Imperial College London

ProfessorAriLaptev

Faculty of Natural SciencesDepartment of Mathematics

Chair in Pure Mathematics
 
 
 
//

Contact

 

+44 (0)20 7594 8499a.laptev Website

 
 
//

Assistant

 

Mr David Whittaker +44 (0)20 7594 8481

 
//

Location

 

680Huxley BuildingSouth Kensington Campus

//

Summary

 

Publications

Publication Type
Year
to

63 results found

LAPTEV AR, READ LARRY, SCHIMMER LUKAS, 2022, Calogero type bounds in two dimensions, Archive for Rational Mechanics and Analysis, Vol: 245, ISSN: 0003-9527

For a Schrödinger operator on the plane R2 with electric potential V and an Aharonov–Bohm magnetic field, we obtain an upper bound on the number of its negative eigenvalues in terms of the L1(R2)-norm of V. Similar to Calogero’s bound in one dimension, the result is true under monotonicity assumptions on V. Our method of proof relies on a generalisation of Calogero’s bound to operator-valued potentials. We also establish a similar bound for the Schrödinger operator (without magnetic field) on the half-plane when a Dirichlet boundary condition is imposed and on the whole plane when restricted to antisymmetric functions.

Journal article

Krejcirik D, Laptev A, Stampach F, 2022, Spectral enclosures and stability for non-self-adjoint discrete Schrodinger operators on the half-line, Bulletin of the London Mathematical Society, ISSN: 0024-6093

We make a spectral analysis of discrete Schrödinger operators on the half-line, subject to complex Robin-type boundary couplings and complex-valued potentials. First, optimal spectral enclosures are obtained for summable potentials. Second, general smallness conditions on the potentials guaranteeing a spectral stability are established. Third, a general identity which allows to generate optimal discrete Hardy inequalities for the discrete Dirichlet Laplacian on the half-line is proved.

Journal article

Laptev A, Loss M, Schimmer L, 2022, On a Conjecture by Hundertmark and Simon, ANNALES HENRI POINCARE, Vol: 23, Pages: 4057-4067, ISSN: 1424-0637

Journal article

Guzu D, Kapitanski L, Laptev A, 2022, Hardy inequalities for discrete magnetic Dirichlet forms, http://yokohamapublishers.jp/

Journal article

Laptev A, Schimmer L, 2021, A sharp lieb-thirring inequality for functional difference operators, Symmetry, Integrability and Geometry: Methods and Applications, Vol: 17, Pages: 1-10, ISSN: 1815-0659

We prove sharp Lieb-Thirring type inequalities for the eigenvalues of a class of one-dimensional functional difference operators associated to mirror curves. We furthermore prove that the bottom of the essential spectrum of these operators is a resonance state.

Journal article

Laptev A, Weth T, 2021, Spectral properties of the logarithmic Laplacian, Analysis and Mathematical Physics, Vol: 11, Pages: 1-24, ISSN: 1664-235X

We obtain spectral inequalities and asymptotic formulae for the discrete spectrum of the operator 12log(−Δ) in an open set Ω∈Rd, d≥2, of finite measure with Dirichlet boundary conditions. We also derive some results regarding lower bounds for the eigenvalue λ1(Ω) and compare them with previously known inequalities.

Journal article

Ibrogimov OO, Krejcirik D, Laptev A, 2021, Sharp bounds for eigenvalues of biharmonic operators with complex potentials in low dimensions, Mathematische Nachrichten, Vol: 294, Pages: 1333-1349, ISSN: 0025-584X

We derive sharp quantitative bounds for eigenvalues of biharmonic operators perturbed by complex-valued potentials in dimensions one, two and three.

Journal article

Guzu D, Hoffmann-Ostenhof T, Laptev A, 2021, On a class of sharp multiplicative Hardy inequalities, St. Petersburg Mathematical Journal, Vol: 32, Pages: 523-530, ISSN: 0234-0852

A class of weighted Hardy inequalities is treated. The sharp constants depend on the lowest eigenvalues of auxiliary Schrödinger operators on a sphere. In particular, for some block radial weights these sharp constants are given in terms of the lowest eigenvalue of a Legendre type equation.

Journal article

Bonheure D, Dolbeault J, Esteban MJ, Laptev A, Loss Met al., 2021, Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3, Reviews in Mathematical Physics: a journal for survey and expository articles in the field of mathematical physics, Vol: 33, Pages: 1-29, ISSN: 0129-055X

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schrödinger operators involving Aharonov–Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions 2 and 3. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in the presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

Journal article

Hoffmann-Ostenhof T, Laptev A, 2021, Hardy inequality for antisymmetric functions, Functional Analysis and Its Applications, Vol: 55, Pages: 122-129, ISSN: 0016-2663

We consider Hardy inequalities on antisymmetric functions. Such inequalities have substantially better constants. We show that they depend on the lowest degree of an antisymmetric harmonic polynomial. This allows us to obtain some Caffarelli–Kohn–Nirenberg-type inequalities that are useful for studying spectral properties of Schrödinger operators.

