Prof Leonid Parnovski

Professor in Pure Mathematics

Department of Mathematics, University College London
Gower Street, London WC1E 6BT, UK

leonid (at) math.ucl.ac.uk

Research interests

Spectral theory of differential and pseudo-differential operators
Periodic, almost-periodic and random spectral problems
A priori estimates of eigenvalues
Trapped modes in waveguides
Geometric characteristics of null variety
Geometric spectral and scattering theory

Teaching

Current:

6401: Mathematics for Students of Economics, Statistics and Related Disciplines (Term 1, 2010-11)

Office hour: Mondays 11-12.

Answers to some of your questions concerning exams.

M111 (Spectral Theory)

Recent publications

Submitted papers:

L.Parnovski, R.Shterenberg: Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schroedinger operators, to appear in Ann. of Math.

Published papers:

L.Parnovski, A.Sobolev: On the Bethe-Sommerfeld conjecture for the polyharmonic operator, Duke Math. J., 107(2) (2001), 209-238.
M.Levitin, L.Parnovski: Commutators,Spectral Trace Identities, and Universal Estimates for Eigenvalues, Journal of Functional Analysis, 192 (2002), 425-445.
L.Parnovski: Spectral asymptotics of Laplace operators on surfaces with cusps, Mathematische Annalen, 303 (1995), 281-296.
E.B.Davies, L.Parnovski: Trapped modes in acoustic waveguides, Quaterly Journal of Machanics and Applied Mathematics, 51(3) (1998), 477-492.
R.Hill, L.Parnovski The Variance of the Hyperbolic Lattice Point Counting Function, Russian Journal of Mathematical Physics, Vol. 12, No. 4, 2005, pp. 472-482 (special volume dedicated to the memory of B.M.Levitan).
Michael Levitin, Leonid Parnovski, Iosif Polterovich Isospectral domains with mixed boundary conditions, Journal of Physics A: Mathematical and General 39 (2006), 2073-2082.
E R Johnson, M Levitin, and L Parnovski Existence of eigenvalues of a linear operator pencil in a curved waveguide — localized shelf waves on a curved coast, SIAM Journal on Mathematical Analysis 37(5) (2006), 1465-1481
M.Levitin, L.Parnovski On the principal eigenvalue of a Robin problem with a large parameter, Mathematische Nachrichten, 281(2) (2008), 272-281.
L.Parnovski: Bethe-Sommerfeld Conjecture, Annales Henri Poincare, 9(3) (2008), 457-508.
L.Parnovski, R.Shterenberg: Asymptotic expansion of the integrated density of states of a two-dimensional periodic Schroedinger operator, Invent.Math., 176(2) (2009), 275–323.
G.Barbatis, L.Parnovski: Bethe - Sommerfeld conjecture for pseudo-differential perturbation, Comm.P.D.E., 34(4) (2009), 383-418.
R.Benguria, M.Levitin, L.Parnovski: Fourier transform, null variety, and Laplacian's eigenvalues, J.Func.Anal., 257(7) (2009), 2088-2123.
L.Parnovski, A.Sobolev: Bethe-Sommerfeld conjecture for periodic operators with strong perturbations, Invent.Math., 181(3) (2010), 467 - 540.
L.Parnovski, N.Sidorova: Critical Dimensions for counting Lattice Points in Euclidean Annuli, Mathematical Modelling of Natural Phenomena, Vol. 5 No. 4, (2010), 293-316 (special volume dedicated to the memory of M.S.Birman)
S.Morozov, L.Parnovski, I.Pchelintseva: Lower Bound on the Density of States for Periodic Schroedinger Operators, Amer. Math. Soc. Transl., 231 (2010), 161-171 (special volume dedicated to the memory of V.B.Lidskii)

Presentations:

My inaugural lecture, 24 November 2010.

Talk at Caltech "Bethe-Sommerfeld Conjecture", 16 February 2011

Talk at Western States Meeting "Integrated density of states of periodic and almost-periodic Schroedinger operators". Caltech, 21 February 2011

Links

Last modified: 15 January 2011 by Leonid Parnovski