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Diomba Sambou

Publications and source records attributed to Diomba Sambou.

At least 19 recordsLinked to original sources
arXiv2025

On the singularities of the spectral shift function for some tight-binding models

We consider perturbed discrete tight-binding models in $\ell^2(\mathbb{Z_h},\mathcal{G})$ describing union of quantum particles with localized interactions, where $\mathbb{Z_h}$ is the 1D lattice $h\mathbb{Z_h}$, $h > 0$, and $\mathcal G$ is a separable Hilbert space. The perturbations play the role of self-adjoint relatively compact (matrix-valued) electric potentials with $\mathcal B(\mathcal G)$-valued coefficients decaying polynomially at infinity. We analyze the Spectral Shift Function (SSF) associated to the pair of the perturbed and the unperturbed operators. On the one hand, we show that the SSF is bounded near the spectral thresholds of the essential spectrum if $\dim(\mathcal G) < +\infty$. On the other hand, if $\dim(\mathcal G) = +\infty$, we show that it may have singularities at some thresholds points $μ$ of the essential spectrum. In particular, new mechanisms allowing the SSF to have singularities at the thresholds are exhibited, based on the degeneracy of the spectrum of the unperturbed operator. Moreover, we give the main terms of the asymptotic behaviors of the SSF near $μ$ described in terms of some explicit effective Berezin-Toeplitz type operators. These results are completed by Levinson type formulas and examples of eigenvalues asymptotics for power-like and exponential decay potentials.

math.SP↗
arXiv2025

Clusters of resonances for a non-selfadjoint multichannel discrete Schrödinger operator

We study the distribution of resonances for discrete Hamiltonians of the form $H_0+V$ near the thresholds of the spectrum of $H_0$. Here, the unperturbed operator $H_0$ is a multichannel Laplace type operator on $\ell^2(\mathbb Z; \mathbb C^N) \cong \ell^2(\mathbb Z)\otimes \mathbb C^N$ and $V$ is a non-selfadjoint compact perturbation. We compute the exact number of resonances and give a precise description on their location in clusters around some special points in the complex plane.

math-ph↗
arXiv2020

On the spectral properties of non-selfadjoint discrete Schrödinger operators

Let $H_0$ be a purely absolutely continuous selfadjoint operator acting on some separable infinite-dimensional Hilbert space and $V$ be a compact non-selfadjoint perturbation. We relate the regularity properties of $V$ to various spectral properties of the perturbed operator $H_0+V$. The structure of the discrete spectrum and the embedded eigenvalues are analysed jointly with the existence of limiting absorption principles in a unified framework. Our results are based on a suitable combination of complex scaling techniques, resonance theory and positive commutators methods. Various results scattered throughout the literature are recovered and extended. For illustrative purposes, the case of the one-dimensional discrete Laplacian is emphasized.

math.SP↗
arXiv2018

Quantum time delay for unitary operators: general theory

We present a suitable framework for the definition of quantum time delay in terms of sojourn times for unitary operators in a two-Hilbert spaces setting. We prove that this time delay defined in terms of sojourn times (time-dependent definition) exists and coincides with the expectation value of a unitary analogue of the Eisenbud-Wigner time delay operator (time-independent definition). Our proofs rely on a new summation formula relating localisation operators to time operators and on various tools from functional analysis such as Mackey's imprimititvity theorem, Trotter-Kato Formula and commutator methods for unitary operators. Our approach is general and model-independent.

math-ph↗
arXiv2018

Spectral non-self-adjoint analysis of complex Dirac, Pauli and Schrödinger operators of full rank with constant magnetic fields

We consider Dirac, Pauli and Schrödinger quantum magnetic Hamiltonians of full rank in ${\rm L}^2 \big(\mathbb{R}^{2d} \big)$, $d \ge 1$, perturbed by non-self-adjoint (matrix-valued) potentials. On the one hand, we show the existence of non-self-adjoint perturbations, generating near each point of the essential spectrum of the operators, infinitely many (complex) eigenvalues. In particular, we establish point spectrum analogous of Bögli results [Bög17] obtained for non-magnetic Laplacians, and hence showing that classical Lieb-Thirring inequalities cannot hold for our magnetic models. On the other hand, we give asymptotic behaviours of the number of the (complex) eigenvalues. In particular, for compactly supported potentials, our results establish non-self-adjoint extensions of Raikov-Warzel [RW02] and Melgaard-Rozenblum [MR03] results. So, we show how the (complex) eigenvalues converge to the points of the essential spectrum asymptotically, i.e., up to a multiplicative explicit constant, as $$ \frac{1}{d!} \Bigg(\frac{\vert \ln r \vert}{\ln \vert \ln r \vert} \Bigg)^d, \quad r \searrow 0, $$ in small annulus of radius $r > 0$ around the points of the essential spectrum.

math-ph↗
arXiv2017

Resonances on regular tree graphs

We investigate the distribution of the resonances near spectral thresholds of Laplace operators on regular tree graphs with $k$-fold branching, $k \geq 1$, perturbed by nonself-adjoint exponentially decaying potentials. We establish results on the absence of resonances which in particular involve absence of discrete spectrum near some sectors of the essential spectrum of the operators.