Journal article

Laptev A, 2021, On factorisation of a class of Schrodinger operators, Complex Variables and Elliptic Equations: an international journal, Vol: 66, Pages: 1100-1107, ISSN: 1747-6933

The aim of this paper is to find inequalities for 3/2 moments of the negative eigenvalues of Schrödinger operators on half-line that have a ‘Hardy term’ by using the commutator method.

Journal article

Ilyin A, Laptev A, Zelik S, 2020, Lieb-Thirring constant on the sphere and on the torus, Journal of Functional Analysis, Vol: 279, Pages: 1-18, ISSN: 0022-1236

We prove on the sphere S2 and on the torus T2the Lieb–Thirring inequalities with improved constants for orthonormal familiesof scalar and vector functions.

Journal article

Ilyin AA, Laptev AA, 2020, Magnetic lieb-thirring inequality for periodic functions, Russian Mathematical Surveys, Vol: 75, Pages: 779-781, ISSN: 0036-0279

Journal article

Fanelli L, Krejcirik D, Laptev A, Vega Let al., 2020, On the improvement of the Hardy inequality due to singular magnetic fields, Communications in Partial Differential Equations, Vol: 45, Pages: 1-11, ISSN: 0360-5302

We establish magnetic improvements upon the classical Hardy inequality for two specific choices of singular magnetic fields. First, we consider the Aharonov-Bohm field in all dimensions and establish a sharp Hardy-type inequality that takes into account both the dimensional as well as the magnetic flux contributions. Second, in the three-dimensional Euclidean space, we derive a non-trivial magnetic Hardy inequality for a magnetic field that vanishes at infinity and diverges along a plane.

Journal article

Bonheure D, Dolbeault J, Esteban MJ, Laptev A, Loss Met al., 2020, Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields, Communications in Mathematical Physics, Vol: 375, Pages: 2071-2087, ISSN: 0010-3616

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schrödinger operator involving an Aharonov–Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller–Lieb–Thirring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy–Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.

Journal article

Ilyin A, Laptev A, 2020, Lieb-Thirring inequalities on the sphere, St. Petersburg Mathematical Journal, Vol: 31, Pages: 479-493, ISSN: 0234-0852

On the sphere $ \mathbb{S}^2$, the Lieb-Thirring inequalities are proved for orthonormal families of scalar and vector functions both on the whole sphere and on proper domains on $ \mathbb{S}^2$. By way of applications, an explicit estimate is found for the dimension of the attractor of the Navier-Stokes system on a domain on the sphere with Dirichlet nonslip boundary conditions.

Journal article

Ferrulli F, Laptev A, 2020, To Vladimir Maz'ya with respect and admiration, Rendiconti Lincei - Matematica e Applicazioni, Vol: 31, Pages: 1-13, ISSN: 1120-6330

We derive some bounds on the location of complex eigenvalues for a family of Schrödinger operators H0,ν defined on the positive half line and subject to integrable complex potential. We generalise the results obtained in [14] where the operator does not have a Hardy term and also include the analysis for potentials belonging to weighted Lp spaces. Some information on the geometry of the complex region which bounds the eigenvalues of the radial Schrödinger multidimensional operator are then recovered.

Journal article

Hassannezhad A, Laptev A, 2020, Eigenvalue bounds of mixed Steklov problems, Communications in Contemporary Mathematics, Vol: 22, Pages: 1-23, ISSN: 0219-1997

. The Steklov–Neumann eigenvalue problem is also called the sloshing problem. We obtain two-term asymptotically sharp lower bounds on the Riesz means of the sloshing problem and also provide an asymptotically sharp upper bound for the Riesz means of mixed Steklov–Dirichlet problem. The proof of our results for the sloshing problem uses the average variational principle and monotonicity of sloshing eigenvalues. In the case of Steklov–Dirichlet eigenvalue problem, the proof is based on a well-known bound on the Riesz means of the Dirichlet fractional Laplacian, and an inequality between the Dirichlet and Navier fractional Laplacian. The two-term asymptotic results for the Riesz means of mixed Steklov eigenvalue problems are discussed in the Appendix which in particular show the asymptotic sharpness of the bounds we obtain.

Journal article

Zelik SV, Ilyin AA, Laptev AA, 2019, On the Lieb-Thirring Constant on the Torus, Mathematical Notes, Vol: 106, Pages: 1019-1023, ISSN: 0001-4346

Journal article

Laptev A, Schimmer L, Takhtajan LA, 2019, Weyl asymptotics for perturbed functional difference operators, Journal of Mathematical Physics, Vol: 60, Pages: 1-10, ISSN: 0022-2488

We consider the difference operator HW = U + U−1 + W, where U is the self-adjoint Weyl operator U = e−bP, b > 0, and the potential W is of the form W(x) = x2N + r(x) with N∈ℕ and |r(x)| ≤ C(1 + |x|2N−ɛ) for some 0 < ɛ ≤ 2N − 1. This class of potentials W includes polynomials of even degree with leading coefficient 1, which have recently been considered in Grassi and Mariño [SIGMA Symmetry Integrability Geom. Methods Appl. 15, 025 (2019)]. In this paper, we show that such operators have discrete spectrum and obtain Weyl-type asymptotics for the Riesz means and for the number of eigenvalues. This is an extension of the result previously obtained in Laptev et al. [Geom. Funct. Anal. 26, 288–305 (2016)] for W = V + ζV−1, where V = e2πbx, ζ > 0.

