arXiv subjects

Vincent Bruneau

Publications and source records attributed to Vincent Bruneau.

At least 19 recordsLinked to original sources

arXiv2026

The Landau-Dirac operator with shell interactions: self-adjointness and clustering

We consider the two-dimensional Dirac operator with constant magnetic field that is perturbed by a combination of electrostatic and Lorentz-scalar delta interactions with variable coefficients supported on a smooth closed curve. Self-adjointness is studied in the so called non critical and critical cases. In the non-critical case the essential spectrum is unchanged - it remains to be the set of the Landau-Dirac levels, the eigenvalues of infinite multiplicity of the unperturbed operator - while in the critical case an additional interval of essential spectrum emerges in the spectral gap containing zero. Our main result concerns the discrete spectrum in the non-critical case: using the pseudodifferential properties of the involved boundary integral operators, we show that the eigenvalues accumulate at each Landau-Dirac level at a rate governed by the logarithmic capacity of the curve. A novel and surprising phenomenon is the change in the side of the accumulation depending on the position relative to the critical value. As a byproduct, clusters of eigenvalues for a family of exterior boundary value problems are obtained via confining couplings; the infinite-mass boundary condition arises as a special case.

math-ph↗

arXiv2026

The Spectral Shift Function for Non-Self-Adjoint Perturbations

This paper is devoted to the definition and analysis of the spectral shift function (SSF) associated with non-self-adjoint perturbations of self-adjoint operators. Motivated by applications in scattering theory, we consider both trace-class and relatively trace-class perturbations. We extend the Lifshits-Kre__n trace formula to non-self-adjoint operators under suitable assumptions on the spectrum and the behavior of the resolvent. The role of spectral singularities is carefully analyzed, and we provide a generalization of the SSF using functional calculus. Finally, we apply our results to Schr{\"o}dinger operators with complex-valued short-range potentials in dimension three. Toy models illustrate properties that one might hope to extend to general cases. In particular, they suggest that the SSF carries information on the presence of complex eigenvalues.

math-ph↗

arXiv2025

Spectrum of the perturbed Landau-Dirac operator

In this article, we consider the Dirac operator with constant magnetic field in $\mathbb R^2$. Its spectrum consists of eigenvalues of infinite multiplicities, known as the Landau-Dirac levels. Under compactly supported perturbations, we study the distribution of the discrete eigenvalues near each Landau-Dirac level. Similarly to the Landau (Schrödinger) operator, we demonstrate that a three-terms asymptotic formula holds for the eigenvalue counting function. One of the main novelties of this work is the treatment of some perturbations of variable sign. In this context we explore some remarkable phenomena related to the finiteness or infiniteness of the discrete eigenvalues, which depend on the interplay of the different terms in the matrix perturbation.

math.SP↗

arXiv2022

A Poincaré-Steklov map for the MIT bag model

The purpose of this paper is to introduce and study Poincaré-Steklov (PS) operators associated to the Dirac operator $D_m$ with the so-called MIT bag boundary condition. In a domain $Ω\subset\mathbb{R}^3$, for a complex number $z$ and for $U_z$ a solution of $(D_m-z)U_z=0$, the associated PS operator maps the value of $Γ_- U_z$, the MIT bag boundary value of $U_z$, to $Γ_+ U_z$, where $Γ_\pm$ are projections along the boundary $\partialΩ$ and $(Γ_ - + Γ_+) = t_{\partialΩ}$ is the trace operator on $\partialΩ$. In the first part of this paper, we show that the PS operator is a zero-order pseudodifferential operator and give its principal symbol. In the second part, we study the PS operator when the mass $m$ is large, and we prove that it fits into the framework of $1/m$-pseudodifferential operators, and we derive some important properties, especially its semiclassical principal symbol. Subsequently, we apply these results to establish a Krein-type resolvent formula for the Dirac operator $H_M= D_m+ Mβ1_{\mathbb{R}^3\setminus\overlineΩ}$ for large masses $M>0$, in terms of the resolvent of the MIT bag operator on $Ω$. With its help, the large coupling convergence with a convergence rate of $\mathcal{O}(M^{-1})$ is shown.

