# Available student project - Exact Bohr-Sommerfeld quantisation and Conformal Field Theory

## Research fields

- Theoretical Physics

## Project details

It is well known that the quasiclassical quantisation of the harmonic oscillator leads to its exact quantum mechanical spectrum. Remarkably, this result can be generalized to various anharmonic systems via mysterious connections to Conformal Field Theory. The idea is that the exact quantisation conditions take the form of the so-called Bethe Ansatz equations, which are the key transcendental equations in the theory of integrable quantum system. This phenomenon is known as the ODE/IM correspondence. (ODE stands for "ordinary differential equations" and IM for "integrable models").

At the honours level the project will consist of an advanced study of the quasiclassical approximation to the Schroedinger equation in quantum mechanics.

At the Masters and PhD levels the project will aim to construct and study new cases of the ODE/IM correspondence for systems with extended conformal symmetry, connected to W-algebras. Remarkably the problem have important applications to mesoscopic quantum systems, e.g. Kondo impurities and ultra-small Josephson junctions.

**Literature**

- V.V.Bazhanov, L.S.Lukyanov, A.B.Zamolodchikov, “Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory”, J. Stat. Phys.
**102**(2001) 587-576, arXiv:hep-th/9812247 - P.Dorey, C.Dunning and R.Tateo, "Topical review: The ODE/IM Correspondence,'' J. Phys. A,
**40**(2007) R205, arXiv:hep-th/0703066

## Required background

Advanced level courses in maths and theoretical physics.

## Project suitability

- Honours project
- Phd or Masters