000089750 001__ 89750 000089750 005__ 20210902121709.0 000089750 0247_ $$2doi$$a10.3390/math8040527 000089750 0248_ $$2sideral$$a117850 000089750 037__ $$aART-2020-117850 000089750 041__ $$aeng 000089750 100__ $$0(orcid)0000-0002-3066-7418$$aEstrada, Ernesto 000089750 245__ $$ad-Path Laplacians and Quantum Transport on Graphs 000089750 260__ $$c2020 000089750 5060_ $$aAccess copy available to the general public$$fUnrestricted 000089750 5203_ $$aWe generalize the Schrödinger equation on graphs to include long-range interactions (LRI) by means of the Mellin-transformed d-path Laplacian operators. We find analytical expressions for the transition and return probabilities of a quantum particle at the nodes of a ring graph. We show that the average return probability in ring graphs decays as a power law with time when LRI is present. In contrast, we prove analytically that the transition and return probabilities on a complete and start graphs oscillate around a constant value. This allowed us to infer that in a barbell graph-a graph consisting of two cliques separated by a path-the quantum particle get trapped and oscillates across the nodes of the path without visiting the nodes of the cliques. We then compare the use of the Mellin-transformed d-path Laplacian operators versus the use of fractional powers of the combinatorial Laplacian to account for LRI. Apart from some important differences observed at the limit of the strongest LRI, the d-path Laplacian operators produces the emergence of new phenomena related to the location of the wave packet in graphs with barriers, which are not observed neither for the Schrödinger equation without LRI nor for the one using fractional powers of the Laplacian. 000089750 540__ $$9info:eu-repo/semantics/openAccess$$aby$$uhttp://creativecommons.org/licenses/by/3.0/es/ 000089750 590__ $$a2.258$$b2020 000089750 591__ $$aMATHEMATICS$$b24 / 330 = 0.073$$c2020$$dQ1$$eT1 000089750 592__ $$a0.495$$b2020 000089750 593__ $$aMathematics (miscellaneous)$$c2020$$dQ2 000089750 655_4 $$ainfo:eu-repo/semantics/article$$vinfo:eu-repo/semantics/publishedVersion 000089750 773__ $$g8, 4 (2020), 527 [16 pp.]$$pMathematics (Basel)$$tMATHEMATICS$$x2227-7390 000089750 8564_ $$s561903$$uhttps://zaguan.unizar.es/record/89750/files/texto_completo.pdf$$yVersión publicada 000089750 8564_ $$s374405$$uhttps://zaguan.unizar.es/record/89750/files/texto_completo.jpg?subformat=icon$$xicon$$yVersión publicada 000089750 909CO $$ooai:zaguan.unizar.es:89750$$particulos$$pdriver 000089750 951__ $$a2021-09-02-09:19:08 000089750 980__ $$aARTICLE