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We describe the geometric and probabilistic properties of a noncommutative -torus in a magnetic field. We study the volume invariance, integrated scalar curvature, and the volume form by using the operator method of perturbation by an inner derivation of the magnetic Laplacian operator on the noncommutative -torus. We then analyze the magnetic stochastic process describing the motion of a particle subject to a uniform magnetic field on the noncommutative -torus, and discuss the related main properties. [ABSTRACT FROM AUTHOR]

The bicomplex magnetic Laplacian (shortly bc‐magnetic Laplacian) is defined as a couple of magnetic Laplacians on two separate complex planes. In the present paper, we provide an explicit characterization of its L2$$ {L}^2 $$‐eigenspaces when acting on the so‐called bicomplex p‐Hilbert space. The common eigenfunction problem for the bc‐magnetic Laplacian and its †$$ \dagger $$‐conjugate is also tackled. The corresponding eigenspaces are described and the explicit expression of their reproducing kernels is given. [ABSTRACT FROM AUTHOR]

In this paper, we consider the semiclassical 2D magnetic Schrödinger operator in the case where the magnetic field vanishes along a smooth closed curve. Assuming that this curve has an axis of symmetry, we prove that semiclassical tunneling occurs. The main result is an expression of the splitting of the first two eigenvalues and an explicit tunneling formula. [ABSTRACT FROM AUTHOR]

In this paper, we prove the multiplicity and concentration behavior of complex-valued solutions for the following Kirchhoff–Schrödinger equation with magnetic field - (a ε 2 + b ε [ u ] A / ε 2) Δ A / ε u + V (x) u = f (| u | 2) u , x ∈ R 2 , where ε > 0 is a small parameter, the nonlinearity f is involved in critical exponential growth in the sense of Trudinger–Moser inequality and both V : R 2 → R and A : R 2 → R 2 are continuous potential and magnetic potential, respectively. Imposing a local constraint of potential V(x) first introduced from del Pino and Felmer, we get the multiplicity of solutions by way of the relationship between the number of the solutions and the topology of the set with V attaining the minimum. Our strategy of main proof is based on the variational methods combined with the penalization technique, the Trudinger–Moser inequality and Ljusternik–Schnirelmann theory, and our result is still new even without magnetic effect. [ABSTRACT FROM AUTHOR]

We address the concrete spectral analysis of an invariant magnetic Schrödinger operator, which acts on one-dimensional L 2 -mixed automorphic functions associated with a given equivariant pair (ρ , τ) and a discrete subgroup of the semi-direct group U (1) ⋉ C . To achieve this, we employ a lifting theorem to the classical automorphic functions associated with a specific pseudo-character. In addition, we offer a partial characterization of the equivariant pairs relative to our setting and discuss possible generalization to higher dimensions. [ABSTRACT FROM AUTHOR]

We prove an analogue of Pólya's conjecture for the eigenvalues of the magnetic Schrödinger operator with Aharonov-Bohm potential on the disk for Dirichlet and magnetic Neumann boundary conditions. This answers a question posed by R. L. Frank and A. M. Hansson. [ABSTRACT FROM AUTHOR]

We consider a novel bounded integral transform with a kernel function being the n-th polyanalytic Intissar–Hermite polynomial. We provide a concrete description of its range, shown to be a reproducing kernel Hilbert space for which we provide an explicit closed formula of its reproducing kernel. The limit of these ranges leads, in a precise sense, to the so-called van Eijndhoven–Meyers Bargmann space, which also corresponds to the limit case. The null space of the considered transform is also characterized. It gives rise in particular to new integral representation of the true-polyanalytic Bargmann space by means of the polyanalytic Intissar–Hermite integral transform. [ABSTRACT FROM AUTHOR]

We present the basic physical and mathematical ideas (P. Curie, Darboux, Poincaré, Dirac) that led to the concept of magnetic charge, the general construction of magnetic Laplacians for magnetic monopoles on Riemannian manifolds, and the results of Kordyukov and the author on the quasiclassical approximation for eigensections of these operators. [ABSTRACT FROM AUTHOR]

