Magnetic Laplacian Research Articles

The magnetic discrete Laplacian inferred from the Gauß–Bonnet operator and application

We consider the notion of $$\chi$$ -completeness of a locally finite graph and we extend this notion to the weighted magnetic graph. Moreover, we establish a link between the magnetic adjacency matrix on line graph and the magnetic discrete Laplacian on 1-forms.

Schrödinger Equation in Moving Domains

We consider the Schr\''odinger equation \begin{equation}\label{eq_abstract} i\partial_t u(t)=-\Delta u(t)~~~~~\text{ on }\Omega(t) \tag{$\ast$} \end{equation}where $\Omega(t)\subset\mathbb{R}$ is a moving domain depending on the time $t\in [0,T]$. The aim of this work is to provide a meaning to the solutions of such an equation. We use the existence of a bounded reference domain $\Omega_0$ and a specific family of unitary maps $h^\sharp(t): L^2(\Omega(t),\mathbb{C})\longrightarrow L^2(\Omega_0,\mathbb{C})$. We show that the conjugation by $h^\sharp$ provides a newequation of the form \begin{equation}\label{eq_abstract2}i\partial_t v= h^\sharp(t)H(t)h_\sharp(t) v~~~~~\text{ on }\Omega_0\tag{$\ast\ast$} \end{equation} where $h_\sharp=(h^\sharp)^{-1}$. The Hamiltonian $H(t)$ is a magnetic Laplacian operator of the form$$H(t)=-(div+iA)\circ(grad+iA)-|A|^2$$where $A$ is an explicit magnetic potential depending on the deformation of the domain $\Omega(t)$. The formulation \eqref{eq_abstract2} enables to ensure the existence of weak and strong solutions of the initial problem \eqref{eq_abstract} on $\Omega(t)$ endowed with Dirichlet boundary conditions. In addition, it also indicates that the correct Neumann type boundary conditions for \eqref{eq_abstract} are not the homogeneous but the magnetic ones$$\partial_\nu u(t)+i\langle\nu| A\rangle u(t)=0,$$even though \eqref{eq_abstract} has no magnetic term. All the previous results are also studied in presence of diffusion coefficients as well as magnetic and electric potentials. Finally, we prove some associated byproducts as an adiabatic result for slow deformations of the domain and atime-dependent version of the so-called ``Moser's trick''. We use this outcome in order to simplify Equation \eqref{eq_abstract2} and to guarantee the well-posedness for slightly less regular deformations of $\Omega(t)$.

Magnetic fractional order Orlicz–Sobolev spaces

In this paper we define the notion of nonlocal magnetic Sobolev spaces with non-standard growth for Lipschitz magnetic fields. In this context we prove a Bourgain - Brezis - Mironescu type formula for functions in this space as well as for sequences of functions. Finally, we deduce some consequences such as the $\Gamma-$convergence of modulars and convergence of solutions for some non-local magnetic Laplacian allowing non-standard growth laws to its local counterpart.

Ricci curvature and eigenvalue estimates for the magnetic Laplacian on manifolds

In this paper, we present a Lichnerowicz type estimate and (higher order) Buser type estimates for the magnetic Laplacian on a closed Riemannian manifold with a magnetic potential. These results relate eigenvalues, magnetic fields, Ricci curvature, and Cheeger type constants.

Edge States for the Magnetic Laplacian in Domains with Smooth Boundary

We are interested in the spectral properties of the magnetic Schrödinger operator $H_\varepsilon$ in a domain $\Omega \subset \mathbb{R}^2$ with compact boundary and with magnetic field of intensity $\varepsilon^{-2}$. We impose Dirichlet boundary conditions on $\partial\Omega$. Our main focus is the existence and description of the so-called edge states, namely eigenfunctions for $H_{\varepsilon}$ whose mass is localized at scale $\varepsilon$ along the boundary $\partial\Omega$. When the intensity of the magnetic field is large (i.e., $\varepsilon <<1$), we show that such edge states exist. Furthermore, we give a detailed description of their localization close to $\partial\Omega$, as well as how their mass is distributed along it.

Self-adjointness of perturbed bi-Laplacians on infinite graphs

We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.

