Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian

Olgiati, Alessandro

doi:10.1007/978-3-319-58904-6_15

Cited by 10 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…A generalization of the technique used in the proof allows one to cover also the case of one-body Hamiltonians more general than −∆ . This has been pointed out in the single component case in Remark 2.1 in [18]; we refer the reader to [16] for a more detailed analysis of what is needed in order to adapt the argument to the relevant case of the magnetic Laplacian ∆ A = (∇ − iA) 2 Remark 5. Assumption (B1) on the potential is crucial in this formalism; with different techniques (see [2]) it is possible to consider potentials with some singularity and unbounded support.…”

Section: Remarkmentioning

confidence: 92%

Section: Remarkmentioning

confidence: 92%

See 1 more Smart Citation

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Olgiati

2017

Advances in Quantum Mechanics

Self Cite

View full text Add to dashboard Cite

We report on a recent result concerning the effective dynamics for a mixture of Bose-Einstein condensates, a class of systems much studied in physics and receiving a large amount of attention in the recent literature in mathematical physics; for such models, the effective dynamics is described by a coupled system of non-linear Schödinger equations. After reviewing and commenting our proof in the mean-field regime from a previous paper, we collect the main details needed to obtain the rigorous derivation of the effective dynamics in the Gross-Pitaevskii scaling limit.

show abstract

Section: Remarkmentioning

confidence: 92%

Section: Remarkmentioning

confidence: 92%

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Olgiati

2017

Advances in Quantum Mechanics

Self Cite

View full text Add to dashboard Cite

show abstract

“…The relevance of equation (1.1) is hard to underestimate, both for the interest it deserves per se, given the variety of techniques that have been developed for its study, and for the applications in various contexts in physics. Among the latter, (1.1) is the typical effective evolution equation for the quantum dynamics of an interacting Bose gas subject to an external magnetic field, and as such it can be derived in suitable scaling limits of infinitely many particles [33,41,3,39]: in this context the |u| γ−1 u term with γ = 3 (resp., γ = 5) arises as the self-interaction term due to a two-body (resp., three-body) inter-particle interaction of short scale, whereas the (| · | −α * |u| 2 )u term accounts for a two-body interaction of meanfield type, whence its non-local character. On the other hand (1.1) arises also as an effective equation for the dynamics of quantum plasmas.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

Antonelli¹,

Michelangeli²,

Scandone³

2018

Z. Angew. Math. Phys.

View full text Add to dashboard Cite

We prove the existence of weak solutions in the space of energy for a class of non-linear Schrödinger equations in the presence of a external, rough, time-dependent magnetic potential. Under our assumptions it is not possible to study the problem by means of usual arguments like resolvent techniques or Fourier integral operators, for example. We use a parabolic regularisation and we solve the approximating Cauchy problem. This is achieved by obtaining suitable smoothing estimates for the dissipative evolution. The total mass and energy bounds allow to extend the solution globally in time. We then infer sufficient compactness properties in order to produce a global-in-time finite energy weak solution to our original problem.

show abstract

“…It can be considered as a discrete magnetic Schrödinger operator on graphs. The magnetic Laplacian is widely used in mathematics Colin de Verdière ( 2013) and physics Olgiati (2017). Since it is a complexvalued Hermitian matrix, its eigenvalues are real-valued and eigenvectors are orthonormal.…”

Section: Related Workmentioning

confidence: 99%

MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs

Zhang¹,

Hui²,

Harn³

et al. 2021

Preprint

View full text Add to dashboard Cite

Recent advancements in Graph Neural Networks have led to state-of-the-art performance on representation learning of graphs for node classification. However, the majority of existing works process directed graphs by symmetrization, which may cause loss of directional information. In this paper, we propose the magnetic Laplacian that preserves edge directionality by encoding it into complex phase as a deformation of the combinatorial Laplacian. In addition, we design an Auto-Regressive Moving-Average (ARMA) filter that is capable of learning global features from graphs. To reduce time complexity, Taylor expansion is applied to approximate the filter. We derive complex-valued operations in graph neural network and devise a simplified Magnetic Graph Convolution network, namely sMGC. Our experiment results demonstrate that sMGC is a fast, powerful, and widely applicable GNN.

show abstract

Remarks on the Derivation of Gross-Pitaevskii Equation with Magnetic Laplacian

Cited by 10 publications

References 11 publications

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Effective Non-linear Dynamics of Binary Condensates and Open Problems

Global, finite energy, weak solutions for the NLS with rough, time-dependent magnetic potentials

MGC: A Complex-Valued Graph Convolutional Network for Directed Graphs

Contact Info

Product

Resources

About