math.AP↗

arXiv2019

Threshold singularities of the spectral shift function for geometric perturbations of magnetic Hamiltonians

We consider the 3D Schrödinger operator $H_0$ with constant magnetic field $B$ of scalar intensity $b>0$, and its perturbations $H_+$ (resp., $H_-$) obtained by imposing Dirichlet (resp., Neumann) conditions on the boundary of the bounded domain $Ω_{\rm in} \subset {\mathbb R}^3$. We introduce the Krein spectral shift functions $ξ(E;H_\pm,H_0)$, $E \geq 0$, for the operator pairs $(H_\pm,H_0)$, and study their singularities at the Landau levels $Λ_q : = b(2q+1)$, $q \in {\mathbb Z}_+$, which play the role of thresholds in the spectrum of $H_0$. We show that $ξ(E;H_+,H_0)$ remains bounded as $E \uparrow Λ_q$, $q \in {\mathbb Z}_+$ being fixed, and obtain three asymptotic terms of $ξ(E;H_-,H_0)$ as $E \uparrow Λ_q$, and of $ξ(E;H_\pm,H_0)$ as $E \downarrow Λ_q$. The first two terms are independent of the perturbation while the third one involves the {\em logarithmic capacity} of the projection of $Ω_{\rm in}$ onto the plane perpendicular to $B$.

math.SP↗

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↗

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

Spectral Properties of Harmonic Toeplitz Operators and Applications to the Perturbed Krein Laplacian

We consider harmonic Toeplitz operators $T_V = PV:{\mathcal H}(Ω) \to {\mathcal H}(Ω)$ where $P: L^2(Ω) \to {\mathcal H}(Ω)$ is the orthogonal projection onto ${\mathcal H}(Ω) = \left\{u \in L^2(Ω)\,|\,Δu = 0 \; \mbox{in}\;Ω\right\}$, $Ω\subset {\mathbb R}^d$, $d \geq 2$, is a bounded domain with $\partial Ω\in C^\infty$, and $V: Ω\to {\mathbb C}$ is a suitable multiplier. First, we complement the known criteria which guarantee that $T_V$ is in the $p$th Schatten-von Neumann class $S_p$, by sufficient conditions which imply $T_V \in S_{p, {\rm w}}$, the weak counterpart of $S_p$. Next, we assume that $Ω$ is the unit ball in ${\mathbb R}^d$, and $V = \overline{V}$ is radially symmetric, and investigate the eigenvalue asymptotics of $T_V$ if $V$ has a power-like decay at $\partial Ω$ or $V$ is compactly supported in $Ω$. Further, we consider general $Ω$ and $V \geq 0$ which is regular in $Ω$, and admits a power-like decay of rate $γ> 0$ at $\partial Ω$, and we show that in this case $T_V$ is unitarily equivalent to a pseudo-differential operator of order $-γ$, self-adjoint in $L^2(\partial Ω)$. Using this unitary equivalence, we obtain the main asymptotic term of the eigenvalue counting function for the operator $T_V$. Finally, we introduce the Krein Laplacian $K \geq 0$, self-adjoint in $L^2(Ω)$; it is known that ${\rm Ker}\,K = {\mathcal H}(Ω)$, and the zero eigenvalue of $K$ is isolated. We perturb $K$ by $V \in C(\overlineΩ;{\mathbb R})$, and show that $σ_{\rm ess}(K+V) = V(\partial Ω)$. Assuming that $V \geq 0$ and $V{|\partial Ω} = 0$, we study the asymptotic distribution of the eigenvalues of $K \pm V$ near the origin, and find that the effective Hamiltonian which governs this distribution is the Toeplitz operator $T_V$.

math.SP↗

arXiv2017

Threshold Singularities of the Spectral Shift Function for a Half-Plane Magnetic Hamiltonian

