## Publications

57 results found

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.

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.

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

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

Bonheure D, Dolbeault J, Esteban MJ,
et 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.

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.

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.

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

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.

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