Publications
65 results found
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.
Dolbeault J, Esteban MJ, Laptev A, et al., 2014, One-dimensional Gagliardo Nirenberg Sobolev inequalities: remarks on duality and flows, Journal of the London Mathematical Society-Second Series, Vol: 90, Pages: 525-550, ISSN: 1469-7750
his paper is devoted to one-dimensional interpolation Gagliardo–Nirenberg–Sobolev inequalities. We study how various notions of duality, transport and monotonicity of functionals along flows defined by some non-linear diffusion equations apply.We start by reducing the inequality to a much simpler dual variational problem using mass transportation theory. Our second main result is devoted to the construction of a Lyapunov functional associated with a non-linear diffusion equation, that provides an alternative proof of the inequality. The key observation is that the inequality on the line is equivalent to Sobolev's inequality on the sphere, at least when the dimension is an integer, or to the critical interpolation inequality for the ultraspherical operator in the general case. The time derivative of the functional along the flow is itself very interesting. It explains the machinery of some rigidity estimates for non-linear elliptic equations and shows how eigenvalues of a linearized problem enter into the computations. Notions of gradient flows are then discussed for various notions of distances.Throughout this paper, we shall deal with two classes of inequalities corresponding either to p>2p>2 or to 1<p<21<p<2. The algebraic part in the computations is very similar in both cases, although the case 1<p<21<p<2 is definitely less standard.
Dolbeault J, Esteban MJ, Laptev A, 2014, Spectral estimates on the sphere, Analysis & PDE, Vol: 7, Pages: 435-460, ISSN: 1948-206X
In this article we establish optimal estimates for the first eigenvalue of Schrödinger operators on the dd-dimensional unit sphere. These estimates depend on LpLp norms of the potential, or of its inverse, and are equivalent to interpolation inequalities on the sphere. We also characterize a semiclassical asymptotic regime and discuss how our estimates on the sphere differ from those on the Euclidean space.
Exner P, Laptev A, Usman M, 2014, On Some Sharp Spectral Inequalities for Schrodinger Operators on Semiaxis, Communications in Mathematical Physics, Vol: 326, Pages: 531-541, ISSN: 0010-3616
In this paper we obtain sharp Lieb–Thirring inequalities for a Schrödingeroperator on semiaxis with a matrix potential and show how they can be used to otherrelated problems. Among them are spectral inequalities on star graphs and spectralinequalities for Schrödinger operators on half-spaces with Robin boundary conditions.
Dolbeault J, Esteban MJ, Laptev A, et al., 2013, Spectral properties of Schrodinger operators on compact manifolds: Rigidity, flows, interpolation and spectral estimates, Comptes Rendus Mathematique, Vol: 351, Pages: 437-440, ISSN: 1631-073X
This note is devoted to optimal spectral estimates for Schrödinger operators on compact connected Riemannian manifolds without boundary. These estimates are based on the use of appropriate interpolation inequalities and on some recent rigidity results for nonlinear elliptic equations on those manifolds.
Cuenin J-C, Laptev A, Tretter C, 2013, Eigenvalue Estimates for Non-Selfadjoint Dirac Operators on the Real Line, Annales Henri Poincaré, Vol: 15, Pages: 707-736, ISSN: 1424-0637
We show that the non-embedded eigenvalues of the Dirac operatoron the real line with complex mass and non-Hermitian potentialV lie in the disjoint union of two disks, provided that the L1-norm of Vis bounded from above by the speed of light times the reduced Planckconstant. The result is sharp; moreover, the analogous sharp result forthe Schr¨odinger operator, originally proved by Abramov, Aslanyan andDavies, emerges in the nonrelativistic limit. For massless Dirac operators,the condition on V implies the absence of non-real eigenvalues. Our resultsare further generalized to potentials with slower decay at infinity. Asan application, we determine bounds on resonances and embedded eigenvaluesof Dirac operators with Hermitian dilation-analytic potentials.
Laptev A, Solomyak M, 2013, On spectral estimates for two-dimensional Schrodinger operators, Journal of Spectral Theory, Vol: 3, Pages: 505-515, ISSN: 1664-0403
For the two-dimensional Schrödinger operator HαV=−Δ−αV, V≥0HαV=−Δ−αV, V≥0, we study the behavior of the number N−(HαV)N−(HαV) of its negative eigenvalues (bound states), as the coupling parameter αα tends to infinity. A wide class of potentials is described, for which N−(HαV)N−(HαV) has the semi-classical behavior, i.e. N−(HαV)=O(α)N−(HαV)=O(α). For the potentials from this class, the necessary and sufficient condition is found for the validity of the Weyl asymptotic law.