We consider the Schrödinger operator with constant magnetic field defined on the half-plane with a Dirichlet boundary condition, $H_0$, and a decaying electric perturbation $V$. We analyze the spectral density near the Landau levels, which are thresholds in the spectrum of $H_0,$ by studying the Spectral Shift Function (SSF) associated to the pair $(H_0+V,{H_0})$. For perturbations of a fixed sign, we estimate the SSF in terms of the eigenvalue counting function for certain compact operators. If the decay of $V$ is power-like, then using pseudodifferential analysis, we deduce that there are singularities at the thresholds and we obtain the corresponding asymptotic behavior of the SSF. Our technique gives also results for the Neumann boundary condition.

math.SP↗

arXiv2016

Convergence of a Vector Penalty Projection Scheme for the Navier-Stokes Equations with moving body

In this paper, we analyse a Vector Penalty Projection Scheme (see [1]) to treat the displacement of a moving body in incompressible viscous flows in the case where the interaction of the fluid on the body can be neglected. The presence of the obstacle inside the computational domain is treated with a penalization method introducing a parameter $\eta$. We show the stability of the scheme and that the pressure and velocity converge towards a limit when the penalty parameter $\epsilon$, which induces a small divergence and the time step $\delta$t tend to zero with a proportionality constraint $\epsilon$ = $\lambda$$\delta$t. Finally, when $\eta$ goes to 0, we show that the problem admits a weak limit which is a weak solution of the Navier-Stokes equations with no-sleep condition on the solid boundary. R{\'e}sum{\'e} Dans ce travail nous analysons un sch{\'e}ma de projection vectorielle (voir [1]) pour traiter le d{\'e}placement d'un corps solide dans un fluide visqueux incompressible dans le cas o` u l'interaction du fluide sur le solide est n{\'e}gligeable. La pr{\'e}sence de l'obstacle dans le domaine solide est mod{\'e}lis{\'e}e par une m{\'e}thode de p{\'e}nalisation. Nous montrons la stabilit{\'e} du sch{\'e}ma et la convergence des variables vitesse-pression vers une limite quand le param etre $\epsilon$ qui assure une faible divergence et le pas de temps $\delta$t tendent vers 0 avec une contrainte de proportionalit{\'e} $\epsilon$ = $\lambda$$\delta$t. Finalement nous montrons que leprob{\`i} eme converge au sens faible vers une solution des equations de Navier-Stokes avec une condition aux limites de non glissement sur lafront{\`i} ere immerg{\'e}e quand le param etre de p{\'e}nalisation $\eta$ tend vers 0.

math.NA↗

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↗

arXiv2015

Counting function of magnetic resonances for exterior problems

We study the asymptotic distribution of the resonances near the Landau levels $Λ\_q =(2q+1)b$, $q \in \mathbb{N}$, of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of a compact domain of $\mathbb{R}^3$ of the 3D Schr{ö}dinger operator with constant magnetic field of scalar intensity $b\textgreater{}0$. We investigate the corresponding resonance counting function and obtain the main asymptotic term. In particular, we prove the accumulation of resonances at the Landau levels and the existence of resonance free sectors. In some cases, it provides the discreteness of the set of embedded eigenvalues near the Landau levels.

math.SP↗

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↗

arXiv2012

Dirichlet and Neumann Eigenvalues for Half-Plane Magnetic Hamiltonians

Let $H_{0, D}$ (resp., $H_{0,N}$) be the Schroedinger operator in constant magnetic field on the half-plane with Dirichlet (resp., Neumann) boundary conditions, and let $H_\ell : = H_{0, \ell} - V$, $\ell =D,N$, where the scalar potential $V$ is non negative, bounded, does not vanish identically, and decays at infinity. We compare the distribution of the eigenvalues of $H_D$ and $H_N$ below the respective infima of the essential spectra. To this end, we construct effective Hamiltonians which govern the asymptotic behaviour of the discrete spectrum of $H_\ell$ near $\inf σ_{ess}(H_\ell) = \inf σ(H_{0,\ell})$, $\ell = D,N$. Applying these Hamiltonians, we show that $σ_{disc}(H_D)$ is infinite even if $V$ has a compact support, while $σ_{disc}(H_N)$ could be finite or infinite depending on the decay rate of $V$.

