Related papers: Spectral-Variation Bounds in Hyperbolic Geometry
We adapt Stein's method to isoperimetric and geometric inequalities. The main challenge is the treatment of boundary terms. We address this by using an elliptic PDE with an oblique boundary condition. We apply our geometric formulation of…
We consider the sum of two large Hermitian matrices $A$ and $B$ with a Haar unitary conjugation bringing them into a general relative position. We prove that the eigenvalue density on the scale slightly above the local eigenvalue spacing is…
For smooth bounded domains in $\mathbb{R}$, we prove upper and lower $L^2$ bounds on the boundary data of Neumann eigenfunctions, and prove quasi-orthogonality of this boundary data in a spectral window. The bounds are tight in the sense…
This paper presents a fast and powerful method for the computation of eigenvalue bounds for Hessian matrices $\nabla^2 \varphi(x) $ of nonlinear functions $\varphi: U \subseteq R^n\rightarrow R$ on hyperrectangles $B \subset U$. The method…
In this article we establish optimal estimates for the first eigenvalue of Schr\"odinger operators on the d-dimensional unit sphere. These estimates depend on Lebsgue's norms of the potential, or of its inverse, and are equivalent to…
We develop a method for computing the Bogoliubov transformation experienced by a confined quantum scalar field in a globally hyperbolic spacetime, due to the changes in the geometry and/or the confining boundaries. The method constructs a…
Moved by the need for rigorous and reliable numerical tools for the analysis of peridynamic materials, the authors propose a model able to capture the dispersive features of nonlocal soliton-like solutions obtained by a peridynamic…
The recent development of spectral method has been praised for its high-order convergence in simulating complex physical problems. The combination of embedded boundary method and spectral method becomes a mainstream way to tackle…
Standard perturbation theory of eigenvalue problems consists of obtaining approximations of eigenmodes in the neighborhood of an operator where the corresponding eigenmode is known. Nevertheless, if the corresponding eigenmodes of several…
Achieving resolution in the sub-Rayleigh regime (superresolution) is one of the rapidly developing topics in quantum optics and metrology. Recently, it was shown that perfect measurement based on spatial mode demultiplexing (SPADE) in…
We develop exterior calculus approaches for partial differential equations on radial manifolds. We introduce numerical methods that approximate with spectral accuracy the exterior derivative $\mathbf{d}$, Hodge star $\star$, and their…
We prove trace inequalities for a self-adjoint operator on an abstract Hilbert space. These inequalities lead to universal bounds on spectral gaps and on moments of eigenvalues lambda_k that are analogous to those known for Schroedinger…
One of the great miracles of random matrix theory is that, in the $N \to \infty$ limit, many otherwise intractable matrix problems with horrendously complicated finite-$N$ expressions admit remarkably simple and elegant asymptotic…
In this article, we improve the classical Bukhgeim-Klibanov method presented in [1],which can be used to prove the conditional stability of inverse source problem for a hyperbolic equation from the measurement on the subboundary. A major…
We study the long time behavior of radial solutions to nonlinear Schr\"{o}dinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic…
We consider bounds on the convergence of Ritz values from a sequence of Krylov subspaces to interior eigenvalues of Hermitean matrices. These bounds are useful in regions of low spectral density, for example near voids in the spectrum, as…
Singular source terms expressed as weighted summations of Dirac-delta functions are regularized through approximation theory with convolution operators. We consider the numerical solution of scalar and one-dimensional hyperbolic…
A spectral method is described for solving coupled elliptic problems on an interior and an exterior domain. The method is formulated and tested on the two-dimensional interior Poisson and exterior Laplace problems, whose solutions and their…
Mixed optimal stopping and stochastic control problems define variational inequalities with non-linear Hamilton-Jacobi-Bellman (HJB) operators, whose numerical solution is notoriously difficult and lack of reliable benchmarks. We first use…
Spectral enhancement -- which aims to undo spectral broadening -- leads to integral equations which are ill-posed and require special regularisation techniques for their solution. Even when an optimal regularisation technique is used,…