In 1996 Erdös showed that among planar domains of fixed area, the smallest principal eigenvalue of the Dirichlet Laplacian with a constant magnetic field is uniquely achieved on the disk. We establish a quantitative version of this inequality, with an explicit remainder term depending on the field strength that measures how much the domain deviates from the disk. [ABSTRACT FROM AUTHOR]

Using the generalization of the multidimensional WKB method to magnetic Laplacians corresponding to monopoles, which we proposed earlier, we obtain explicit formulas for quasi-classical approximations of eigenfunctions for the Dirac monopole. [ABSTRACT FROM AUTHOR]

We provide a concrete characterization of the poly-analytic planar automorphic functions, a special class of non analytic planar automorphic functions with respect to the Appell–Humbert automorphy factor, arising as images of the holomorphic ones by means of the creation differential operator. This is closely connected to the spectral theory of the magnetic Laplacian on the complex plane. [ABSTRACT FROM AUTHOR]

We consider the eigenvalues of the magnetic Laplacian on a bounded domain Ω of R 2 with uniform magnetic field β > 0 and magnetic Neumann boundary conditions. We find upper and lower bounds for the ground state energy λ 1 and we provide semiclassical estimates in the spirit of Kröger for the first Riesz mean of the eigenvalues. We also discuss upper bounds for the first eigenvalue for non-constant magnetic fields β = β (x) on a simply connected domain in a Riemannian surface. In particular: we prove the upper bound λ 1 < β for a general plane domain for a constant magnetic field, and the upper bound λ 1 < max x ∈ Ω ‾ ⁡ | β (x) | for a variable magnetic field when Ω is simply connected. For smooth domains, we prove a lower bound of λ 1 depending only on the intensity of the magnetic field β and the rolling radius of the domain. The estimates on the Riesz mean imply an upper bound for the averages of the first k eigenvalues which is sharp when k → ∞ and consists of the semiclassical limit 2 π k | Ω | plus an oscillating term. We also construct several examples, showing the importance of the topology: in particular we show that an arbitrarily small tubular neighborhood of a generic simple closed curve has lowest eigenvalue bounded away from zero, contrary to the case of a simply connected domain of small area, for which λ 1 is always small. [ABSTRACT FROM AUTHOR]

In this paper we introduce the magnetic Hodge Laplacian, which is a generalization of the magnetic Laplacian on functions to differential forms. We consider various spectral results, which are known for the magnetic Laplacian on functions or for the Hodge Laplacian on differential forms, and discuss similarities and differences of this new "magnetic-type" operator. [ABSTRACT FROM AUTHOR]

Signed directed graphs contain both sign and direction information on their edges, providing richer information about real-world phenomena compared to unsigned or undirected graphs. However, analyzing such graphs is more challenging due to their complexity, and the limited availability of existing methods. Consequently, despite their potential uses, signed directed graphs have received less research attention. In this paper, we propose a novel spectral graph convolution model that effectively captures the underlying patterns in signed directed graphs. To this end, we introduce a complex Hermitian adjacency matrix that can represent both sign and direction of edges using complex numbers. We then define a magnetic Laplacian matrix based on the adjacency matrix, which we use to perform spectral convolution. We demonstrate that the magnetic Laplacian matrix is positive semi-definite (PSD), which guarantees its applicability to spectral methods. Compared to traditional Laplacians, the magnetic Laplacian captures additional edge information, which makes it a more informative tool for graph analysis. By leveraging the information of signed directed edges, our method generates embeddings that are more representative of the underlying graph structure. Furthermore, we showed that the proposed method has wide applicability for various graph types and is the most generalized Laplacian form. We evaluate the effectiveness of the proposed model through extensive experiments on several real-world datasets. The results demonstrate that our method outperforms state-of-the-art techniques in signed directed graph embedding. • This paper introduces a novel complex Hermitian adjacency matrix that encodes signed directed graphs using complex numbers. • It proposes a magnetic Laplacian matrix via the Hermitian and proves its positive semi-definite property. • With the magnetic Laplacian, this paper define a spectral graph convolution operation. • The proposed convolution enjoys wide applicability for several graph types from vanilla to signed directed. • SD-GCN shows superior performance in several datasets compare to other SOTAs. [ABSTRACT FROM AUTHOR]

