Related papers: Orthonormal bases on $L^2(\mathbb{R}^+)$

We introduce the problem of constructing weighted complex projective 2-designs from the union of a family of orthonormal bases. If the weight remains constant across elements of the same basis, then such designs can be interpreted as…

Given an n-dimensional natural Hamiltonian L on a Riemannian or pseudo-Riemannian manifold, we call "extension" of L the n+1 dimensional Hamiltonian $H=\frac 12 p_u^2+\alpha(u)L+\beta(u)$ with new canonically conjugated coordinates…

Let $X$ be a complex normed space. Based on the right norm derivative $\rho_{_{+}}$, we define a mapping $\rho_{_{\infty}}$ by \begin{equation*} \rho_{_{\infty}}(x,y) = \frac1\pi\int_0^{2\pi}e^{i\theta}\rho_{_{+}}(x,e^{i\theta}y)d\theta…

Let X be a Riemannian symmetric space of non-compact type. We prove a theorem of holomorphic extension for eigenfunctions of the Laplace-Beltrami operator on X, by techniques from the theory of partial differential equations.

For general non-Hermitian random matrices $X$ and deterministic deformation matrices $A$, we prove that the local eigenvalue statistics of $A+X$ close to the typical edge points of its spectrum are universal. Furthermore, we show that under…

Suppose that $M$ is a compact Riemannian manifold with boundary and $u$ is an $L^2$-normalized Dirichlet eigenfunction with eigenvalue $\lambda$. Let $\psi$ be its normal derivative at the boundary. Scaling considerations lead one to expect…

This paper constructs polynomial bases that capture the structure of the de Rham complex with boundary conditions in disks and cylinders (both periodic and finite) in a way that respects rotational symmetry. The starting point is explicit…

In this paper, we consider Schr\"odinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that…

Factorization method is developed for a family of discretely spiked harmonic oscillators. Two sets of intertwining and ladder operators are presented to algebraically generate eigenstates with energies isomorphic to those of the ordinary…

We consider a constant-size subset of left and right eigenvectors of an $N\times N$ i.i.d. complex non-Hermitian matrix associated with the eigenvalues with pairwise distances at least $N^{-\frac12+\epsilon}$. We show that arbitrary…

For a family of elliptic operators with rapidly oscillating periodic coefficients, we study the convergence rates for Dirichlet eigenvalues and bounds of the normal derivatives of Dirichlet eigenfunctions. The results rely on an…

We propose and analyse numerical algorithms based on weighted least squares for the approximation of a real-valued function on a general bounded domain $\Omega \subset \mathbb{R}^d$. Given any $n$-dimensional approximation space $V_n…

Let $H$ be a Hermitian random matrix whose entries $H_{xy}$ are independent, centred random variables with variances $S_{xy} = \mathbb E|H_{xy}|^2$, where $x, y \in (\mathbb Z/L\mathbb Z)^d$ and $d \geq 1$. The variance $S_{xy}$ is…

We consider the discrete spectrum of the two-dimensional Hamiltonian $H=H_0+V$, where $H_0$ is a Schr\"odinger operator with a non-constant magnetic field $B$ that depends only on one of the spatial variables, and $V$ is an electric…

For a long time it has been a challenging goal to identify all orthogonal polynomial systems that occur as eigenfunctions of a linear differential equation. One of the widest classes of such eigenfunctions known so far, is given by…

We present an $L^2$-extension theorem with an estimate depending on the weight functions for domains in $\mathbb{C}$. When the Hartogs domain defined by the weight function is strictly pseudoconvex, this estimate is strictly sharper than…

M.Levitin and E.Shargorodsky purposed in a recent article, [math.SP/0212087], the use of the so called ``second order relative spectrum'', to find eigenvalues of self-adjoint operators in gaps of the essential spectrum. Let $M$ be a…

We analyze the (discrete) spectrum of the semirelativistic ``spinless-Salpeter'' Hamiltonian H = \beta \sqrt{m^2 + p^2} + V(r), beta > 0, where V(r) represents an attractive, spherically symmetric potential in three dimensions. In order to…

We present a general Fourier analytic technique for constructing orthonormal basis expansions of translation-invariant kernels from orthonormal bases of $\mathscr{L}_2(\mathbb{R})$. This allows us to derive explicit expansions on the real…

Recently the Gelfand-Tsetlin construction of orthogonal bases has been explicitly described for the spaces of k-homogeneous polynomial solutions of the Hodge-de Rham system in the Euclidean space R^m which take values in the space of…