Related papers: Heisenberg Uniqueness Pairs and the wave equation
We compute the bipartite entanglement properties of the spin-half square-lattice Heisenberg model by a variety of numerical techniques that include valence bond quantum Monte Carlo (QMC), stochastic series expansion QMC, high temperature…
The Heisenberg uncertainty principle shows that no one can specify the values of the non-commuting canonically conjugated variables simultaneously. However, the uncertainty relation is usually applied to two incompatible measurements. We…
In [FI09], Friedlander and Iwaniec studied the so-called Hyperbolic Prime Number Theorem, which asks for an infinitude of elements gamma in SL(2,Z) such that the norm squared |gamma|^2 = a^2 + b^2 + c^2 + d^2 = p, a prime. Under the…
In this paper we revisit nonnegative kernels in the first Heisenberg group $\He$, and in particular we further study the family $$K_\alpha(x,y,z)= \frac{|z|^{\alpha/2}}{\|(x,y,z)\|_{H}^{\alpha+1}}, \quad \alpha>0,$$ which was introduced in…
Let $\Lambda$ be a numerical semigroup, $\mathcal{C}\subseteq \mathbb{A}^n$ the monomial curve singularity associated to $\Lambda$, and $\mathcal{T}$ its tangent cone. In this paper we provide a sharp upper bound for the least positive…
We study the interplay between the recently defined concept of minimum homotopy area and the classical topic of self-overlapping curves. The latter are plane curves which are the image of the boundary of an immersed disk. Our first…
In this paper we study the quantitative isoperimetric inequality in the plane. We prove the existence of a set $\Omega$, different from a ball, which minimizes the ratio $\delta(\Omega)/\lambda^2(\Omega)$, where $\delta$ is the…
In this paper we consider the problem of determining an unknown pair $\lambda$, $\mu$ of piecewise constant Lam\'{e} parameters inside a three dimensional body from the Dirichlet to Neumann map. We prove uniqueness and Lipschitz continuous…
Let $\Gamma$ be an arbitrary $\mathbb{Z}^n$-periodic metric graph, which does not coincide with a line. We consider the Hamiltonian $\mathcal{H}_\varepsilon$ on $\Gamma$ with the action $-\varepsilon^{-1}{\mathrm{d}^2/\mathrm{d} x^2}$ on…
Given a domain $\Omega\subset\Bbb R^d$ with positive and finite Lebesgue measure and a discrete set $\Lambda\subset \Bbb R^d$, we say that $(\Omega, \Lambda)$ is a {\it frame spectral pair} if the set of exponential functions $\mathcal…
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane, using a multiscale sum of what is now known as Jones $\beta$-numbers, numbers measuring flatness in a given scale and location. This work was…
Let $\mathbb{H}$ be the sub-Riemannian Heisenberg group. That $\mathbb{H}$ supports a rich family of quasiconformal mappings was demonstrated by Kor\'{a}nyi and Reimann using the so-called flow method. Here we supply further evidence of the…
An interferometer - no matter how clever the design - cannot reveal both the wave and particle behavior of a quantum system. This fundamental idea has been captured by inequalities, so-called wave-particle duality relations (WPDRs), that…
In this paper, we define Mannheim partner curves in a three dimensional Lie group G with a bi-invariant metric. And then the main result in this paper is given as (Theorem 3.3): A curve {\alpha} with the Frenet apparatus…
We study infinite dimensional generalisations of the Heisenberg doubles of the Borel half of $U_q(sl(2))$ and of $U_q(osp(1|2))$ and find associated canonical elements which satisfy pentagon equation. The former reproduces the canonical…
The smallest nonzero eigenvalue of the normalized Laplacian matrix of a graph has been extensively studied and shown to have many connections to properties of the graph. We here study a generalization of this eigenvalue, denoted $\lambda(G,…
Heisenberg in 1929 introduced the "collapse of the wavepacket" into quantum theory. We review here an experiment at Berkeley which demonstrated several aspects of this idea. In this experiment, a pair of daughter photons was produced in an…
After formulating the pressure wave equation in half-space minus a crack with a zero Neumann condition on the top plane, we introduce a related inverse problem. That inverse problem consists of identifying the crack and the unknown forcing…
The explicit formula for the hyperbolic metric $\lambda_{\alpha,\,\beta,\,\gamma}(z)|dz|$ on the thrice-punctured sphere $\mathbb{P} \backslash \{z_1,\,z_2,\,z_3\}$ with singularities of order $\alpha,\,\beta,\,\gamma \leq 1$ with…
Based on the analytic model of Feshbach resonances in harmonic traps described in Phys. Rev. A 83, 030701 (2011) a Bose-Hubbard model is introduced that provides an accurate description of two atoms in an optical lattice at a Feshbach…