the spectrum of the Hill operator with a PT-symmetric periodic optical potential. We prove that the second critical point, after which the real parts of the first and second bands disappear, is a number between 0.8884370025 and 0. Read More


We show that there is generically non-uniqueness for the anisotropic Calder\'on problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n $\ge$ 3, there exist in the conformal class of g an infinite number of Riemannian metrics $\tilde g$ metrics such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data $\Gamma$D and Neumann data $\Gamma$N are measured on disjoint sets and satisfy $\Gamma$D $\cup$ $\Gamma$N = $\partial$M. The conformal factors that lead to these non-uniqueness results for the anisotropic Calder\'on problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (M, g) and are associated to a natural but subtle gauge invariance of the anisotropic Calder\'on problem with data on disjoint sets. Read More


This paper is devoted to dimensional reductions via the norm resolvent convergence. We derive explicit bounds on the resolvent difference as well as spectral asymptotics. The efficiency of our abstract tool is demonstrated by its application on seemingly different PDE problems from various areas of mathematical physics; all are analysed in a unified manner now, known results are recovered and new ones established. Read More


We search for the best fit in Frobenius norm of $A \in {\mathbb C}^{m \times n}$ by a matrix product $B C^*$, where $B \in {\mathbb C}^{m \times r}$ and $C \in {\mathbb C}^{n \times r}$, $r \le m$ so $B = \{b_{ij}\}$, ($i=1, \dots, m$,~ $j=1, \dots, r$) definite by some unknown parameters $\sigma_1, \dots, \sigma_k$, $k << mr$ and all partial derivatives of $\displaystyle \frac{\delta b_{ij}}{\delta \sigma_l}$ are definite, bounded and can be computed analytically. We show that this problem transforms to a new minimization problem with only $k$ unknowns, with analytical computation of gradient of minimized function by all $\sigma$. The complexity of computation of gradient is only 4 times bigger than the complexity of computation of the function, and this new algorithm needs only $3mr$ additional memory. Read More


In this paper, we study the backward problem of determining initial condition for some class of nonlinear parabolic equations in multidimensional domain where data are given under random noise. This problem is ill-posed, i.e. Read More


We give the description of self-adjoint regular Dirac operators, on $[0, \pi]$, with the same spectra. Read More


In terms of triples of Banach spaces, we define a large class of boundary problems for ordinary differential equations (of arbitrary order) with singular coefficients. Read More


Let $\Omega$ be a bounded domain in $\mathbb{R}^n$, $n\geq 2$, and $V\in L^\infty(\Omega)$ be a potential function. Consider the following transmission eigenvalue problem for nontrivial $v, w\in L^2(\Omega)$ and $k\in\mathbb{R}_+$, \[(\Delta+k^2)v= 0 \quad \text{in } \Omega,\] \[(\Delta+k^2(1+V))w= 0 \quad \text{in } \Omega,\] \[w-v \in H^2_0(\Omega), \quad \lVert v \rVert_{L^2(\Omega)}=1. \] We show that the transmission eigenfunctions $v$ and $w$ carry the geometric information of $\mathrm{supp}(V)$. Read More


We study spectral properties of a class of global infinite order pseudo-differential operators. We obtain formulae concerning the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. Read More


We prove a limiting eigenvalue distribution theorem (LEDT) for suitably scaled eigenvalue clusters around the discrete negative eigenvalues of the hydrogen atom Hamiltonian formed by the perturbation by a weak constant magnetic field. We study the hydrogen atom Zeeman Hamiltonian $H_V(h,B) = (1/2)( - i h {\mathbf \nabla} - {\mathbf A}(h))^2 - |x|^{-1}$, defined on $L^2 (R^3)$, in a constant magnetic field ${\mathbf B}(h) = {\mathbf \nabla} \times {\mathbf A}(h)=(0,0,\epsilon(h)B)$ in the weak field limit $\epsilon(h) \rightarrow 0$ as $h\rightarrow{0}$. We consider the Planck's parameter $h$ taking values along the sequence $h=1/(N+1)$, with $N=0,1,2,\ldots$, and $N\rightarrow\infty$. Read More


We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schr\"odinger operator with an attractive $\delta$-interaction of a fixed strength the support of which is a star graph with finitely many edges of an equal length $L \in (0,\infty]$. Under the constraint of fixed number of the edges and fixed length of them, we prove that the lowest eigenvalue is maximized by the fully symmetric star graph. The proof relies on the Birman-Schwinger principle, properties of the Macdonald function, and on a geometric inequality for polygons circumscribed into the unit circle. Read More


This paper is concerned with regular approximations of spectra of singular discrete linear Hamiltonian systems with one singular endpoint. For any given self-adjoint subspace extension (SSE) of the corresponding minimal subspace, its spectrum can be approximated by eigenvalues of a sequence of induced regular SSEs, generated by the same difference expression on smaller finite intervals. It is shown that every SSE of the minimal subspace has a pure discrete spectrum, and the k-th eigenvalue of any given SSE is exactly the limit of the k-th eigenvalues of the induced regular SSEs; that is, spectral exactness holds, in the limit circle case. Read More


We complete the classical Schoenberg representation theorem for radial positive definite functions. We apply this result to study spectral properties of self-adjoint realizations of two- and three-dimensional Schr\"odinger operators with point interactions on a finite set. In particular, we prove that any realization has purely absolutely continuous non-negative spectrum. Read More


We consider minimal non-negative Jacobi operator with $p\times p-$matrix entries. Using the technique of boundary triplets and the corresponding Weyl functions, we describe the Friedrichs and Krein extensions of the minimal Jacobi operator. Moreover, we parameterize the set of all non-negative self-adjoint extensions in terms of boundary conditions. Read More