math-ph↗
arXiv2016

On eigenvalue accumulation for non-self-adjoint magnetic operators

In this work, we use regularized determinant approach to study the discrete spectrum generated by relatively compact non-self-adjoint perturbations of the magnetic Schrödinger operator $(-i\nabla - \textbf{\textup{A}})^{2} - b$ in dimension $3$ with constant magnetic field of strength $b>0$. The situation near the Landau levels $2bq$, $q \in \mathbb{N}$, is more interesting since they play the role of thresholds of the spectrum of the free operator. First, we obtain sharp upper bounds on the number of the complex eigenvalues near the Landau levels. Under appropriate hypothesis, we then prove the presence of an infinite number of complex eigenvalues near each Landau level $2bq$, $q \in \mathbb{N}$, and the existence of sectors free of complex eigenvalues. We also prove that the eigenvalues are localized in certain sectors adjoining the Landau levels. In particular, we provide an adequate answer to the open problem from [34] about the existence of complex eigenvalues accumulating near the Landau levels. Furthermore, we prove that the Landau levels are the only possible accumulation points of the complex eigenvalues.

math.SP↗
arXiv2016

A simple criterion for the existence of nonreal eigenvalues for a class of 2D and 3D Pauli operators

In this work, we investigate the discrete spectrum generated by complex matrix-valued perturbations for a class of 2D and 3D Pauli operators with nonconstant magnetic fields. We establish a simple criterion for the potentials to produce discrete spectrum near the low ground energy of the operators. Moreover, in case of creation of nonreal eigenvalues, this criterion specifies also their location.

math.SP↗
arXiv2016

A criterion for the existence of non-real eigenvalues for a Dirac operator

The aim of this work is to explore the discrete spectrum generated by complex perturbations in $L^{2}(\mathbb{R}^3,\mathbb{C}^4)$ of the $3d$ Dirac operator $α\cdot (-i\nabla - \textbf{A}) + m β$ with variable magnetic field. Here, $α:= (α_1,α_2,α_3)$ and $β$ are $4 \times 4$ Dirac matrices, and $m > 0$ is the mass of a particle. We give a simple criterion for the potentials to generate discrete spectrum near $\pm m$. In the case of creation of non-real eigenvalues, this criterion gives also their location.

math.SP↗
arXiv2016

Quantitative bounds on the discrete spectrum of non self-adjoint quantum magnetic Hamiltonians

We establish Lieb-Thirring type inequalities for non self-adjoint relatively compact perturbations of certain operators of mathematical physics. We apply our results to quantum Hamiltonians of Schr{ö}dinger and Pauli with constant magnetic field of strength $b\textgreater{}0$. In particular, we use these bounds to obtain some information on the distribution of the eigenvalues of the perturbed operators in the neighborhood of their essential spectrum.

math-ph↗
arXiv2016

Counting function of magnetic eigenvalues for non-definite sign perturbations

We consider the perturbed operator $H(b,V) := H(b,0) + V$, where $H(b,0)$ is the $3$d Hamiltonian of Pauli with non-constant magnetic field, and $V$ is \textit{a non-definite sign electric potential} decaying exponentially with respect to the variable along the magnetic field. We prove that the only resonances of $H(b,V)$ near the low ground energy zero of $H(b,0)$ are its eigenvalues and are concentrated in the semi axis $(-\infty,0)$. Further, we establish new asymptotic expansions, upper and lower bounds on their number near zero.

math-ph↗
arXiv2015

Spectral shift function and Resonances near the low ground state for Pauli and Schrödinger operators

We study the spectral shift function (SSF) $ξ(λ)$ and the resonances of the operator $H_V := \big( σ\cdot (-i\nabla - \textbf{A}) \big)^{2} + V$ in $L^2(\mathbb{R}^3)$ near the origin. Here $σ:= (σ_1,σ_2,σ_3)$ are the $2 \times 2$ Pauli matrices and $V$ is a hermitian potential decaying exponentially in the direction of the magnetic field $\textbf{B} := \text{curl} \hspace{0.6mm} \textbf{A}$. We give a representation of the derivative of the SSF as a sum of the imaginary part of a holomorphic function and a harmonic measure related to the resonances of $H_V$. This representation warrant the Breit-Wigner approximation moreover we deduce information about the singularities of the SSF at the origin and a local trace formula.

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↗
arXiv2013

Lieb-Thirring type inequalities for non self-adjoint perturbations of magnetic Schrödinger operators

Let $H := H_{0} + V$ and $H_{\perp} := H_{0,\perp} + V$ be respectively perturbations of the free Schrödinger operators $H_{0}$ on $L^{2}\big(\mathbb{R}^{2d+1}\big)$ and $H_{0,\perp}$ on $L^{2}\big(\mathbb{R}^{2d}\big)$, $d \geq 1$ with constant magnetic field of strength $b>0$, and $V$ is a complex relatively compact perturbation. We prove Lieb-Thirring type inequalities for the discrete spectrum of $H$ and $H_{\perp}$. In particular, these estimates give $a\, priori$ information on the distribution of the discrete eigenvalues around the Landau levels of the operator, and describe how fast sequences of eigenvalues converge.

math-ph↗