We devote this paper to the multiplicity and concentration behavior of complex-valued solutions for the following Schrödinger–Poisson equation with magnetic Laplacian (ε i ∇ - A (x) ) 2 u + V (x) u - ϕ (x) | u | 3 u = f (| u | 2 ) u + | u | 4 u in R 3 , - ε 2 Δ ϕ = | u | 5 in R 3 , where ε > 0 is a small parameter, the nonlinearity f ∈ C 1 (R , R) , both V : R 3 → R and A : R 3 → R 3 are continuous potential and magnetic potential, respectively. Imposing a global condition on potential V(x) first introduced from Rabinowitz, to get the multiplicity of solutions we investigate the relationship between the number of solutions and the topology of a set with V attaining the minimum based on the variational methods and Lusternik–Schnirelmann theory. Moreover, we prove that the problem under consideration admits the solutions with exponential decay at infinity and concentrating around global minimum of V(x), for ε > 0 small. We would like to remark that a key ingredient is how to apply a delicate analysis to overcome the difficulty due to the presence of double critical growths and magnetic field. [ABSTRACT FROM AUTHOR]

For a given discrete subgroup Γ = α Z of (C , +) , we consider the Landau Hamiltonian acting on the space of (L 2 , Γ) -automorphic functions, perturbed by an electric potential with compact support on C / Γ . We investigate the asymptotic behaviour of the discrete spectrum near Landau levels. [ABSTRACT FROM AUTHOR]

We give an explicit expansion series and an integral representation for the heat kernel associated with the magnetic Laplacian on the quantized Riemann sphere. We also derive the asymptotic expansion of the associated heat operator. [ABSTRACT FROM AUTHOR]

In this paper, we estimate the magnetic Laplacian energy norm in appropriate planar domains under a weak regularity hypothesis on the magnetic field. Our main contribution is an averaging estimate, valid in small cells, allowing us to pass from non-uniform to uniform magnetic fields. As a matter of application, we derive new upper and lower bounds of the lowest eigenvalue of the Dirichlet Laplacian which match in the regime of large magnetic field intensity. Furthermore, our averaging technique allows us to estimate the nonlinear Ginzburg–Landau energy, and as a byproduct, yields a non-Gaussian trial state for the Dirichlet magnetic Laplacian. [ABSTRACT FROM AUTHOR]

Higher-order networks encode the many-body interactions existing in complex systems, such as the brain, protein complexes, and social interactions. Simplicial complexes are higher-order networks that allow a comprehensive investigation of the interplay between topology and dynamics. However, simplicial complexes have the limitation that they only capture undirected higher-order interactions while in real-world scenarios, often there is a need to introduce the direction of simplices, extending the popular notion of direction of edges. On graphs and networks the Magnetic Laplacian, a special case of connection Laplacian, is becoming a popular operator to address edge directionality. Here we tackle the challenge of handling directionality in simplicial complexes by formulating higher-order connection Laplacians taking into account the configurations induced by the simplices’ directions. Specifically, we define all the connection Laplacians of directed simplicial complexes of dimension two and we discuss the induced higher-order diffusion dynamics by considering instructive synthetic examples of simplicial complexes. The proposed higher-order diffusion processes can be adopted in real scenarios when we want to consider higher-order diffusion displaying non-trivial frustration effects due to conflicting directionalities of the incident simplices.

We discuss isoperimetric inequalities for the magnetic Laplacian on bounded domains of R 2 endowed with an Aharonov–Bohm potential. When the flux of the potential around the pole is not an integer, the lowest eigenvalue for the Neumann and the Steklov problems is positive. We establish isoperimetric inequalities for the lowest eigenvalue in the spirit of the classical inequalities of Szegö–Weinberger, Brock and Weinstock, the model domain being a disk with the pole at its center. We consider more generally domains in the plane endowed with a rotationally invariant metric, which include the spherical and the hyperbolic case. [ABSTRACT FROM AUTHOR]

We compute the trace formula for the magnetic Laplacian on a compact hyperbolic surface of constant curvature with a constant magnetic field for energies above the Mane critical level of the corresponding magnetic geodesic flow. We discuss the asymptotic behavior of the coefficients of the trace formula when the energy approaches the Mane critical level. [ABSTRACT FROM AUTHOR]

