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
Geometric spectral and scattering theory

Teaching

Current:

M0071 (Spectral Theory)

Some publications

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)
L.Parnovski, R.Shterenberg: Complete asymptotic expansion of the integrated density of states of multidimensional almost-periodic Schroedinger operators, Ann. of Math., 176 (2) (2012) 1039 - 1096.
Andrei Artemev, Leonid Parnovski and Iosif Polterovich, Inverse electrostatic and elasticity problems for checkered distributions, Inverse Problems, 29 (2013), 075010, 16pp.
P. Exner, A. Minakov, and L. Parnovski Asymptotic eigenvalue estimates for a Robin problem with a large parameter, Portugaliae Mathematica 71(2) (2014), 141-156.
S. Morozov, L. Parnovski, R. Shterenberg, Complete Asymptotic Expansion of the Integrated Density of States of Multidimensional Almost-Periodic Pseudo-Differential Operators. ANNALES HENRI POINCARE , 15 (2) (2014) 263 - 312.
L.Parnovski, R.Shterenberg: Complete asymptotic expansion of the spectral function of multidimensional almost-periodic Schroedinger operators, Duke Mathematical Journal, Vol. 165, No. 3, 2016, 509-562.
A. Girouard, L. Parnovski, I. Polterovich, and D. Sher The Steklov spectrum of surfaces: asymptotics and invariants, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 157, Issue 3 (2014), pp. 379-389.
J.Lagace, L.Parnovski A generalised Gauss circle problem and integrated density of states, J.Spec.Theor., Volume 6, Issue 4, 2016, pp. 859-879.
L.Parnovski, R.Shterenberg: Perturbation theory for spectral gap edges of 2D periodic Schroedinger operators, J.Funct.Anal., Volume 273, Issue 1, 1 July 2017, Pages 444-470
L.Parnovski, R.Shterenberg: Perturbation Theory for Almost-Periodic Potentials I: One-Dimensional Case , Communications in Mathematical Physics, March 2019, Volume 366, Issue 3, pp 1229-1257
M. Levitin, L. Parnovski, I. Polterovich, D. A. Sher Sloshing, Steklov and corners: Asymptotics of sloshing eigenvalues, Journal d'Analyse Mathematique, 146, pages 65-125 (2022)
S. Krymski, M. Levitin, L. Parnovski, I. Polterovich, D. A. Sher Inverse Steklov spectral problem for curvilinear polygons, IMRN, Volume 2021, Issue 1, January 2021, 1-37
Yulia Karpeshina, Leonid Parnovski, Roman Shterenberg Ballistic Transport for Schroedinger Operators with Quasi-periodic Potentials , Journal of Mathematical Physics, Vol.62, Issue 5 (special volume dedicated to the memory of J.Bourgain), 063509 (2021)
Ilya Kachkovskiy, Stanislav Krymski, Leonid Parnovski, Roman Shterenberg Perturbative diagonalisation for Maryland-type quasiperiodic operators with flat pieces , Journal of Mathematical Physics Vol.62, Issue 5 (special volume dedicated to the memory of J.Bourgain), 063509 (2021)
M. Levitin, L. Parnovski, I. Polterovich, D. A. Sher Sloshing, Steklov and corners: Asymptotics of Steklov eigenvalues for curvilinear polygons, Proceedings of the London Mathematical Society, 125(3), 2022, 359-487.
Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg Convergence of perturbation series for unbounded monotone quasiperiodic operators , Advances in Math., 409(B), 2022, 108647
Jean Lagace, Sergey Morozov, Leonid Parnovski, Bernhard Pfirsch, Roman Shterenberg The almost periodic gauge transform --- An abstract scheme with applications to Dirac Operators , Annales Henri Lebesgue, to appear
Jeffrey Galkowski, Leonid Parnovski, and Roman Shterenberg Classical Wave Mathods and Modern Gauge Transforms: Spectral Asymptotics in the One Dimensional Case , GAFA, Volume 33, pages 1454-1538, (2023)
Yulia Karpeshina, Leonid Parnovski, Roman Shterenberg Bethe-Sommerfeld Conjecture and Absolutely Continuous Spectrum of Multi-Dimensional Quasi-Periodic Schroedinger Operators, Ann. of Math., to appear
Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg Perturbative diagonalization and spectral gaps of quasiperiodic operators on $\ell^2(\Z^d)$ with monotone potentials , Comm. Math. Phys., to apppear.
Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg On gaps in the spectra of quasiperiodic Schroedinger operators with discontinuous monotone potentials , IMRN, to appear.

Some 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: April 2025 by Leonid Parnovski