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Ahmed Intissar

Publications and source records attributed to Ahmed Intissar.

8 recordsLinked to original sources
arXiv2017

On concrete spectral properties of a twisted-Laplacian associated to a central extension of the real Heisenberg group

We consider the magnetic Laplacian $Δ_{ν,μ}$ on $\mathbb{R}^{2n}=\mathbb{C}^n$ given by $$ Δ_{ν,μ}= 4\sum\limits_{j=1}\limits^{n}\frac{\partial^2 }{\partial z_j \partial \overline{z_j}} +2iν(E+ \overline{E} +n) +2μ(E- \overline{E} ) -(ν^2+μ^2)|z|^2. $$ We show that $Δ_{ν,μ}$ is connected to the sub-Laplacian of a group of Heisenberg type given by $\mathbb{C}\times_ω\mathbb{C}^n$ realized as a central extension of the real Heisenberg group $H_{2n+1}$. We also discuss invariance properties of $Δ_{ν,μ}$ and give some of their explicit spectral properties.

math.CV↗
arXiv2015

Weighted Bergman-Dirichlet and Bargmann-Dirichlet spaces of order $m$ in high dimensions

We introduce and study a generalization of the classical weighted Bergman and Dirichlet spaces on the unit ball in high dimension, the Bergman-Dirichlet spaces. Their counterparts on the whole $n$-complex space, the Bargmann-Dirichlet spaces, are also introduced and studied. Mainly, we give a complete description of the considered spaces, including orthonormal basis and the explicit formulas for their reproducing kernel functions. Moreover, we investigate their asymptotic behavior when the curvature goes to $0$.

math.CV↗
arXiv2015

On holomorphic theta functions associated to rank $r$ isotropic discrete subgroups of a $g$-dimensional complex space

We are interested in the $L^2$-holomorphic automorphic functions on a $g$-dimensional complex space $V^g_{\mathbb{C}}$ endowed with a positive definite hermitian form and associated to isotropic discrete subgroups $Γ$ of rank $2\leq r \leq g$. We show that they form an infinite reproducing kernel Hilbert space which looks like a tensor product of a theta Fock-Bargmann space on $V^{r}_{\mathbb{C}}=Span_{\mathbb{C}}(Γ)$ and the classical Fock-Bargmann space on $V^{g-r}_{\mathbb{C}}$. Moreover, we provide an explicit orthonormal basis using Fourier series and we give the expression of its reproducing kernel function in terms of Riemann theta function of several variables with special characteristics.

math.CV↗
arXiv2012

Construction of concrete orthonormal basis for (L^2,Γ,χ)-theta functions associated to discrete subgroups of rank one in (C,+)

Let χbe a character on a discrete subgroup Γof rank one of the additive group (C,+). We construct a complete orthonormal basis of the Hilbert space of (L^2,Γ,χ)-theta functions. Furthermore, we show that it possesses a Hilbertian orthogonal decomposition involving the L^2-eigenspaces of the Landau operator Δ_ν; ν>0, associated to the eigenvalues νm. For m=0, the associated L^2-eigenspace is the Hilbert subspace of entire (L^2,Γ,χ)-theta functions. Corresponding orthonormal basis are constructed and the corresponding reproducing kernel can be expressed in terms of the generalized theta function of characteristic [α,0].

math.CV↗
arXiv2010

Landau operator on the quaternionic field

The Landau operator on the quaternionic field is defined as the partial Fourier transform of the sub-Laplacian on the quaternionic Heisenberg group. This operator is viewed as the Hamiltonian of two harmonic oscillators on the two dimensional complex space with a uniform magnetic field.

math-ph↗
arXiv2008

Landau (Γ,χ)-automorphic functions on \mathbb{C}^n of magnitude ν

We investigate the spectral theory of the invariant Landau Hamiltonian $\La^ν$ acting on the space ${\mathcal{F}}^ν_{Γ,χ}$ of $(Γ,χ)$-automotphic functions on $\C^n$, for given real number $ν>0$, lattice $Γ$ of $\C^n$ and a map $χ:Γ\to U(1)$ such that the triplet $(ν,Γ,χ)$ satisfies a Riemann-Dirac quantization type condition. More precisely, we show that the eigenspace $ {\mathcal{E}}^ν_{Γ,χ}(λ)=\set{f\in {\mathcal{F}}^ν_{Γ,χ}; \La^νf = ν(2λ+n) f}$; $λ\in\C,$ is non trivial if and only if $λ=l=0,1,2, ...$. In such case, ${\mathcal{E}}^ν_{Γ,χ}(l)$ is a finite dimensional vector space whose the dimension is given explicitly. We show also that the eigenspace ${\mathcal{E}}^ν_{Γ,χ}(0)$ associated to the lowest Landau level of $\La^ν$ is isomorphic to the space, ${\mathcal{O}}^ν_{Γ,χ}(\C^n)$, of holomorphic functions on $\C^n$ satisfying $$ g(z+γ) = χ(γ) e^{\frac ν2 |γ|^2+ν\scal{z,γ}}g(z), \eqno{(*)} $$ that we can realize also as the null space of the differential operator $\sum\limits_{j=1}\limits^n(\frac{-\partial^2}{\partial z_j\partial \bar z_j} + ν\bar z_j \frac{\partial}{\partial \bar z_j})$ acting on $\mathcal C^\infty$ functions on $\C^n$ satisfying $(*)$.

math.SP↗