arXiv subjects

Nicolas Popoff

Publications and source records attributed to Nicolas Popoff.

At least 19 recordsLinked to original sources

arXiv2019

Eigenvalue and Resonance Asymptotics in perturbed periodically twisted tubes: Twisting versus Bending

We consider the Dirichlet Laplacian in a three-dimensional waveguide that is a small deformation of a periodically twisted tube. The deformation is given by a bending and an additional twisting of the tube, both parametrized by a coupling constant $δ$. We expand the resolvent of the perturbed operator near the bottom of its essential spectrum and we show the existence of exactly one resonance, in the asymptotic regime of $δ$ small. We are able to perform the asymptotic expansion of the resonance in $δ$, which in particular permits us to give a quantitative geometric criterion for the existence of a discrete eigenvalue below the essential spectrum. In the particular case of perturbations of straight tubes, we are able to show the existence of resonances not only near the bottom of the essential spectrum but near each threshold in the spectrum. We also obtain the asymptotic behavior of the resonances in this situation, which is generically different from the first case.

math.SP↗

arXiv2019

Magnetic fields and boundary conditions in spectral and asymptotic analysis

This memoir is devoted to a part of the results from the author about two topics: in the first part, the asymptotics of the low-lying eigenvalues of Schrödinger operators in domains that may have corners, and in the second part, the analysis of the thresholds of a class of fibered operators. The main common object is the magnetic Laplacian, and the two parts are connected through the study of model problems in unbounded domains.

math.SP↗

arXiv2018

Band functions of Iwatsuka models : power-like and flat magnetic fields

In this note we consider the Iwatsuka model with a postive increasing magnetic field having finite limits. The associated magnetic Laplacian is fibred through partial Fourier transform, and, for large frequencies, the band functions tend to the Landau levels, which are thresholds in the spectrum. The asymptotics of the band functions is already known when the magnetic field converge polynomially to its limits. We complete this analysis by giving the asymptotics for a regular magnetic field which is constant at infinity, showing that the band functions converge now exponentially fast toward the thresholds. As an application, we give a control on the current of quantum states localized in energy near a threshold.

math.SP↗

arXiv2018

Plummeting and blinking eigenvalues of the Robin Laplacian in a cuspidal domain

We consider the Robin Laplacian in the domains $Ω$ and $Ω^\varepsilon$, $\varepsilon >0$, with sharp and blunted cusps, respectively. Assuming that the Robin coefficient $a$ is large enough, the spectrum of the problem in $Ω$ is known to be residual and to cover the whole complex plane, but on the contrary, the spectrum in the Lipschitz domain $Ω^\varepsilon$ is discrete. However, our results reveal the strange behavior of the discrete spectrum as the blunting parameter $\varepsilon$ tends to 0: we construct asymptotic forms of the eigenvalues and detect families of "hardly movable" and "plummeting" ones. The first type of the eigenvalues do not leave a small neighborhood of a point for any small $\varepsilon > 0$ while the second ones move at a high rate $O(|\ln \varepsilon|)$ downwards along the real axis $\mathbb{R}$ to $ -\infty$. At the same time, any point $λ\in \mathbb{R}$ is a "blinking eigenvalue", i.e., it belongs to the spectrum of the problem in $Ω^\varepsilon$ almost periodically in the $|\ln \varepsilon|$-scale. Besides standard spectral theory, we use the techniques of dimension reduction and self-adjoint extensions to obtain these results.

math.AP↗

arXiv2018

Low-lying eigenvalues of semiclassical Schrödinger operator with degenerate wells

In this article, we consider the semiclassical Schrödinger operator $P = - h^{2} Δ+ V$ in $\mathbb{R}^{d}$ with confining non-negative potential $V$ which vanishes, and study its low-lying eigenvalues $λ_{k} ( P )$ as $h \to 0$. First, we give a necessary and sufficient criterion upon $V^{-1} ( 0 )$ for $λ_{1} ( P ) h^{- 2}$ to be bounded. When $d = 1$ and $V^{-1} ( 0 ) = \{ 0 \}$, we are able to control the eigenvalues $λ_{k} ( P )$ for monotonous potentials by a quantity linked to an interval $I_{h}$, determined by an implicit relation involving $V$ and $h$. Next, we consider the case where $V$ has a flat minimum, in the sense that it vanishes to infinite order. We give the asymptotic of the eigenvalues: they behave as the eigenvalues of the Dirichlet Laplacian on $I_{h}$. Our analysis includes an asymptotic of the associated eigenvectors and extends in particular cases to higher dimensions.