math.SP↗

arXiv2011

Counting function of characteristic values and magnetic resonances

We consider the meromorphic operator-valued function 1-K(z) = 1-A(z)/z where A(z) is holomorphic on the domain D, and has values in the class of compact operators acting in a given Hilbert space. Under the assumption that A(0) is a selfadjoint operator which can be of infinite rank, we study the distribution near the origin of the characteristic values of 1-K(z), i.e. the complex numbers w for which the operator 1-K(w) is not invertible, and we show that generically the characteristic values of 1-K(z) converge to 0 with the same rate as the eigenvalues of A(0). We apply our abstract results to the investigation of the resonances of the operator H = H_0 + V where H_0 is the shifted 3D Schrödinger operator with constant magnetic field of scalar intensity b>0, and V is a real electric potential which admits a suitable decay at infinity. It is well known that the spectrum of H_0 is purely absolutely continuous, coincides with [0,+\infty[, and the so-called Landau levels 2bq with integer q, play the role of thresholds in the spectrum of H_0. We study the asymptotic distribution of the resonances near any given Landau level, and under generic assumptions obtain the main asymptotic term of the corresponding resonance counting function, written explicitly in the terms of appropriate Toeplitz operators.

math.SP↗

arXiv2010

Discrete Spectrum of Quantum Hall Effect Hamiltonians I. Monotone Edge Potential

We consider the unperturbed operator $H_0 : = (-i \nabla - A)^2 + W$, self-adjoint in $L^2(\R^2)$. Here $A$ is a magnetic potential which generates a constant magnetic field $b>0$, and the edge potential $W$ is a non-decreasing non constant bounded function depending only on the first coordinate $x \in \R$ of $(x,y) \in \R^2$. Then the spectrum of $H_0$ has a band structure and is absolutely continuous; moreover, the assumption $\lim_{x \to \infty}(W(x) - W(-x)) < 2b$ implies the existence of infinitely many spectral gaps for $H_0$. We consider the perturbed operators $H_{\pm} = H_0 \pm V$ where the electric potential $V \in L^{\infty}(\R^2)$ is non-negative and decays at infinity. We investigate the asymptotic distribution of the discrete spectrum of $H_\pm$ in the spectral gaps of $H_0$. We introduce an effective Hamiltonian which governs the main asymptotic term; this Hamiltonian involves a pseudo-differential operator with generalized anti-Wick symbol equal to $V$. Further, we restrict our attention on perturbations $V$ of compact support and constant sign. We establish a geometric condition on the support of $V$ which guarantees the finiteness of the eigenvalues of $H_{\pm}$ in any spectral gap of $H_0$. In the case where this condition is violated, we show that, generically, the convergence of the infinite series of eigenvalues of $H_+$ (resp. $H_-$) to the left (resp. right) edge of a given spectral gap, is Gaussian.

math-ph↗

arXiv2008

Dynamical resonances and SSF singularities for a magnetic Schroedinger operator

We consider the Hamiltonian $H$ of a 3D spinless non-relativistic quantum particle subject to parallel constant magnetic and non-constant electric field. The operator $H$ has infinitely many eigenvalues of infinite multiplicity embedded in its continuous spectrum. We perturb $H$ by appropriate scalar potentials $V$ and investigate the transformation of these embedded eigenvalues into resonances. First, we assume that the electric potentials are dilation-analytic with respect to the variable along the magnetic field, and obtain an asymptotic expansion of the resonances as the coupling constant $\varkappa$ of the perturbation tends to zero. Further, under the assumption that the Fermi Golden Rule holds true, we deduce estimates for the time evolution of the resonance states with and without analyticity assumptions; in the second case we obtain these results as a corollary of suitable Mourre estimates and a recent article of Cattaneo, Graf and Hunziker \cite{cgh}. Next, we describe sets of perturbations $V$ for which the Fermi Golden Rule is valid at each embedded eigenvalue of $H$; these sets turn out to be dense in various suitable topologies. Finally, we assume that $V$ decays fast enough at infinity and is of definite sign, introduce the Krein spectral shift function for the operator pair $(H+V, H)$, and study its singularities at the energies which coincide with eigenvalues of infinite multiplicity of the unperturbed operator $H$.

math.SP↗