We study the magnetic Laplacian on the Lieb lattice and prove the Cantor spectrum for arbitrary irrational magnetic flux. We also provide a complete spectral analysis for the reduced one-dimensional Hamiltonian, proving Cantor spectra for all irrational frequencies, and sharp arithmetic phase transitions. Part of our analysis reveals a novel coexistence phenomenon of point spectrum and absolutely/singular continuous spectrum.

We investigate a two-dimensional magnetic Laplacian with two radially symmetric magnetic wells. Its spectral properties are determined by the tunneling between them. If the tunneling is weak and the wells are mirror symmetric, the two lowest eigenfunctions are localized in both wells being distributed roughly equally. In this note, we show that an exponentially small symmetry violation can in this situation have a dramatic effect, making each of the eigenfunctions localized dominantly in one well only. This is reminiscent of the 'flea on the elephant' effect for Schrödinger operators; our result shows that it has a purely magnetic counterpart.

Bicomplex Polyholomorphic Bergman Spaces Associated with a Bicomplex Magnetic Laplacian on the Discus

This paper aims to show that, in the limit of strong magnetic fields, the optimal domains for eigenvalues of magnetic Laplacians tend to exhibit symmetry. We establish several asymptotic bounds on magnetic eigenvalues to support this conclusion. Our main result implies that if, for a bounded simply-connected planar domain, the n-th eigenvalue of the magnetic Dirichlet Laplacian with uniform magnetic field is smaller than the corresponding eigenvalue for a disk of the same area, then the Fraenkel asymmetry of that domain tends to zero in the strong magnetic field limit. Comparable results are also derived for the magnetic Dirichlet Laplacian on rectangles, as well as the magnetic Dirac operator with infinite mass boundary conditions on smooth domains. As part of our analysis, we additionally provide a new estimate for the torsion function on rectangles.

Emotions play a foundational role in human cognition and social interactions, influencing interpersonal relationships, communication effectiveness, decision-making, and the customization of health and technology services. Accurately interpreting these internal emotional states from physiological signals is methodologically challenging, especially when it comes to decoding emotions from neurophysiological data, which presents significant signal-processing and representational difficulties. Electroencephalography (EEG) has been a primary modality for decoding emotions, but it is complicated by unique complexities that hinder reliable interpretation. Additionally, existing literature has mainly focused on broad emotional distinctions, such as positive versus negative emotions, and has given relatively little attention to classification of multiple emotional states, like distinguishing between nine discrete emotional categories. We propose FCMagnet, a fully complex-valued magnetic graph convolutional network for nine-class EEG emotion recognition. Directed effective connectivity is estimated via MVAR modeling and Partial Directed Coherence and encoded as Hermitian (magnetic) Laplacians. FCMagnet applies complex spectral filtering, native complex linear and convolutional layers with polar-tanh activations, and a complex-valued label encoding that maps the arousal–valence circumplex to prototype points in the complex plane. Evaluated on the FACED dataset, FCMagnet achieves 33.4 ± 4.0% nine-class accuracy and improved F1 relative to classical real-valued GNNs while remaining compact. Results demonstrate that fully complex spectral filtering of effective connectivity yields richer, phase-aware representations for fine-grained emotion decoding.

Precisely identifying cancer drivers helps us to understand the molecular mechanisms of cancer, offering critical targets for early diagnosis. Despite the increasing application of graph neural networks in predicting cancer driver genes, existing approaches do not fully leverage the information from gene networks, and are unable to effectively extract node features from directed graphs. To this end, we propose MDIGNN, a novel deep learning model designed to identify cancer driver genes by integrating directed gene networks with multi-omics data. First, we construct a directed graph through the integration of existing gene networks from diverse databases and multi-omics data. Then, to encode the edge directionality, we develop a graph neural network based on the magnetic Laplacian, which relies on a complex Hermitian matrix for representing the directed graph structure. Next, we apply the channel attention and spatial attention mechanisms to improve the model’s feature representation ability. Finally, MDIGNN uses a fully connected layer to compute the cancer driver probability for each gene. In a comparative evaluation, MDIGNN outperforms existing state-of-the-art methods in the field, and it is capable of detecting potential cancer driver genes.