math.SP↗

arXiv2017

Self-adjoint and skew-symmetric extensions of the Laplacian with singular Robin boundary condition

We study the Laplacian in a smooth bounded domain, with a varying Robin boundary condition singular at one point. The associated quadratic form is not semi-bounded from below, and the corresponding Laplacian is not self-adjoint, it has the residual spectrum covering the whole complex plane. We describe its self-adjoint extensions and exhibit a physically relevant skew-symmetric one. We approximate the boundary condition, giving rise to a family of self-adjoint operators, and we describe their eigenvalues by the method of matched asymptotic expansions. These eigenvalues acquire a strange behaviour when the small perturbation parameter $\varepsilon>0$ tends to zero, namely they become almost periodic in the logarithmic scale $|\ln ε|$ and, in this way, "wander" along the real axis at a speed $O(\eps^{-1})$.

math.SP↗

arXiv2017

Resonances near Thresholds in slightly Twisted Waveguides

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior

math-ph↗

arXiv2017

Spectrum of the Iwatsuka Hamiltonian at thresholds

We consider the bi-dimensional Schrödinger operator with unidirectionally constant magnetic field, $H_0$, sometimes known as the "Iwatsuka Hamiltonian". This operator is analytically fibered, with band functions converging to finite limits at infinity. We first obtain the asymptotic behavior of the band functions and its derivatives. Using this results we give estimates on the current and on the localization of states whose energy value is close to a given \emph{threshold} in the spectrum of $H_0$. In addition, for a non-negative electric perturbation $V$ we study the spectral density of $H_0\pm V$ by considering the Spectral Shift Function associated to the operator pair $(H_0\pm V,H_0)$. We describe the continuity and boundedness properties of the spectral shift function, and we compute the asymptotic behavior at the thresholds, which are the only points where it can grows to infinity.

math.SP↗

arXiv2016

Magnetic Laplacian in sharp three dimensional cones

The core result of this paper is an upper bound for the ground state energyof the magnetic Laplacian with constant magnetic field on cones that are contained in ahalf-space. This bound involves a weighted norm of the magnetic field related to momentson a plane section of the cone. When the cone is sharp, i.e. when its section is small, thisupper bound tends to 0. A lower bound on the essential spectrum is proved for familiesof sharp cones, implying that if the section is small enough the ground state energy is aneigenvalue. This circumstance produces corner concentration in the semi-classical limit forthe magnetic Schrödinger operator when such sharp cones are involved.

math.SP↗

arXiv2016

Eigenvalue counting function for Robin Laplacians on conical domains

We study the discrete spectrum of the Robin Laplacian $Q^Ω_α$ in $L^2(Ω)$, \[ u\mapsto -Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on }\partialΩ, \] where $Ω\subset \mathbb{R}^{3}$ is a conical domain with a regular cross-section $Θ\subset \mathbb{S}^2$, $n$ is the outer unit normal, and $α>0$ is a fixed constant. It is known from previous papers that the bottom of the essential spectrum of $Q^Ω_α$ is $-α^2$ and that the finiteness of the discrete spectrum depends on the geometry of the cross-section. We show that the accumulation of the discrete spectrum of $Q^Ω_α$ is determined by the discrete spectrum of an effective Hamiltonian defined on the boundary and far from the origin. By studying this model operator, we prove that the number of eigenvalues of $Q^Ω_α$ in $(-\infty,-α^2-λ)$, with $λ>0$, behaves for $λ\to0$ as \[ \dfrac{α^2}{8πλ} \int_{\partialΘ} κ_+(s)^2d s +o\left(\frac{1}λ\right), \] where $κ_+$ is the positive part of the geodesic curvature of the cross-section boundary.