In this paper we construct a Birkhoff normal form for a semiclassical magnetic Schrödinger operator with non-degenerate magnetic field, and discrete magnetic well, defined on an even-dimensional Riemannian manifold M. We use this normal form to get an expansion of the first eigenvalues in powers of ...1/2, and semiclassicalWeyl asymptotics for this operator. [ABSTRACT FROM AUTHOR]

In this paper we consider singularly perturbed nonlinear Schrödinger equations with electromagnetic potentials and involving continuous nonlinearities with subcritical, critical or supercritical growth. By means of suitable variational techniques, truncation arguments and Lusternik–Schnirelman theory, we relate the number of nontrivial complex-valued solutions with the topology of the set where the electric potential attains its minimum value. [ABSTRACT FROM AUTHOR]

We define the random magnetic Laplacian with spatial white noise as magnetic field on the two-dimensional torus using paracontrolled calculus. It yields a random self-adjoint operator with pure point spectrum and domain a random subspace of nonsmooth functions in L 2. We give sharp bounds on the eigenvalues which imply an almost sure Weyl-type law. [ABSTRACT FROM AUTHOR]

We obtain upper bounds for the first eigenvalue of the magnetic Laplacian associated to a closed potential 1-form (hence, with zero magnetic field) acting on complex functions of a planar domain Ω , with magnetic Neumann boundary conditions. It is well known that the first eigenvalue is positive whenever the potential admits at least one non-integral flux. By gauge invariance, the lowest eigenvalue is simply zero if the domain is simply connected; then, we obtain an upper bound of the ground state energy depending only on the ratio between the number of holes and the area; modulo a numerical constant the upper bound is sharp and we show that in fact equality is attained (modulo a constant) for Aharonov-Bohm-type operators acting on domains punctured at a maximal ϵ -net. In the last part, we show that the upper bound can be refined, provided that one can transform the given domain in a simply connected one by performing a number of cuts with sufficiently small total length; we thus obtain an upper bound of the lowest eigenvalue by the ratio between the number of holes and the area, multiplied by a Cheeger-type constant, which tends to zero when the domain is metrically close to a simply connected one. [ABSTRACT FROM AUTHOR]

We are interested in the spectral properties of the magnetic Schrödinger operator He in a domain Ω ⊆ ℝ² with compact boundary and with magnetic field of intensity ε -2. We impose Dirichlet boundary conditions on ∂Ω. Our main focus is the existence and description of the so-called edge states, namely eigenfunctions for Hε whose mass is localized at scale ε along the boundary ∂Ω. When the intensity of the magnetic field is large (i.e., ε << 1), we show that such edge states exist. Furthermore, we give a detailed description of their localization close to ∂Ω, as well as how their mass is distributed along it. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Many earlier works were devoted to the study of the breakdown of superconductivity in type-II superconducting bounded planar domains, submitted to smooth magnetic fields. In the present contribution, we consider a new situation where the applied magnetic field is piecewise-constant, and the discontinuity jump occurs along a smooth curve meeting the boundary transversely. To handle this situation, we perform a detailed spectral analysis of a new effective model. Consequently, we establish the monotonicity of the transition from a superconducting to a normal state. Moreover, we determine the location of superconductivity in the sample just before it disappears completely. Interestingly, the study shows similarities with the case of corner domains subjected to constant fields. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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; m = 0 , 1 , .... 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. [ABSTRACT FROM AUTHOR]

We deal with the following nonlinear Schrödinger equation with magnetic field and critical growth: { (ε ı ∇ − A (x)) 2 u + V (x) u = f (| u | 2) u + | u | 2 ⁎ − 2 u in R N , u ∈ H 1 (R N , C) , where ε > 0 is a small parameter, N ≥ 3 , 2 ⁎ = 2 N N − 2 is the critical Sobolev exponent, A ∈ C 1 (R N , R N) is a magnetic vector potential, V : R N → R is a continuous positive potential having a local minimum and f : R → R is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we investigate the existence and concentration of nontrivial solutions for ε > 0 small enough. [ABSTRACT FROM AUTHOR]

