Professor Leonid Parnovski, MSc, PhD
Head of Pure Mathematics Group

Room 709
Tel: 020-7679-2847
E-mail: leonidmath.ucl.ac.uk
Fax: 020-7383-5519

Research Interests
Spectral Theory of Partial Differential Operators, Scattering Theory, Spectral Geometry

If M is a compact manifold and A is an elliptic differential operator acting on L²(M), then the spectrum of M consists of eigenvalues, and the asymptotic behaviour of the high energy eigenvalues is relatively well understood. However, if M is non-compact, the spectrum of A is not necessarily purely discrete, and the question of the quantitative (and, in certain cases, qualitative) behaviour of the high energy spectrum is not understood at all. In some cases, scattering theory makes it possible to 'count' continuous spectrum using either the determinant of the scattering matrix, or the poles of the analytical continuation of the resolvent of A; in these cases, it is sometimes possible to study the asymptotic behaviour of the counting function of the spectrum of A.

I also work on the periodic differential operators acting in the Euclidean space Rⁿ. If A = – Delta + V is the Schrödinger operator with the smooth periodic electric potential V(x), the celebrated Bethe-Sommerfeld conjecture states that the number of gaps in the spectrum is finite if n > 1. Until recently, this conjecture has been proved only if n = 2,3,4, or for arbitrary n, provided the lattice of periods of V is rational. I have proven the conjecture in all dimensions d > 1 without any assumptions on the lattice of periods. The challenging problem is to extend the proof to the case of Schrödinger operators with periodic magnetic potentials. Asymptotic behaviour of the integrated density of states is another related problem of much interest. The generalisation of this setting to the case of periodic operators acting on arbitrary manifolds (with non-commutative group of periods) is a problem which has much more questions than answers.

Problems described above are examples of 'hard' analysis; to tackle them, one requires quite elaborated technique. I also work on slightly easier problems, like the conditions for the existence of trapped modes in waveguides, or the estimates of small eigenvalues of differential operators using abstract trace identities.