arXiv subjects

Martin A. Hallnäs

Publications and source records attributed to Martin A. Hallnäs.

3 recordsLinked to original sources

arXiv2018

Quantization and explicit diagonalization of new compactified trigonometric Ruijsenaars-Schneider systems

Recently, Fehér and Kluck discovered, at the level of classical mechanics, new compactified trigonometric Ruijsenaars-Schneider $n$-particle systems, with phase space symplectomorphic to the $(n-1)$-dimensional complex projective space. In this article, we quantize the so-called type (i) instances of these systems and explicitly solve the joint eigenvalue problem for the corresponding quantum Hamiltonians by generalising previous results of van Diejen and Vinet. Specifically, the quantum Hamiltonians are realized as discrete difference operators acting in a finite-dimensional Hilbert space of complex-valued functions supported on a uniform lattice over the classical configuration space, and their joint eigenfunctions are constructed in terms of discretized $A_{n-1}$ Macdonald polynomials with unitary parameters.

math-ph↗

arXiv2017

On the Spectra of Real and Complex Lamé Operators

We study Lamé operators of the form $$L = -\frac{d^2}{dx^2} + m(m+1)ω^2\wp(ωx+z_0),$$ with $m\in\mathbb{N}$ and $ω$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $ω$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lamé operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lamé spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.

math.SP↗

arXiv2015

Complex exceptional orthogonal polynomials and quasi-invariance

Consider the Wronskians of the classical Hermite polynomials $$H_{λ, l}(x):=\mathrm{Wr}(H_l(x),H_{k_1}(x),\ldots, H_{k_n}(x)), \quad l \in \mathbb Z_{\geq 0},$$ where $k_i=λ_i+n-i, \,\, i=1,\dots, n$ and $λ=(λ_1, \dots, λ_n)$ is a partition. Gómez-Ullate et al showed that for a special class of partitions the corresponding polynomials are orthogonal and dense among all polynomials with certain inner product, but in contrast to the usual case have some degrees missing (so called exceptional orthogonal polynomials). We generalise their results to all partitions by considering complex contours of integration and non-positive Hermitian products. The corresponding polynomials are orthogonal and dense in a finite-codimensional subspace of $\mathbb C[x]$ satisfying certain quasi-invariance conditions. A Laurent version of exceptional orthogonal polynomials, related to monodromy-free trigonometric Schrödinger operators, is also presented.

math-ph↗