math.SP↗

arXiv2016

On the negative spectrum of the Robin Laplacian in corner domains

For a bounded corner domain $Ω$, we consider the Robin Laplacian in $Ω$ with large Robin parameter. Exploiting multiscale analysis and a recursive procedure, we have a precise description of the mechanism giving the ground state of the spectrum. It allows also the study of the bottom of the essential spectrum on the associated tangent structures given by cones. Then we obtain the asymptotic behavior of the principal eigenvalue for this singular limit in any dimension, with remainder estimates. The same method works for the Schrödinger operator in $\mathbb{R}^n$ with a strong attractive delta-interaction supported on $\partialΩ$. Applications to some Erhling's type estimates and the analysis of the critical temperature of some superconductors are also provided.

math.SP↗

arXiv2016

Ground state energy of the magnetic Laplacian on corner domains

The asymptotic behavior of the first eigenvalues of magnetic Laplacian operators with large magnetic fields and Neumann realization in smooth three-dimensional domains is characterized by model problems inside the domain or on its boundary. In two-dimensional polygonal domains, a new set of model problems on sectors has to be taken into account. In this paper, we consider the class of general corner domains. In dimension 3, they include as particular cases polyhedra and axisymmetric cones. We attach model problems not only to each point of the closure of the domain, but also to a hierarchy of "tangent substructures" associated with singular chains. We investigate properties of these model problems, namely continuity, semi-continuity, existence of generalized eigenfunctions satisfying exponential decay. We prove estimates for the remainders of our asymptotic formula. Lower bounds are obtained with the help of an IMS partition based on adequate two-scale coverings of the corner domain, whereas upper bounds are established by a novel construction of quasimodes, qualified as sitting or sliding according to spectral properties of local model problems. A part of our analysis extends to any dimension.

math.SP↗

arXiv2015

An effective Hamiltonian for the eigenvalue asymptotics of a Robin Laplacian with a large parameter

We consider the Laplacian on a class of smooth domains $Ω\subset \mathbb{R}^ν$, $ν\ge 2$, with attractive Robin boundary conditions: \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \ α>0, \] where $n$ is the outer unit normal, and study the asymptotics of its eigenvalues $E_{j}(Q^Ω_α)$ as well as some other spectral properties for $α\to+\infty$ We work with both compact domains and non-compact ones with a suitable behavior at infinity. For domains with compact $C^2$ boundaries and fixed $j$, we show that \[ E_{j}(Q^Ω_α)=-α^2+μ_j(α)+{\mathcal O}(\log α), \] where $μ_j(α)$ is the $j^{\mbox{th}}$ eigenvalue, as soon as it exists, of $-Δ_{S}-(ν-1)αH$ with $(-Δ_{S})$ and $H$ being respectively the positive Laplace-Beltrami operator and the mean curvature on $\partialΩ$. Analogous results are obtained for a class of domains with non-compact boundaries. In particular, we discuss the existence of eigenvalues in non-compact domains and the existence of spectral gaps for periodic domains. We also show that the remainder estimate can be improved under stronger regularity assumptions. The effective Hamiltonian $-Δ_{S}-(ν-1)αH$ enters the framework of semi-classical Schrödinger operators on manifolds, and we provide the asymptotics of its eigenvalues in the limit $α\to+\infty$ under various geometrical assumptions. In particular, we describe several cases for which our asymptotics provides gaps between the eigenvalues of $Q^Ω_α$ for large $α$.

math.SP↗

arXiv2015

Band functions in the presence of magnetic steps

We complete the analysis of the band functions for two-dimensional magnetic Schrödinger operators with piecewise constant magnetic fields. The discontinuity of the magnetic field can create edge currents that ow along the discontinuity that have been described by physicists. Properties of these edge currents are directly related to the behavior of the band functions. The effective potential of the fiber operator is an asymmetric double well (eventually degenerated) and the analysis of the splitting of the bands incorporates the asymmetry. If the magnetic field vanishes, the reduced operator has essential spectrum and we provide an explicit description of the band functions located below the essential spectrum. For non degenerate magnetic steps, we provide an asymptotic expansion of the band functions at infinity. We prove that when the ratio of the two magnetic fields is rational, a splitting of the band functions occurs and has a natural order, predicted by numerical computations.