We show that allowing magnetic fields to be complex-valued leads to an improvement in the magnetic Hardy-type inequality due to Laptev and Weidl. The proof is based on the study of momenta on the circle with complex magnetic fields, which is of independent interest in the context of PT-symmetric and quasi-Hermitian quantum mechanics. We study basis properties of the non-self-adjoint momenta and derive closed formulae for the similarity transforms relating them to self-adjoint operators. [ABSTRACT FROM AUTHOR]

In this note, we consider the Iwatsuka model with a positive 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 an estimate on the current of quantum states localized in energy near a threshold. [ABSTRACT FROM AUTHOR]

We consider a Riemannian cylinder Ω endowed with a closed potential 1-form A and study the magnetic Laplacian Δ A with magnetic Neumann boundary conditions associated with those data. We establish a sharp lower bound for the first eigenvalue and show that the equality characterizes the situation where the metric is a product. We then look at the case of a planar domain bounded by two closed curves and obtain an explicit lower bound in terms of the geometry of the domain. We finally discuss sharpness of this last estimate. [ABSTRACT FROM AUTHOR]

On a compact Riemannian manifold M with boundary, we give an estimate for the eigenvalues (λk(τ,α))k of the magnetic Laplacian with Robin boundary conditions. Here, τ is a positive number that defines the Robin condition and α is a real differential 1-form on M that represents the magnetic field. We express these estimates in terms of the mean curvature of the boundary, the parameter τ, and a lower bound of the Ricci curvature of M (see Theorem 1.3 and Corollary 1.5). The main technique is to use the Bochner formula established in Egidi et al. (Ricci curvature and eigenvalue estimates for the magentic Laplacian on manifolds, arXiv:1608.01955v1) for the magnetic Laplacian and to integrate it over M (see Theorem 1.2). In the last part, we compare the eigenvalues λk(τ,α) with the first eigenvalue λ1(τ)=λ1(τ,0) (i.e. without magnetic field) and the Neumann eigenvalues λk(0,α) (see Theorem 1.6) using the min-max principle. [ABSTRACT FROM AUTHOR]

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 threshold in the spectrum of H 0 . In addition, for non-negative electric perturbations V we study the spectral properties of H 0 ± V , by considering the Spectral Shift Function associated to the operator pair ( H 0 ± V , H 0 ) . We compute the asymptotic behavior of the Spectral Shift Function at the thresholds, which are the only points where it can grow to infinity. [ABSTRACT FROM AUTHOR]

We propose a framework for the visualization of directed networks relying on the eigenfunctions of the magnetic Laplacian, called here Magnetic Eigenmaps. The magnetic Laplacian is a complex deformation of the well-known combinatorial Laplacian. Features such as density of links and directionality patterns are revealed by plotting the phases of the first magnetic eigenvectors. An interpretation of the magnetic eigenvectors is given in connection with the angular synchronization problem. Illustrations of our method are given for both artificial and real networks. [ABSTRACT FROM AUTHOR]

We give 2 widest Mehler's formulas for the univariate complex Hermite polynomials [ABSTRACT FROM AUTHOR]

We note that building a magnetic Laplacian from the Markov transition matrix, rather than the graph adjacency matrix, yields several benefits for the magnetic eigenmaps algorithm. The two largest benefits are that the embedding becomes more stable as a function of the rotation parameter g , and the principal eigenvector of the magnetic Laplacian now converges to the page rank of the network as a function of diffusion time. We show empirically that this normalization improves the phase and real/imaginary embeddings of the low-frequency eigenvectors of the magnetic Laplacian. [ABSTRACT FROM AUTHOR]

We estimate the ground state energy for the magnetic Laplacian with a Robin boundary condition. In a special asymptotic limit, we find that the magnetic field does not contribute to the two-term expansion of the ground state energy, thereby proving that the Robin boundary condition weakens diamagnetism. We discuss a semi-classical version of the operator and prove that the ground states concentrate near the boundary points of maximal curvature. [ABSTRACT FROM AUTHOR]

The aim of the paper is to derive spectral estimates into several classes of magnetic systems. They include three-dimensional regions with Dirichlet boundary as well as a particle in confined by a local change of the magnetic field. We establish two-dimensional Berezin-Li-Yau and Lieb-Thirring-type bounds in the presence of magnetic fields and, using them, get three-dimensional estimates for the eigenvalue moments of the corresponding magnetic Laplacians. [ABSTRACT FROM AUTHOR]