d -Path Laplacians and Quantum Transport on Graphs

Ernesto Estrada
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Ernesto Estrada: Institute of Applied Mathematics (IUMA), Universidad de Zaragoza, Pedro Cerbuna 12, E-50009 Zaragoza, Spain

Mathematics, 2020, vol. 8, issue 4, 1-16

Abstract: We generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d -path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph—a graph consisting of two cliques separated by a path—the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d -path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d -path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian.

Keywords: Schrödinger equation; long-range interactions; nonlocality; d-path Laplacian; quantum mechanics on graphs (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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