Reference number: JC/MA/2020
Start date of studentship: 1 October 2020
Closing date of advert: 14 February 2020
Primary supervisor: Jean-Claude Cuenin
The interrelation between quantum mechanics and spectral theory of self-adjoint linear operators in Hilbert spaces is one of the prime examples for the mutual synergy between mathematics and physics. Spectral properties of self-adjoint operators have been studied intensively for many decades. The key result here is the so-called spectral theorem which is a generalization of the theorem on the diagonalization of hermitian matrices to infinite dimensional Hilbert spaces. While self-adjoint operators play a distinguished role mainly in quantum mechanics, the need to study their non-self-adjoint counterparts is substantiated by the vast diversity of physical applications. These include dissipation phenomena in quantum and nuclear physics, quantum chromodynamics, disordered systems, neural networks and turbulence phenomena in hydrodynamics.
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Full Project Detail
The Schrödinger equation describes the motion of quantum mechanical particles. The spatial part of the equation is governed by a so-called Schrödinger operator, a partial differential operator that can be viewed as a quantization of the classical Hamiltonian. The eigenvalues (or more generally, the spectrum) of this operator are the possible energies of the system. Conservation of energy requires that the Schrödinger operator is self-adjoint (“symmetric/hermitian”); in particular, the potential must be real-valued in this case. However, many interesting physical phenomena (resonances, dissipation of energy etc.) are modelled by Schrödinger operators with complex-valued potentials. Mathematically, complex potentials pose a significant challenge, and the theory is much less developed than its classical counterpart dealing only with real-valued potentials. Even though rapid progress has been made in recent years, there is a need for more (counter)-examples.
The aim of this project is to construct explicit (counter)-examples of complex potentials leading to spectral behaviour that is “unexpected” from the point of view of the classical theory. The candidate will have the opportunity to participate in workshops of our LMS Joint Research Group “Challenges in Non-Self-Adjoint Spectral Theory”.
Find out more
Challenges in Non-Self-Adjoint Spectral Theory https://sites.google.com/view/lms-jrg-nsa/home
Open problems http://nsa.fjfi.cvut.cz/problems.html
Review of E.B. Davies’ book “Linear Operators and their Spectra” https://www.math.ru.nl/~landsman/Davies.pdf https://www.lboro.ac.uk/science/study/postgraduate-research/studentships/
Applicants should have a Master’s degree in Mathematics or Physics. A strong background in analysis or PDE is of advantage.
Name: Jean-Claude Cuenin
Email address: [email protected]
Telephone number: +44 (0)1509 223243
How to apply
All applications should be made online at https://www.lboro.ac.uk/study/postgraduate/research-degrees/research-opportunities/
under program name Mathematical Sciences.
Please quote reference number: JC/MA/2020.