## Publications

51 results found

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

© 2020 Elsevier Inc. We prove on the sphere S2 and on the torus T2 the Lieb–Thirring inequalities with improved constants for orthonormal families of scalar and vector functions.

Fanelli L, Krejcirik D, Laptev A,
et 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.

Bonheure D, Dolbeault J, Esteban MJ,
et 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.

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.

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.

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.

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

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.

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.

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.

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.

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

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.

Dolbeault J, Esteban MJ, Laptev A,
et 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.

Dolbeault J, Esteban MJ, Laptev A,
et 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.

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

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.

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

Laptev A, Sasane SM, 2017, Perturbations of embedded eigenvalues for a magnetic Schrodinger operator on a cylinder, *Journal of Mathematical Physics*, Vol: 58, ISSN: 0022-2488

Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper, we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C2 with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.

Laptev A, Kapitanski L, 2016, On continuous and discrete Hardy inequalities, *Journal of Spectral Theory*, Vol: 6, Pages: 837-858, ISSN: 1664-039X

We obtain a number of Hardy type inequalities for continuous anddiscrete operators.

Laptev A, Peicheva A, Shlapunov A, 2016, Finding Eigenvalues and Eigenfunctions of the Zaremba Problem for the Circle, *COMPLEX ANALYSIS AND OPERATOR THEORY*, Vol: 11, Pages: 895-926, ISSN: 1661-8254

We consider Zaremba type boundary value problem for the Laplace operator in the unit circle on the complex plane. Using the theorem on the exponential representation for solutions to equations with constant coefficients we indicate a way to find eigenvalues of the problem and to construct its eigenfunctions.

Ashbaugh MS, Gesztesy F, Laptev A,
et al., 2016, A bound for the eigenvalue counting function for Krein-von Neumann and Friedrichs extensions., *Advances in Mathematics*, Vol: 304, Pages: 1108-1155, ISSN: 0001-8708

For an arbitrary open, nonempty, bounded set , , and sufficiently smooth coefficients , we consider the closed, strictly positive, higher-order differential operator in defined on , associated with the differential expression (equations missing) and its Krein–von Neumann extension in . Denoting by , , the eigenvalue counting function corresponding to the strictly positive eigenvalues of , we derive the bound (equations missing)where (with ) is connected to the eigenfunction expansion of the self-adjoint operator in defined on , corresponding to . Here denotes the (Euclidean) volume of the unit ball in (equations missing).Our method of proof relies on variational considerations exploiting the fundamental link between the Krein–von Neumann extension and an underlying abstract buckling problem, and on the distorted Fourier transform defined in terms of the eigenfunction transform ofin (equations missing)We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension in of (equations missing).

Frank RL, Laptev A, Safronov O, 2016, On the number of eigenvalues of Schrödinger operators with complex potentials, *Journal of the London Mathematical Society*, Vol: 94, Pages: 377-390, ISSN: 0024-6107

We study the eigenvalues of Schrödinger operators with complex potentials in odd space dimensions. We obtain bounds on the total number of eigenvalues in the case where VV decays exponentially at infinity.

Laptev A, Schimmer L, Takhtajan LA, 2016, Weyl Type Asymptotics and Bounds for the Eigenvalues of Functional-Difference Operators for Mirror Curves, *Geometric and Functional Analysis*, Vol: 26, Pages: 288-305, ISSN: 1420-8970

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are H(ζ)=U+U−1+V+ζV−1H(ζ)=U+U−1+V+ζV−1 and Hm,n=U+V+q−mnU−mV−nHm,n=U+V+q−mnU−mV−n, where UU and VV are self-adjoint Weyl operators satisfying UV=q2VUUV=q2VU with q=eiπb2q=eiπb2, b>0b>0 and ζ>0ζ>0, m,n∈Nm,n∈N. We prove that H(ζ)H(ζ) and Hm,nHm,n are self-adjoint operators with purely discrete spectrum on L2(R)L2(R). Using the coherent state transform we find the asymptotical behaviour for the Riesz mean ∑j≥1(λ−λj)+∑j≥1(λ−λj)+ as λ→∞λ→∞ and prove the Weyl law for the eigenvalue counting function N(λ)N(λ) for these operators, which imply that their inverses are of trace class.

Ilyin A, Laptev A, Loss M,
et al., 2016, One-dimensional interpolation inequalities, Carlson-Landau inequalities, and magnetic Schrodinger operators, *International Mathematics Research Notices*, Vol: 2016, Pages: 1190-1222, ISSN: 1073-7928

In this paper, we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson–Landau-type inequalities and to magnetic Schrödinger operators. We also obtain Lieb–Thirring inequalities for magnetic Schrödinger operators on multi-dimensional cylinders.

Ilyin AA, Laptev AA, 2015, Lieb-Thirring inequalities on the torus, *Sbornik: Mathematics*, Vol: 207, Pages: 1410-1434, ISSN: 1064-5616

We consider the Lieb-Thirring inequalities on the $d$-dimensional torus with arbitrary periods. In the space of functions with zero average with respect to the shortest coordinate we prove the Lieb-Thirring inequalities for the $\gamma$-moments of the negative eigenvalues with constants independent of ratio of the periods. Applications to the attractors of the damped Navier-Stokes system are given.

Ekholm T, Kovarik H, Laptev A, 2015, Hardy inequalities for p-Laplacians with Robin boundary conditions, *Nonlinear Analysis-Theory Methods & Applications*, Vol: 128, Pages: 365-379, ISSN: 0362-546X

In this paper we study the best constant in a Hardy inequality for the p-Laplace operator on convex domains with Robin boundary conditions. We show, in particular, that the best constant equals ((p−1)/p)p whenever Dirichlet boundary conditions are imposed on a subset of the boundary of non-zero measure. We also discuss some generalizations to non-convex domains.

Hoffmann-Ostenhof T, Laptev A, 2015, Hardy inequalities with homogeneous weights, *Journal of Functional Analysis*, Vol: 268, Pages: 3278-3289, ISSN: 0022-1236

In this paper we obtain some sharp Hardy inequalities with weight functions that may admit singularities on the unit sphere. In order to prove the main results of the paper we use some recent sharp inequalities for the lowest eigenvalue of Schrödinger operators on the unit sphere obtained in the paper [3].

Zelik SV, Ilyin AA, Laptev AA, 2015, Sharp interpolation inequalities for discrete operators, *Doklady Mathematics*, Vol: 91, Pages: 215-219, ISSN: 1531-8362

Ilyin A, Laptev A, Zelik S, 2014, Sharp interpolation inequalities for discrete operators and applications, *Bulletin of Mathematical Sciences*, Vol: 5, Pages: 19-57, ISSN: 1664-3615

We consider interpolation inequalities for imbeddings of the l2-sequencespaces over d-dimensional lattices into the l∞0 spaces written as interpolation inequalitybetween the l2-norm of a sequence and its difference. A general method is developedfor finding sharp constants, extremal elements and correction terms in this type ofinequalities. Applications to Carlson’s inequalities and spectral theory of discreteoperators are given.

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