math.SP↗

arXiv2014

Limiting absorption principle for the Magnetic Dirichlet Laplacian in a half-plane

We consider the Dirichlet Laplacian in the half-plane with constant magnetic field. Due to the translational invariance this operator admits a fiber decomposition and a family of dis- persion curves, that are real analytic functions. Each of them is simple and monotically decreasing from positive infinity to a finite value, which is the corresponding Landau level. These finite limits are thresholds in the purely absolutely continuous spectrum of the magnetic Laplacian. We prove a limiting absorption principle for this operator both outside and at the thresholds. Finally, we establish analytic and decay properties for functions lying in the absorption spaces. We point out that the analysis carried out in this paper is rather general and can be adapted to a wide class of fibered magnetic Laplacians with thresholds in their spectrum that are finite limits of their band functions.

math.SP↗

arXiv2014

Mean curvature bounds and eigenvalues of Robin Laplacians

We consider the Laplacian with attractive Robin boundary conditions, \[ Q^Ω_αu=-Δu, \quad \dfrac{\partial u}{\partial n}=αu \text{ on } \partialΩ, \] in a class of bounded smooth domains $Ω\in\mathbb{R}^ν$; here $n$ is the outward unit normal and $α>0$ is a constant. We show that for each $j\in\mathbb{N}$ and $α\to+\infty$, the $j$th eigenvalue $E_j(Q^Ω_α)$ has the asymptotics \[ E_j(Q^Ω_α)=-α^2 -(ν-1)H_\mathrm{max}(Ω)\,α+{\mathcal O}(α^{2/3}), \] where $H_\mathrm{max}(Ω)$ is the maximum mean curvature at $\partial Ω$. The discussion of the reverse Faber-Krahn inequality gives rise to a new geometric problem concerning the minimization of $H_\mathrm{max}$. In particular, we show that the ball is the strict minimizer of $H_\mathrm{max}$ among the smooth star-shaped domains of a given volume, which leads to the following result: if $B$ is a ball and $Ω$ is any other star-shaped smooth domain of the same volume, then for any fixed $j\in\mathbb{N}$ we have $E_j(Q^B_α)>E_j(Q^Ω_α)$ for large $α$. An open question concerning a larger class of domains is formulated.

math.SP↗

arXiv2014

Characterization of bulk states in one-edge quantum Hall systems

We study magnetic quantum Hall systems in a half-plane with Dirichlet boundary conditions along the edge. Much work has been done on the analysis of the currents associated with states whose energy is located between Landau levels. These edge states are localized near the boundary and they carry a non-zero current. In this article, we study the behavior of states with energy close to a Landau level that are referred to as bulk states in the physics literature. The magnetic Schrödinger operator is invariant with respect to translations in the direction of the edge and is a direct integral of operators indexed by a real wave number. We analyse the fiber operators and prove new asymptotics on the band functions and their first derivative as the wave number goes to infinity. We apply these results to prove that the current carried by a bulk state is small compared to the current carried by an edge state. We also prove that the bulk states are exponentially small near the edge.

math-ph↗

arXiv2014

On the ground state of the Laplacian in presence of a magnetic field created by a rectilinear current

We consider the three-dimensional Laplacian with a magnetic field created by an infinite rectilinear current bearing a constant current. The spectrum of the associated hamiltonian is the positive half-axis as the range of an infinity of band functions all decreasing toward 0. We make a precise asymptotics of the band function near the ground energy and we exhibit a semi-classical behavior. We perturb the hamiltonian by an electric potential. Helped by the analysis of the band functions, we show that for slow decaying potential, an infinite number of negative eigenvalues are created whereas only finite number of eigenvalues appears for fast decaying potential. Our results show different borderline type conditions that in the case where there is no magnetic field.

math.AP↗