Abstract We introduce an operator-theoretic definition of a proper graph Laplacian as any matrix associated with a given graph that can be expressed as the composition of a divergence and a gradient operator, with the gradient acting between graph-related spaces and annihilating constant functions. This provides a unified framework for determining whether a matrix represents a genuine diffusive operator on a graph. Within this framework, we prove that the standard Laplacian, the fractional Laplacian, the d d -path Laplacians, and the degree-attracting and degree-repelling Laplacians are all proper diffusive Laplacians . In contrast, the in-degree and out-degree Laplacians correspond to advection operators, while the signed, signless, magnetic, and deformed Laplacians are improper , as they cannot be written as the composition of a divergence and a true gradient. The magnetic Laplacian is shown to arise as the Schur complement of an extended proper Laplacian defined on a higher-dimensional space, a property also inherited by the signless Laplacian. The Lerman-Ghosh Laplacian is identified as a nonconservative diffusive operator coupled to an external reservoir. Finally, we prove that the Moore-Penrose pseudoinverse of the Laplacian is itself a proper Laplacian. This classification establishes a rigorous operator-theoretic foundation for distinguishing proper, nonconservative, and improper graph Laplacians.

Multi-scale signed graph convolutional network based on framelet.

Abstract This paper investigates ground states of the following magnetic nonlinear Choquard equation: − Δ A u + V ( x ) u = μ u + ( I 2 ∗ | u | 2 ) u in R 3 , where u ∈ H 1 ( R 3 , C ) satisfies ‖ u ‖ 2 2 = m > 0 , V ( x ) ∈ L ∞ ( R 3 ) , I 2 := 1 4 π | x | is the Riesz potential and Δ A denotes the magnetic Laplacian. By considering the associated constrained minimisation problem e A ( m ) , we prove the existence of ground states by employing the concentration compactness principle. Furthermore, we provide a refined description of the asymptotic behaviour of ground states as m → ∞ by establishing the expansion of e A ( m ) up to the second order. Additionally, we conclude that, there exists a unique ground state (up to a constant phase), which must be real-valued and cylindrically symmetric when V(x) is cylindrically symmetric and m is sufficiently large.

Abstract We consider the magnetic Steklov eigenvalue problem on compact Riemannian manifolds with boundary for generic magnetic potentials and establish various results concerning the spectrum. We provide equivalent characterizations of magnetic Steklov operators which are unitarily equivalent to the classical Steklov operator and study bounds for the smallest eigenvalue. We prove a Cheeger–Jammes‐type lower bound for the first eigenvalue by introducing magnetic Cheeger constants. We also obtain an analogue of an upper bound for the first magnetic Neumann eigenvalue due to Colbois, El Soufi, Ilias, and Savo. In addition, we compute the full spectrum in the case of the Euclidean 2‐ball and 4‐ball for a particular choice of magnetic potential given by Killing vector fields, and discuss the behavior. Finally, we establish a comparison result for the magnetic Steklov operator associated with the manifold and the square root of the magnetic Laplacian on the boundary, which generalizes the uniform geometric upper bounds for the difference of the corresponding eigenvalues in the nonmagnetic case due to Colbois, Girouard, and Hassannezhad.

Abstract Influence maximization is a fundamental problem in network analysis, focusing on identifying a subset of nodes in a social network to maximize the spread of influence. In this paper, we present an approach for tackling the Influence Maximization (IM) problem, integrating Deep Reinforcement Learning (DRL) techniques with attentive Graph Neural Networks (GATs). Our study builds upon a prior algorithm (S2V-DQN-IM) and progressively refines it towards IM-GNN, ultimately achieving competitive performance against state-of-the-art methods on classic IM. Through experiments on benchmark datasets, we empirically validate the effectiveness of graph attention mechanisms and positional encoding, using the graph magnetic Laplacian, to reach state-of-the-art performance in terms of influence spread. Building on this success, we extend our IM-GNN framework to incorporate topic-awareness in TIM-GNN, recognizing the inherent topical nature of real-world diffusions. By harnessing probabilistic techniques, we construct topic-aware social graphs using real cascades and assess the effectivenesss of TIM-GNN on them. Our extensive experimental results validate the utility of our topic-aware approach, demonstrating significant advances over existing topic-aware IM methods. Finally, in order to improve upon performance (latency) at query time, we develop a variant of TIM-GNN, called TIM-GNN $$^x$$ , by using cross -attention mechanisms. We show it maintains comparable overall spread performance as its predecessor, while achieving a 10x-20x speed-up.