Laptev A, Solomyak M, 2012, On the Negative Spectrum of the Two-Dimensional Schrodinger Operator with Radial Potential, Communications in Mathematical Physics, Vol: 314, Pages: 229-241, ISSN: 1432-0916
For a two-dimensional Schrödinger operatorHαV = −−αV with the radialpotential V(x) = F(|x|), F(r) ≥ 0, we study the behavior of the number N−(HαV )of its negative eigenvalues, as the coupling parameter α tends to infinity. We obtain thenecessary and sufficient conditions for the semi-classical growth N−(HαV ) = O(α)and for the validity of the Weyl asymptotic law.
Kovarik H, Laptev A, 2012, Hardy inequalities for robin laplacians, Journal of Functional Analysis, Vol: 262, Pages: 4972-4985, ISSN: 1096-0783
In this paper we establish a Hardy inequality for Laplace operators with Robin boundary conditions. For convex domains, in particular, we show explicitly how the corresponding Hardy weight depends on the coefficient of the Robin boundary conditions. We also study several extensions to non-convex and unbounded domains.
Aermark L, Laptev A, 2012, Hardy inequalities for a magnetic Grushin operator with aharonov-bohm type magnetic field, St Petersburg Mathematical Journal, Vol: 23, Pages: 203-208, ISSN: 1061-0022
A version of the Aharonov–Bohm magnetic field for a Grushin subellipticoperator is introduced; then its quadratic form is shown to satisfy an improvedHardy inequality.
Geisinger L, Laptev A, Weidl T, 2011, Geometrical Versions of improved Berezin-Li-Yau Inequalities, Journal of Spectral Theory, Vol: 1, Pages: 87-109, ISSN: 1664-039X
We study the eigenvalues of the Dirichlet Laplace operator on an arbitrary bounded, open set in RdRd, d≥2d≥2. In particular, we derive upper bounds on Riesz means of order σ≥3/2σ≥3/2, that improve the sharp Berezin inequality by a negative second term. This remainder term depends on geometric properties of the boundary of the set and reflects the correct order of growth in the semi-classical limit.Under certain geometric conditions these results imply new lower bounds on individual eigenvalues, which improve the Li–Yau inequality.
Frank RL, Laptev A, Seiringer R, 2011, A Sharp Bound on Eigenvalues of Schrodinger Operators on the Half-line with Complex-valued Potentials, Conference on Operator Theory, Analysis and Mathematical Physics (OTAMP 2008), Publisher: BIRKHAUSER VERLAG AG, Pages: 39-+, ISSN: 0255-0156
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- Citations: 20
Frank RL, Laptev A, 2010, Inequalities between Dirichlet and Neumann Eigenvalues on the Heisenberg Group, International Mathematics Research Notices, Vol: 15, Pages: 2889-2902, ISSN: 1687-0247
We prove that for any domain in the Heisenberg group the (k + 1)th Neumann eigenvalue of the sub-Laplacian is strictly less than the kth Dirichlet eigenvalue. As a byproduct, we obtain similar inequalities for the Euclidean Laplacian with a homogeneous magnetic field.
Laptev A, Safronov O, 2009, Eigenvalue estimates for Schrodinger operators with complex potentials, Communications in Mathematical Physics, Vol: 292, Pages: 29-54, ISSN: 1432-0916
We discuss properties of eigenvalues of non-self-adjoint Schrödinger operatorswith complex-valued potential V. Among our results are estimates of the sum ofpowers of imaginary parts of eigenvalues by the L p-norm of ℑV.
Frank RL, Laptev A, Molchanov S, 2008, Eigenvalue estimates for magnetic Schrodinger operators in domains, PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, Vol: 136, Pages: 4245-4255, ISSN: 0002-9939
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- Citations: 9
Gordon A, Holt J, Laptev A, et al., 2008, On the Simon-Spencer theorem, JOURNAL OF MATHEMATICAL PHYSICS ANALYSIS GEOMETRY, Vol: 4, Pages: 108-120, ISSN: 1812-9471
Laptev A, Dolbeaut J, Loss M, 2007, Lieb-Thirring inequalities with improved constants.
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