Green’s function approach for quantum graphs: An overview

Andrade, Fabiano M.; Schmidt, Alexandre G. M.; Vicentini, Eduardo; Cheng, Bin Kang; Luz, M. G. E. da

doi:10.1016/j.physrep.2016.07.001

Cited by 28 publications

(24 citation statements)

References 215 publications

(401 reference statements)

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“…It is well-known that Green's functions on metric graphs are closely related to scattering problem at the branching points [23]. In our case, the component G ij , (i = j) describes, for example, heat flow conduction from i-th bond to j-th bond, while G ii describes heat flow reflection (thermal reflection) on i-th bond.…”

Section: Discussionmentioning

confidence: 93%

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Sobirov¹,

Rakhimov²,

Ergashov³

2021

Nanosystems: Phys. Chem. Math.

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This work devoted to construction of the matrix-Green's functions of initial-boundary value problems for the time-fractional diffusion equation on the metric star graph with equal bonds. In the case of Dirichlet and mixed boundary conditions we constructed Green's functions explicitly. The uniqueness of the solutions of the considered problems were proved by the method of energy integrals. Some possible applications in branched nanostructures were discussed.

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Section: Discussionmentioning

confidence: 93%

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Sobirov¹,

Rakhimov²,

Ergashov³

2021

Nanosystems: Phys. Chem. Math.

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“…The result above brings us to the following interesting and useful result. So, the wave functions for the quantum star graph can be obtained from the GFA directly from p λ Sn , without the need to resort to the calculation of the residues of the GF [8]. We can now state our main result about the connection between the maps U S Sn (k) and U G Sn (k).…”

Section: Proof Let a λmentioning

confidence: 89%

“…T . The most general BC that are consistent with the self-adjoint condition [8] are totally defined by two d j × d j matrices A j and B j such that [15] A…”

mentioning

confidence: 99%

Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs

Andrade

Severini

2018

Phys. Rev. A

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In a previous work [Andrade et al., Phys. Rep. 647, 1 (2016)], it was shown that the exact Green's function (GF) for an arbitrarily large (although finite) quantum graph is given as a sum over scattering paths, where local quantum effects are taken into account through the reflection and transmission scattering amplitudes. To deal with general graphs, two simplifying procedures were developed: regrouping of paths into families of paths and the separation of a large graph into subgraphs. However, for less symmetrical graphs with complicated topologies as, for instance, random graphs, it can become cumbersome to choose the subgraphs and the families of paths. In this work, an even more general procedure to construct the energy domain GF for a quantum graph based on its adjacency matrix is presented. This new construction allows us to obtain the secular determinant, unraveling a unitary equivalence between the scattering Schrödinger approach and the Green's function approach. It also enables us to write a trace formula based on the Green's function approach. The present construction has the advantage that it can be applied directly for any graph, going from regular to random topologies.

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“…This provides general forms for the representation of the resolvent as an integral operator and for a spectral representation. The representation of the resolvent as an integral operator, m functions, and spectral representations for operators on graphs have appeared in many, more specific situations [1], [2], [3], [5], [8], [10], [11]- [13], [15], [19], [21], [25], [27], [30].…”

Section: Introductionmentioning

confidence: 99%

Canonical systems and quantum graphs

Scarbrough¹

2021

Preprint

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The representation of the resolvent as an integral operator, the m function, and the associated spectral representation are fundamental topics in the spectral theory of self-adjoint ordinary differential operators. Versions of these are developed here for canonical systems Ju ′ = −zHu of arbitrary order. A classical result shows that canonical systems of order two can be used to realize arbitrary spectral data, in the form of m functions from the upper half plane to itself. In this paper, canonical systems on graphs, not necessarily compact but with finitely many vertices, are introduced and proved to be unitarily equivalent to certain higher order canonical systems. It is shown that any Schrödinger operator on a graph is unitarily equivalent to a canonical system on the same graph. Consequently, for an arbitrary canonical system or Schrödinger operator on a graph, a representation of the resolvent as an integral operator and a spectral representation are obtained.

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Green’s function approach for quantum graphs: An overview

Cited by 28 publications

References 215 publications

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Green's function method for time-fractional diffusion equation on the star graph with equal bonds

Unitary equivalence between the Green's function and Schrödinger approaches for quantum graphs

Canonical systems and quantum graphs

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