## Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376

Citations: | 8 - 5 self |

### BibTeX

@MISC{Lenz_stollmann:eigenfunction,

author = {Daniel Lenz and Carsten Schubert and Peter Stollmann},

title = {Stollmann: Eigenfunction expansion for Schrödinger operators on metric graphs (Preprint arXiv:0801.1376},

year = {}

}

### OpenURL

### Abstract

Abstract. We construct an expansion in generalized eigenfunctions for Schrödinger operators on metric graphs. We require rather minimal assumptions concerning the graph structure and the boundary conditions at the vertices.

### Citations

323 | Schrödinger operators, with application to quantum mechanics and global - Cycon, Froese, et al. - 1987 |

137 | Kirchoff's rule for quantum wires
- Kostrykin, Schrader
- 1999
(Show Context)
Citation Context ...in the discussion of random models. See [4, 10, 20] and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see =-=[1, 9, 14, 15, 16, 7, 11, 12, 13]-=- for recent results on quantum graphs). There the authors consider a rather special class of metric graphs, due to the random model they have in mind. We point out, however that part of their discussi... |

88 | Spectral theory and differential operators - Davies - 1995 |

86 |
Quantum graphs: I. Some basic structures
- Kuchment
(Show Context)
Citation Context ...in the discussion of random models. See [4, 10, 20] and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see =-=[1, 9, 14, 15, 16, 7, 11, 12, 13]-=- for recent results on quantum graphs). There the authors consider a rather special class of metric graphs, due to the random model they have in mind. We point out, however that part of their discussi... |

46 | 2005 Quantum graphs: II. Some spectral properties of quantum and combinatorial graphs
- Kuchment
(Show Context)
Citation Context ...in the discussion of random models. See [4, 10, 20] and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see =-=[1, 9, 14, 15, 16, 7, 11, 12, 13]-=- for recent results on quantum graphs). There the authors consider a rather special class of metric graphs, due to the random model they have in mind. We point out, however that part of their discussi... |

34 |
Kirchhoff’s rule for quantum wires. II. The inverse problem with possible applications to quantum computers, Fortschritte der
- Kostrykin, Schrader
(Show Context)
Citation Context |

29 | Absolutely continuous spectra of quantum tree graphs with weak disorder
- Aizenman, Sims, et al.
(Show Context)
Citation Context |

26 |
Graph models for waves in thin structures
- Kuchment
(Show Context)
Citation Context |

23 | A duality between Schrödinger operators on graphs and certain Jacobi matrices, Annales de l’Institut Henri Poincaré. Physique Théorique 66
- Exner
- 1997
(Show Context)
Citation Context |

22 |
Berezanskii; Expansions in eigenfunctions of selfadjoint operators
- M
- 1968
(Show Context)
Citation Context ... the boundary conditions at the vertices. Introduction Expansion in generalized eigenfunctions is a topic that dates back to Fourier’s work, at least. A classical reference is Berezanskii’s monograph =-=[2]-=-. Motivated by examples from Mathematical Physics there has been a steady development involving new models. One trigger of more recent results is the importance of generalized eigenfunction expansions... |

17 |
Laplacians on metric graphs: eigenvalues, resolvents and semigroups, in Quantum Graphs and their Applications
- Kostrykin, Schrader
- 2005
(Show Context)
Citation Context |

10 | Schnol’s theorem for strongly local forms
- Monvel, Lenz, et al.
- 2008
(Show Context)
Citation Context ...s, due to the random model they have in mind. We point out, however that part of their discussion is rather abstract and pretty much equivalent to what had been obtained in [4]. As was pointed out in =-=[3]-=-, the Dirichlet form framework of the latter article applies to a class of quantum graphs with Kirchhoff boundary conditions. The point of the present paper is to establish an expansion in generalized... |

10 | Anderson localization for radial tree-like random quantum graphs
- Hislop, Post
- 2009
(Show Context)
Citation Context ...pansions in the discussion of random models. See [4, 10, 20] and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, =-=[9]-=- (see [1, 9, 14, 15, 16, 7, 11, 12, 13] for recent results on quantum graphs). There the authors consider a rather special class of metric graphs, due to the random model they have in mind. We point o... |

10 | Generalized eigenfunctions for waves in inhomogeneous media
- Klein, Koines, et al.
(Show Context)
Citation Context ... Physics there has been a steady development involving new models. One trigger of more recent results is the importance of generalized eigenfunction expansions in the discussion of random models. See =-=[4, 10, 20]-=- and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see [1, 9, 14, 15, 16, 7, 11, 12, 13] for recent result... |

7 |
Caught by disorder: A course on bound states in random media
- Stollmann
- 2001
(Show Context)
Citation Context ... Physics there has been a steady development involving new models. One trigger of more recent results is the importance of generalized eigenfunction expansions in the discussion of random models. See =-=[4, 10, 20]-=- and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see [1, 9, 14, 15, 16, 7, 11, 12, 13] for recent result... |

5 | Laplace-Operatoren auf Quantengraphen. Diplom Thesis - Schubert - 2006 |

4 |
de Monvel and P. Stollmann, Eigenfunction expansions for generators of Dirichlet forms
- Boutet
(Show Context)
Citation Context ... Physics there has been a steady development involving new models. One trigger of more recent results is the importance of generalized eigenfunction expansions in the discussion of random models. See =-=[4, 10, 20]-=- and the references in there. This was also the background of the first paper that established eigenfunction expansions for quantum graphs, [9] (see [1, 9, 14, 15, 16, 7, 11, 12, 13] for recent result... |

3 |
Expansions in Generalized
- Poerschke, Stolz, et al.
- 1989
(Show Context)
Citation Context ...that have been considered so far. Our discussion is intrinsic and does not require an embedding of the metric graph into an ambient space. As far as methods are concerned, we rely on the results from =-=[17]-=- rather than the approach of [2] that had been used in [9]. However, this is mostly a question of habit. In either approach a main point is to establish certain trace class properties of auxiliary fun... |

2 |
Faris: Self-adjoint operators
- G
- 1975
(Show Context)
Citation Context ... theoretic results for potential perturbations of the operators H0 = HL,P for a quantum graph satisfying assumption (S). For a general background, we refer the reader to [18], Section X. 2 as well as =-=[8]-=-, § 5 and § 6. We are going to consider the class of potentials V ∈ � L2 (e) with M := MV := sup{�VI�2 : I edge segment with length between u and 2u} < ∞. This class will be denoted by L 2 loc,u (X). ... |