In this paper, we study the first eigenvalue of the magnetic Laplacian with Neumann boundary conditions in the unit disk D in R2. There is a rather complete asymptotic analysis when the constant magnetic field tends to +∞ and some inequalities seem to hold for any value of this magnetic field, leading to rather simple conjectures. Our goal is to explore these questions by revisiting a classical picture of the physicist Saint-James theoretically and numerically. On the way, we revisit the asymptotic analysis in light of the asymptotics obtained by Fournais–Helffer, that we can improve by combining them with a formula stated by Saint-James.

While dealing with the J -matrix method for the harmonic oscillator to write down its tridiagonal matrix representation in an orthonormal basis of L 2 ( R ) , we rederive a set of generalized coherent states (GCS) of Perelomov type labelled by points z of the complex plane C and depending on an integer number m ∈ Z + . The number states expansion of these GCS gives rise to coefficients that are complex Hermite polynomials and whose linear superpositions provide eigenfunctions for the two-dimensional magnetic Laplacian, associated with the m th Landau level. We further generalize this construction to the Morse oscillator by introducing a new set of Glauber-type GCS.

Sparsification of the regularized magnetic Laplacian with multi-type spanning forests

This article is devoted to the semiclassical spectral analysis of the Neumann magnetic Laplacian on a smooth bounded domain in three dimensions.Under a generic assumption on the variable magnetic field (involving a localization of the eigenfunctions near the boundary), we establish a semiclassical expansion of the lowest eigenvalues.In particular, we prove that the eigenvalues become simple in the semiclassical limit.

We present numerical minimizers for the first seven eigenvalues of the planar magnetic Dirichlet Laplacian with constant magnetic field in a wide range of field strengths. Adapting an approach by Antunes and Freitas, we use gradient descent for the minimization procedure together with the Method of Fundamental solutions for eigenvalue computation. Remarkably, we observe that when the magnetic flux exceeds the index of the target eigenvalue, the minimizer is always a disk.

The magnetic Laplacian on the disc for strong magnetic fields

Kato’s well-known distributional inequality for the magnetic Laplacian holds equally in the more general setting of non-relativistic quantum electrodynamics (QED), where the wave function is vector-valued and the vector potential is quantized. We give two new applications of this result: First, we show that eigenstates satisfy a subsolution estimate. Second, for general states, with energy distribution strictly below the ionization threshold, we give a short proof of pointwise exponential decay in the electronic configuration.

This article is devoted to the semiclassical spectral analysis of the magnetic Laplacian in two dimensions. Assuming that the magnetic field is positive and has two symmetric radial wells, we establish an accurate tunneling formula, that is a one-term estimate of the spectral gap between the lowest two eigenvalues. This gap is exponentially small when the semiclassical parameter goes to zero, but positive.

We consider the magnetic Laplacian with the homogeneous magnetic field in two and three dimensions. We prove that the (k+1)-th magnetic Neumann eigenvalue of a bounded convex planar domain is not larger than its k-th magnetic Dirichlet eigenvalue for all k∈N. In three dimensions, we restrict our attention to convex domains, which are invariant under rotation by an angle of π around an axis parallel to the magnetic field. For such domains, we prove that the (k+2)-th magnetic Neumann eigenvalue is not larger than the k-th magnetic Dirichlet eigenvalue provided that this Dirichlet eigenvalue is simple. The proofs rely on a modification of the strategy suggested by Payne and developed further by Levine and Weinberger.