Eigenvalues of functional difference operators associated to mirror curves

Toric Calabi–Yau manifolds play an important role in mathematical physics. Their mirror manifolds can be described by algebraic curves and recently it was observed that quantisation of these curves leads to functional-difference operators. The eigenvalues of these operators are conjectured to relate to geometric properties of the associated Calabi–Yau manifolds. Formally, the operators are differential operators of infinite order. Studying properties of their eigenvalues is thus of twofold importance. Firstly, it sheds more light on the conjectured connection to Calabi–Yau manifolds. Secondly, any results obtained can be expected to form limit cases in known spectral results for finite-order differential operators. This project will build on recent progress in establishing eigenvalue bounds for some of these operators. Specifically, it will investigate the asymptotic behaviour of the eigenvalues. Due to the peculiar nature of these operators, several concepts that are well known for finite-order differential operators cannot be directly applied and will need to be modified. The successful candidate will be part of the Analysis and PDE group at Loughborough University, benefitting from a stimulating environment that includes weekly research seminars, diverse expertise in spectral theory and mathematical physics, as well as links with research groups across the UK and EU. 94% of Loughborough’s research impact is rated world-leading or internationally excellent. REF 2021

The standard University IELTS English language requirement is 6.5 overall with 6.0 in each individual element (reading, writing, listening and speaking).

International students: £23,100

UK students: £5,238

UK/Home: £5,238

International: £23,100

October 2026

31 July 2026

• Loughborough, United Kingdom