Polyanalytic reproducing Kernels on the quantized annulus

While dealing with the constant-strength magnetic Laplacian on the annulus, we complete Peetre’s work. In particular, the eigenspaces associated with its discrete spectrum true turns out to be polyanalytic spaces with respect to the invariant Cauchy–Riemann operator, and we write down explicit formulas for their reproducing kernels. When the magnetic field strength is an integer, we compute the limits of the obtained kernels when the outer radius of the annulus tends to infinity and express them by means of the fourth Jacobi theta function and of its logarithmic derivatives. Under the same quantization condition, we also derive their transformation rule under the action of the automorphism group of the annulus.

Quasi-classical approximation for magnetic monopoles

A quasi-classical approximation is constructed to describe the eigenvalues of the magnetic Laplacian on a compact Riemannian manifold in the case when the magnetic field is given by a non-exact 2-form. For this, the multidimensional WKB method in the form of the Maslov canonical operator is applied. In this case, the canonical operator takes values in sections of a non-trivial line bundle. The constructed approximation is demonstrated for the example of the Dirac magnetic monopole on the two-dimensional sphere. Bibliography: 18 titles.

Bivariate Poly-analytic Hermite Polynomials

A new class of bivariate poly-analytic Hermite polynomials is considered. We show that they are realizable as the Fourier–Wigner transform of the univariate complex Hermite functions and form a nontrivial orthogonal basis of the classical Hilbert space on the two-complex space with respect to the Gaussian measure. Their basic properties are discussed, such as their three term recurrence relations, operational realizations and differential equations (Bochner’s property) they obey. Different generating functions of exponential type are obtained. Integral and exponential operational representations are also derived. Some applications in the context of integral transforms and the concrete spectral theory of specific magnetic Laplacians are discussed.

On the semiclassical Laplacian with magnetic field having self-intersecting zero set

This paper is devoted to the spectral analysis of the Neumann realization of the 2D magnetic Laplacian with semiclassical parameter h &gt; 0 in the case when the magnetic field vanishes along a smooth curvewhich crosses itself inside a bounded domain. We investigate the behavior of its eigenpairs in the limit h \to 0 . We show that each crossing point acts as a potential well, generating a new decay scale of h^{3/2} for the lowest eigenvalues, as well as exponential concentration for eigenvectors around the set of crossing points. These properties are consequences of the nature of associated model problems in \mathbb R^2 for which the zero set of the magnetic field is the union of two straight lines. In this paper we also analyze the spectrum of model problems when the angle between the two straight lines tends to 0.

Spectral Decimation of the Magnetic Laplacian on the Sierpinski Gasket: Solving the Hofstadter–Sierpinski Butterfly

The magnetic Laplacian (also called the line bundle Laplacian) on a connected weighted graph is a self-adjoint operator wherein the real-valued adjacency weights are replaced by unit complex-valued weights $\{\omega_{xy}\}_{xy\in E}$, satisfying the condition that $\omega_{xy}=\overline{\omega_{yx}}$ for every directed edge $xy$. When properly interpreted, these complex weights give rise to magnetic fluxes through cycles in the graph. In this paper we establish the spectrum of the magnetic Laplacian, as a set of real numbers with multiplicities, on the Sierpinski gasket graph ($SG$) where the magnetic fluxes equal $\alpha$ through the upright triangles, and $\beta$ through the downright triangles. This is achieved upon showing the spectral self-similarity of the magnetic Laplacian via a 3-parameter map $\mathcal{U}$ involving non-rational functions, which takes into account $\alpha$, $\beta$, and the spectral parameter $\lambda$. In doing so we provide a quantitative answer to a question of Bellissard [Renormalization Group Analysis and Quasicrystals (1992)] on the relationship between the dynamical spectrum and the actual magnetic spectrum. Our main theorems lead to two applications. In the case $\alpha=\beta$, we demonstrate the approximation of the magnetic spectrum by the filled Julia set of $\mathcal{U}$, the Sierpinski gasket counterpart to Hofstadter's butterfly. Meanwhile, in the case $\alpha,\beta\in \{0,\frac{1}{2}\}$, we can compute the determinant of the magnetic Laplacian determinant and the corresponding asymptotic complexity.

Thin domain limit and counterexamples to strong diamagnetism

We study the magnetic Laplacian and the Ginzburg–Landau functional in a thin planar, smooth, tubular domain and with a uniform applied magnetic field. We provide counterexamples to strong diamagnetism, and as a consequence, we prove that the transition from the superconducting to the normal state is non-monotone. In some nonlinear regime, we determine the structure of the order parameter and compute the super-current along the boundary of the sample. Our results are in agreement with what was observed in the Little–Parks experiment, for a thin cylindrical sample.

Characterization and comparison of large directed networks through the spectra of the magnetic Laplacian.

