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Ari Laptev

Publications and source records attributed to Ari Laptev.

At least 37 records · Page 2Linked to original sources
arXiv2019

Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schr{ö}dinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring 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.

math.AP↗
arXiv2019

Spectral inequalities for a class of integral operators

We obtain inequalities for the Riesz means for the discrete spectrum of a class of self-adjoint compact integral operators. Such bounds imply some inequalities for the counting function of the Dirichlet boundary problem for the Laplace operator. The paper is an extension of the results previously obtained in [5].

math.AP↗
arXiv2018

Eigenvalue bounds of mixed Steklov problems

We study bounds on the Riesz means of the mixed Steklov-Neumann and Steklov-Dirichlet eigenvalue problem on a bounded domain $Ω$ in $\mathbb{R}^n$. 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.

math.SP↗
arXiv2018

On the improvement of the Hardy inequality due to singular magnetic fields

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.

math.SP↗
arXiv2018

Magnetic rings

We study functional and spectral properties of perturbations of the magnetic Laplace operator on the circle. This operator appears when considering the restriction to the unit circle of a two-dimensional Schr{ö}dinger operator with the Bohm-Aharonov vector potential. We prove a Hardy-type inequality on the two-dimensional Euclidean space and, on the circle, a sharp interpolation inequality and a sharp Keller-Lieb-Thirring inequality.

math.AP↗
arXiv2018

Lieb--Thirring inequalities on the sphere

We prove on the sphere $\mathbb{S}^2$ the Lieb--Thirring inequalities 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 we obtain an explicit estimate for the dimension of the attractor of the Navier--Stokes system on a domain on the sphere with Dirichlet non-slip boundary conditions.

math.AP↗
arXiv2018

Interpolation inequalities and spectral estimates for magnetic operators

We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring estimates for the principal eigenvalue of magnetic Schr{ö}dinger operators. We establish explicit upper and lower bounds for the best constants and show by numerical methods that our theoretical estimates are accurate.

math.AP↗
arXiv2017

Berezin--Li--Yau inequalities on domains on the sphere

We prove Berezin--Li--Yau inequalities for the Dirichlet and Neumann eigenvalues on domains on the sphere $\mathbb{S}^{d-1}$. The case of $\mathbb{S}^{2}$ is treated in greater detail, including the vector Dirichlet Laplacian and the Stokes operator.

math.SP↗
arXiv2017

Hardy inequalities for Landau Hamiltonian and for Baouendi-Grushin operator with Aharonov-Bohm type magnetic field. Part I

In this paper we prove the Hardy inequalities for the quadratic form of the Laplacian with the Landau Hamiltonian magnetic field. Moreover, we obtain Poincaré type inequality and inequalities with more general families of weights, all with estimates for the remainder terms of these inequalities. Furthermore, we establish weighted Hardy inequalities for the quadratic form of the magnetic Baouendi-Grushin operator for the magnetic field of Aharonov-Bohm type. For these, we show refinements of the known Hardy inequalities for the Baouendi-Grushin operator involving radial derivatives in some of the variables. The corresponding uncertainty type principles are also obtained.

math.FA↗
arXiv2016

A bound for the eigenvalue counting function for Krein--von Neumann and Friedrichs extensions

For an arbitrary open, nonempty, bounded set $Ω\subset \mathbb{R}^n$, $n \in \mathbb{N}$, and sufficiently smooth coefficients $a,b,q$, we consider the closed, strictly positive, higher-order differential operator $A_{Ω, 2m} (a,b,q)$ in $L^2(Ω)$ defined on $W_0^{2m,2}(Ω)$, associated with the higher-order differential expression $$ τ_{2m} (a,b,q) := \bigg(\sum_{j,k=1}^{n} (-i \partial_j - b_j) a_{j,k} (-i \partial_k - b_k)+q\bigg)^m, \quad m \in \mathbb{N}, $$ and its Krein--von Neumann extension $A_{K, Ω, 2m} (a,b,q)$ in $L^2(Ω)$. Denoting by $N(λ; A_{K, Ω, 2m} (a,b,q))$, $λ> 0$, the eigenvalue counting function corresponding to the strictly positive eigenvalues of $A_{K, Ω, 2m} (a,b,q)$, we derive the bound $$ N(λ; A_{K, Ω, 2m} (a,b,q)) \leq C v_n (2π)^{-n} \bigg(1+\frac{2m}{2m+n}\bigg)^{n/(2m)} λ^{n/(2m)} , \quad λ> 0, $$ where $C = C(a,b,q,Ω)>0$ (with $C(I_n,0,0,Ω) = |Ω|$) is connected to the eigenfunction expansion of the self-adjoint operator $\widetilde A_{2m} (a,b,q)$ in $L^2(\mathbb{R}^n)$ defined on $W^{2m,2}(\mathbb{R}^n)$, corresponding to $τ_{2m} (a,b,q)$. Here $v_n := π^{n/2}/Γ((n+2)/2)$ denotes the (Euclidean) volume of the unit ball in $\mathbb{R}^n$. 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 of $\widetilde A_{2} (a,b,q)$ in $L^2(\mathbb{R}^n)$. We also consider the analogous bound for the eigenvalue counting function for the Friedrichs extension $A_{F,Ω, 2m} (a,b,q)$ in $L^2(Ω)$ of $A_{Ω, 2m} (a,b,q)$. No assumptions on the boundary $\partial Ω$ of $Ω$ are made.

math.AP↗
arXiv2016

Weyl type asymptotics and bounds for the eigenvalues of functional-difference operators for mirror curves

We investigate Weyl type asymptotics of functional-difference operators associated to mirror curves of special del Pezzo Calabi-Yau threefolds. These operators are $H(ζ)=U+U^{-1}+V+ζV^{-1}$ and $H_{m,n}=U+V+q^{-mn}U^{-m}V^{-n}$, where $U$ and $V$ are self-adjoint Weyl operators satisfying $UV=q^{2}VU$ with $q=e^{iπb^{2}}$, $b>0$ and $ζ>0$, $m,n\in\mathbb{N}$. We prove that $H(ζ)$ and $H_{m,n}$ are self-adjoint operators with purely discrete spectrum on $L^{2}(\mathbb{R})$. Using the coherent state transform we find the asymptotical behaviour for the Riesz mean $\sum_{j\ge 1}(λ-λ_{j})_{+}$ as $λ\to\infty$ and prove the Weyl law for the eigenvalue counting function $N(λ)$ for these operators, which imply that their inverses are of trace class.

math.SP↗
arXiv2015

Lieb-Thirring inequalities on the torus

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 $γ$-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.

math.AP↗