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 half-periods 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 monodromy-free operators.
A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.