In this paper, we investigated the possibility of using the magnetic Laplacian to characterize directed networks. We address the problem of characterization of network models and perform the inference of the parameters used to generate these networks under analysis. Many interesting results are obtained, including the finding that the community structure is related to rotational symmetry in the spectral measurements for a type of stochastic block model. Due the hermiticity property of the magnetic Laplacian we show here how to scale our approach to larger networks containing hundreds of thousands of nodes using the Kernel Polynomial Method (KPM), a method commonly used in condensed matter physics. Using a combination of KPM with the Wasserstein metric, we show how we can measure distances between networks, even when these networks are directed, large, and have different sizes, a hard problem that cannot be tackled by previous methods presented in the literature.

On the Bochner Laplacian operator on theta line bundle over quasi-tori

In this paper, we consider the Laplcian operator on theta line bundle over the quasi-torus, which is called the Bochner Laplacian. This operator has a canonical realization as a magnetic Laplacian acting on complex valued functions satisfying a functional equation. We study the spectral properties of such Laplacian and we show that its spectrum is reduced to eigenvalues πm; Then, we give a concrete description of each eigenspace in terms of Hermite and complex Hermite polynomials. In particular, an explicit description of the L2-holomorphic sections on the above line bundle is presented as the eigenspace of the magnetic Laplacian corresponding to the least eigenvalue. Also by using the periodization principle, the associated invariant integral operators are discussed.

Holomorphic quantum Hall states in higher Landau levels

Eigenstates of the planar magnetic Laplacian with a homogeneous magnetic field form degenerate energy bands, the Landau levels. We discuss the unitary correspondence between states in higher Landau levels and those in the lowest Landau level, where wave functions are holomorphic. We apply this correspondence to many-body systems; in particular, we represent effective Hamiltonians and particle densities in higher Landau levels by using corresponding quantities in the lowest Landau level.

Квазиклассическое приближение для магнитных монополей

В работе строится квазиклассическое приближение для описания собственных значений магнитного лапласиана на компактном римановом многообразии в случае, когда магнитное поле не задается точной $2$-формой. Для этого применяется многомерный метод ВКБ в форме канонического оператора Маслова. В этом случае канонический оператор принимает значения в сечениях нетривиального линейного расслоения. Построенное приближение продемонстрировано на примере магнитного монополя Дирака на двумерной сфере. Библиография: 18 названий.

A local mountain pass approach for a class of fractional NLS equations with magnetic fields

We complete the recent study started in Ambrosio (2019) concerning the existence and concentration phenomenon of complex-valued solutions for a class of nonlinear Schrödinger equations driven by the fractional magnetic Laplacian (−Δ)As. The proofs are obtained by combining suitable variational methods with a Kato’s approximation argument for (−Δ)As. The approach developed here can be also used to consider other fractional magnetic problems like fractional magnetic Choquard equations, fractional magnetic Kirchhoff problems and fractional magnetic Schrödinger–Poisson equations, in which local conditions on the potential are assumed.

Covering Graphs, Magnetic Spectral Gaps and Applications to Polymers and Nanoribbons

In this article, we analyze the spectrum of discrete magnetic Laplacians (DML) on an infinite covering graph G ˜ → G = G ˜ / Γ with (Abelian) lattice group Γ and periodic magnetic potential β ˜ . We give sufficient conditions for the existence of spectral gaps in the spectrum of the DML and study how these depend on β ˜ . The magnetic potential can be interpreted as a control parameter for the spectral bands and gaps. We apply these results to describe the spectral band/gap structure of polymers (polyacetylene) and nanoribbons in the presence of a constant magnetic field.

Absence of embedded eigenvalues for translationally invariant magnetic Laplacians

Translationally invariant bidimensional magnetic Laplacians are considered. Using an improved version of the harmonic approximation, we establish the absence of a point spectrum under various assumptions on the behavior of the magnetic field.

Trace formula for the magnetic Laplacian

The Guillemin–Uribe trace formula is a semiclassical version of the Selberg trace formula and the more general Duistermaat–Guillemin formula for elliptic operators on compact manifolds, which reflects the dynamics of magnetic geodesic flows in terms of eigenvalues of a natural differential operator (the magnetic Laplacian) associated with the magnetic field. This paper gives a survey of basic notions and results related to the Guillemin–Uribe trace formula and provides concrete examples of its computation for two-dimensional constant curvature surfaces with constant magnetic fields and for the Katok example. Bibliography: 53 titles.