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

It is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by…

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by…

A complete variational treatment is provided for a family of spiked-harmonic oscillator Hamiltonians H = -d^2/dx^2 + B x^2 + lambda/x^alpha, B > 0, lambda > 0, for arbitrary alpha > 0. A compact topological proof is presented that the set S…

Let $g(x)=\chi_B(x)$ be the indicator function of a bounded convex set in $\Bbb R^d$, $d\geq 2$, with a smooth boundary and everywhere non-vanishing Gaussian curvature. Using a combinatorial appraoch we prove that if $d \neq 1 \mod 4$, then…

Vector is a physical quantity and it does not depend on any co-ordinate system. It need to be expanded in some basis for practical calculation and its components do depend on the chosen basis. The expansion in orthonormal basis is…

We consider the non-Hermitian Hamiltonian H= -\frac{d^2}{dx^2}+P(x^2)-(ix)^{2n+1} on the real line, where P(x) is a polynomial of degree at most n \geq 1 with all nonnegative real coefficients (possibly P\equiv 0). It is proved that the…

We study uncertainty principles for orthonormal bases and sequences in $L^2(\R^d)$. As in the classical Heisenberg inequality we focus on the product of the dispersions of a function and its Fourier transform. In particular we prove that…

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions -- the…

The main objective of this paper is to extend certain fundamental inequalities from a single function to a family of orthonormal systems. In the first part of the paper, we consider a non-negative, self-adjoint operator $L$ on $L^2(X,\mu)$,…

We study orthogonal polynomials for a weight function defined over a domain of revolution, where the domain is formed from rotating a two-dimensional region and goes beyond the quadratic domains. Explicit constructions of orthogonal bases…

Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd and let \T be a discrete subset of \Rd. We show that the exponentials $e_t(x):=exp(i2\pi tx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and only if the…

Suppose that $L(\partial_t,\partial_x)$ is a homogeneous constant coefficient strongly hyperbolic partial differential operator on ${\mathbb R}^{1+d}$ and $H$ is a characteristic hyperplane. Suppose that in a conic neighborhood of the…

There exist many ways to build an orthonormal basis of $\mathbb{R}^N$, consisting of the eigenvectors of the discrete Fourier transform (DFT). In this paper we show that there is only one such orthonormal eigenbasis of the DFT that is…

We show that any $N$-dimensional linear subspace of $L^2(\mathbb{T})$ admits an orthonormal system such that the $L^2$ norm of the square variation operator $V^2$ is as small as possible. When applied to the span of the trigonometric…

We consider an eigenvalue problem for an inverted one dimensional harmonic oscillator. We find a complete description for the eigenproblem in $C^{\infty}(\mathbb R)$. The eigenfunctions are described in terms of the confluent hypergeometric…

The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and…

We construct finite-dimensional approximations of solution spaces of divergence form operators with $L^\infty$-coefficients. Our method does not rely on concepts of ergodicity or scale-separation, but on the property that the solution space…

We study the sensitivity of the eigenvectors of random matrices, showing that even small perturbations make the eigenvectors almost orthogonal. More precisely, we consider two deformed Wigner matrices $W+D_1$, $W+D_2$ and show that their…

We consider orthogonal polynomials on the surface of a double cone or a hyperboloid of revolution, either finite or infinite in axis direction, and on the solid domain bounded by such a surface and, when the surface is finite, by…

Let $(M,g)$ be a compact Riemannian surface. Consider a family of $L^2$ normalized Laplace-Beltrami eigenfunctions, written in the semiclassical form $-h_j^2\Delta_g \phi_{h_j} = \phi_{h_j}$, whose eigenvalues satisfy $h h_j^{-1} \in (1, 1…