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PhD Studentship: Schrödinger operators with complex potentials

  • Full or part time
  • Application Deadline
    Friday, February 14, 2020
  • Competition Funded PhD Project (European/UK Students Only)
    Competition Funded PhD Project (European/UK Students Only)

Project Description

Application details:
Reference number: JC/MA/2020
Start date of studentship: 1 October 2020
Closing date of advert: 14 February 2020

Supervisors:
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.

Loughborough University

Loughborough University is a top-ten rated university in England for research intensity (REF2014). In choosing Loughborough for your research, you’ll work alongside academics who are leaders in their field. You will benefit from comprehensive support and guidance from our Doctoral College, including tailored careers advice, to help you succeed in your research and future career.

Loughborough University has a flexible working and maternity/parental leave policy (https://www.lboro.ac.uk/services/hr/leave-absence/family-leave/) and is a Stonewall Diversity Champion providing a supportive and inclusive environment for the LGBT+ community. The University is also a member of the Race Equality Charter which aims to improve the representation, progression and success of minority ethnic staff and students. The School of Science is a recipient of the Athena SWAN bronze award for gender equality.

Find out more: http://www.lboro.ac.uk/study/postgraduate/supporting-you/research/

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/

Entry requirements

Applicants should have a Master’s degree in Mathematics or Physics. A strong background in analysis or PDE is of advantage.

Contact details

Name: Jean-Claude Cuenin
Email address:
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.

Funding Notes

This studentship will be awarded on a competitive basis to applicants who have applied to this project and/or any of the advertised projects prioritised for funding by the School of Science.

The 3-year studentship provides a tax-free stipend of £15,009 (2019 rate) per annum (in line with the standard research council rates) for the duration of the studentship, plus tuition fees at the UK/EU rate. This studentship is only available to those eligible to pay UK/EU fees.

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