Journal article

Safronov O, Laptev A, Ferrulli F, 2019, Eigenvalues of the bilayer graphene operator with a complex valued potential, Analysis and Mathematical Physics, Vol: 9, Pages: 1535-1546, ISSN: 1664-235X

We study the spectrum of a system of second order differential operator Dm perturbed by a non-selfadjoint matrix valued potential V. We prove that eigenvalues of Dm+V are located near the edges of the spectrum of the unperturbed operator Dm.

Journal article

Ilyin A, Laptev A, 2019, Berezin-Li-Yau inequalities on domains on the sphere, Journal of Mathematical Analysis and Applications, Vol: 473, Pages: 1253-1269, ISSN: 0022-247X

We prove Berezin–Li–Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere . A sharp explicit bound for the sums of the Neumann eigenvalues is obtained for all dimensions d. In the case of we also obtain sharp lower bounds with correction terms for the vector Laplacian and the Stokes operator.

Journal article

Frank RL, Laptev A, 2019, Bound on the number of negative eigenvalues of two-dimensional Schrödinger operators on domains, St. Petersburg Mathematical Journal, Vol: 30, Pages: 573-589, ISSN: 1547-7371

A fundamental result of Solomyak says that the number of negative eigenvalues of a Schrödinger operator on a two-dimensional domain is bounded from above by a constant times a certain Orlicz norm of the potential. Here it is shown that in the case of Dirichlet boundary conditions the constant in this bound can be chosen independently of the domain.

Journal article

Laptev A, Ruzhansky M, Yessirkegenov N, 2019, HARDY INEQUALITIES FOR LANDAU HAMILTONIAN AND FOR BAOUENDI-GRUSHIN OPERATOR WITH AHARONOV-BOHM TYPE MAGNETIC FIELD. PART I, MATHEMATICA SCANDINAVICA, Vol: 125, Pages: 239-269, ISSN: 0025-5521

Journal article

Korotyaev E, Laptev A, 2018, Trace formulae for Schrodinger operators with complex-valued potentials on cubic lattices, Bulletin of Mathematical Sciences, Vol: 8, Pages: 453-475, ISSN: 1664-3615

We consider a class of Schrödinger operators with complex decaying potentials on the lattice. Using some classical results from complex analysis we obtain some trace formulae and use them to estimate zeros of the Fredholm determinant in terms of the potential.

Journal article

Dolbeault J, Esteban MJ, Laptev A, Loss Met al., 2018, Magnetic rings, Journal of Mathematical Physics, Vol: 59, ISSN: 1089-7658

We study functional and spectral properties of perturbations of the operator −(∂s+ia)2 in L2( 1). This operator appears when considering the restriction to the unit circle of a two-dimensional Schrödinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on ℝ2 and, on 1, a sharp interpolation inequality and a sharp Keller-Lieb-Thirring inequality.

Journal article

Dolbeault J, Esteban MJ, Laptev A, Loss Met al., 2018, Interpolation Inequalities and Spectral Estimates for Magnetic Operators, ANNALES HENRI POINCARE, Vol: 19, Pages: 1439-1463, ISSN: 1424-0637

We prove magnetic interpolation inequalities and Keller–Lieb–Thirring estimates for the principal eigenvalue of magnetic Schrödinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate.

Journal article

Laptev A, Velicu A, 2018, Bound states of Schrodinger type operators with Heisenberg sub-Laplacian, Conference on Non-linear PDEs, Mathematical Physics and Stochastic Analysis, Publisher: EUROPEAN MATHEMATICAL SOC, Pages: 381-387

Conference paper

Korotyaev EL, Laptev A, 2017, Trace formulas for a discrete Schrodinger operator, Functional Analysis and Its Applications, Vol: 51, Pages: 225-229, ISSN: 0016-2663

The Schrödinger operator with complex decaying potential on a lattice is considered. Trace formulas are derived on the basis of classical results of complex analysis. These formulas are applied to obtain global estimates of all zeros of the Fredholm determinant in terms of the potential.

Journal article

Laptev A, Ashbaugh M, Gesztesy F, Mitrea M, Sukhtaiev Set al., 2017, A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs, Advances in Mathematics, ISSN: 0001-8708

Journal article

This data is extracted from the Web of Science and reproduced under a licence from Thomson Reuters. You may not copy or re-distribute this data in whole or in part without the written consent of the Science business of Thomson Reuters.

Request URL: http://wlsprd.imperial.ac.uk:80/respub/WEB-INF/jsp/search-html.jsp Request URI: /respub/WEB-INF/jsp/search-html.jsp Query String: respub-action=search.html&id=00498016&limit=30&person=true