arXiv subjects

Peter Gibson

Publications and source records attributed to Peter Gibson.

4 recordsLinked to original sources

arXiv2023

Explicit solution of the 1D Schr\"odinger equation

Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schr\"odinger equation. Explicit formulas for the transmission coefficient and $S$-matrix of the classical one-dimensional Schr\"odinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions, and provide a new window on acoustic and quantum scattering in one dimension.

math.AP↗

arXiv2022

Higher dimensional versions of theorems of Euler and Fuss

We present higher dimensional versions of the classical results of Euler and Fuss, both of which are special cases of the celebrated Poncelet porism. Our results concern polytopes, specifically simplices, parallelotopes and cross polytopes, inscribed in a given ellipsoid and circumscribed to another. The statements and proofs use the language of linear algebra. Without loss, one of the ellipsoids is the unit sphere and the other one is also centered at the origin. Let $A$ be the positive symmetric matrix taking the outer ellipsoid to the inner one. If $trace A = 1$, there exists a bijection between the orthogonal group $O(n)$ and the set of such labeled simplices. Similarly, if $trace A^2 = 1$, there are families of parallelotopes and of cross polytopes, also indexed by $O(n)$.

math.MG↗

arXiv2016

A geometry where everything is better than nice

We present a riemannian structure on the disk that has a remarkably rich structure. Geodesics are hypocycloids and the (negative of the) laplacian has integer spectrum with multiplicity the Dirichlet divisor function. Eigenfunctions of the laplacian are orthogonal polynomials naturally suited to the analysis of acoustic scattering in layered media.

math.DG↗

arXiv2002

Spectral distributions and isospectral sets of tridiagonal matrices

We analyze the correspondence between finite sequences of finitely supported probability distributions and finite-dimensional, real, symmetric, tridiagonal matrices. In particular, we give an intrinsic description of the topology induced on sequences of distributions by the usual Euclidean structure on matrices. Our results provide an analytical tool with which to study ensembles of tridiagonal matrices, important in certain inverse problems and integrable systems. As an application, we prove that the Euler characteristic of any generic isospectral set of symmetric, tridiagonal matrices is a tangent number.

math.SP↗