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Spectral properties of integrable Schrodinger operators with singular potentials
The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. A classical example is the Lamé operator L = −d^2/dx^2 + m(m + 1)℘(x), where ℘(z) is the classical Weierstrass elliptic function. We study the spectral properties of it...
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2015

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Online Access:  https://hdl.handle.net/2134/19929 
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author  William HaeseHill 
author_facet  William HaeseHill 
author_sort  William HaeseHill (7158500) 
collection  Figshare 
description  The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. A classical example is the Lamé operator L = −d^2/dx^2 + m(m + 1)℘(x), where ℘(z) is the classical Weierstrass elliptic function. We study the spectral properties of its complex regularisations of the form L = −d^2/dx^2 + m(m + 1)ω^2 ℘(ωx + z_0 ), z_0 ∈ C, where ω is one of the halfperiods of ℘(z). In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromyfree operators. 
format  Default Thesis 
id  rrarticle9374174 
institution  Loughborough University 
publishDate  2015 
record_format  Figshare 
spelling  rrarticle937417420150101T00:00:00Z Spectral properties of integrable Schrodinger operators with singular potentials William HaeseHill (7158500) Other mathematical sciences not elsewhere classified Complex Lamé operators Monodromyfree Schrödinger operators Exceptional orthogonal polynomials Mathematical Sciences not elsewhere classified The integrable Schrödinger operators often have a singularity on the real line, which creates problems for their spectral analysis. A classical example is the Lamé operator L = −d^2/dx^2 + m(m + 1)℘(x), where ℘(z) is the classical Weierstrass elliptic function. We study the spectral properties of its complex regularisations of the form L = −d^2/dx^2 + m(m + 1)ω^2 ℘(ωx + z_0 ), z_0 ∈ C, where ω is one of the halfperiods of ℘(z). In several particular cases we show that all closed gaps lie on the infinite spectral arc. In the second part we develop a theory of complex exceptional orthogonal polynomials corresponding to integrable rational and trigonometric Schrödinger operators, which may have a singularity on the real line. In particular, we study the properties of the corresponding complex exceptional Hermite polynomials related to Darboux transformations of the harmonic oscillator, and exceptional Laurent orthogonal polynomials related to trigonometric monodromyfree operators. 20150101T00:00:00Z Text Thesis 2134/19929 https://figshare.com/articles/thesis/Spectral_properties_of_integrable_Schrodinger_operators_with_singular_potentials/9374174 CC BYNCND 4.0 
spellingShingle  Other mathematical sciences not elsewhere classified Complex Lamé operators Monodromyfree Schrödinger operators Exceptional orthogonal polynomials Mathematical Sciences not elsewhere classified William HaeseHill Spectral properties of integrable Schrodinger operators with singular potentials 
title  Spectral properties of integrable Schrodinger operators with singular potentials 
title_full  Spectral properties of integrable Schrodinger operators with singular potentials 
title_fullStr  Spectral properties of integrable Schrodinger operators with singular potentials 
title_full_unstemmed  Spectral properties of integrable Schrodinger operators with singular potentials 
title_short  Spectral properties of integrable Schrodinger operators with singular potentials 
title_sort  spectral properties of integrable schrodinger operators with singular potentials 
topic  Other mathematical sciences not elsewhere classified Complex Lamé operators Monodromyfree Schrödinger operators Exceptional orthogonal polynomials Mathematical Sciences not elsewhere classified 
url  https://hdl.handle.net/2134/19929 