## A duality between Schrödinger operators on graphs and certain Jacobi matrices (1995)

Citations: | 23 - 5 self |

### BibTeX

@MISC{Exner95aduality,

author = {Pavel Exner},

title = {A duality between Schrödinger operators on graphs and certain Jacobi matrices},

year = {1995}

}

### Years of Citing Articles

### OpenURL

### Abstract

The known correspondence between the Kronig–Penney model and certain Jacobi matrices is extended to a wide class of Schrödinger operators on graphs. Examples include rectangular lattices with and without a magnetic field, or comb–shaped graphs leading to a Maryland–type model.

### Citations

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Citation Context ...and almost periodic systems. The most popular examples are the Harper and related almost Mathieu equation dating back to [13, 14, 15]; for more recent results and an extensive bibliography see, e.g., =-=[16, 17, 18, 19, 20, 21]-=-. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples have also been studied [22, 23]. In case when \Gamma is a line with an array of point interact... |

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Citation Context ...c field, or comb--shaped graphs leading to a Maryland--type model. I Introduction Schrodinger operators on L 2 (\Gamma) , where \Gamma is a graph, were introduced into quantum mechanics long time ago =-=[1]-=-. In recent years we have wittnessed of a renewed interest to them --- see [2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and other, often nonrigorous, studies quoted in these papers --- motivated mostly by the... |

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Citation Context ...are below bounded, one can express for k 2 2 ae(H C ) " ae(H 0 ) the resolvent difference (H C \Gamma k 2 ) \Gamma1 \Gamma (H 0 \Gamma k 2 ) \Gamma1 , where H 0 = H D ff ; H N fi , by Krein's for=-=mula [32, 33, 34]-=-; this expression becomes singular under the condition (3.7) or (3.10), respectively. 2. If two vertices are joined by more than a single link, the theorem can be applied if a vertex with the free ffi... |

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Citation Context ...ion Schrodinger operators on L 2 (\Gamma) , where \Gamma is a graph, were introduced into quantum mechanics long time ago [1]. In recent years we have wittnessed of a renewed interest to them --- see =-=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and other-=-, often nonrigorous, studies quoted in these papers --- motivated mostly by the fact that they provide a natural, if idealized, model of semiconductor "quantum wire" structures. Jacobi matri... |

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Citation Context ...ion Schrodinger operators on L 2 (\Gamma) , where \Gamma is a graph, were introduced into quantum mechanics long time ago [1]. In recent years we have wittnessed of a renewed interest to them --- see =-=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and other-=-, often nonrigorous, studies quoted in these papers --- motivated mostly by the fact that they provide a natural, if idealized, model of semiconductor "quantum wire" structures. Jacobi matri... |

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Citation Context ... operator in question is a Kronig--Penney--type Hamiltonian, there is a bijective correspondence --- dubbed the French connection by B. Simon [16] --- between such systems and certain Jacobi matrices =-=[18, 24, 25, 26, 27, 28]-=-. The aim of this paper is to show that the same duality can be established for a wide class of Schrodinger operators on graphs, including the case of a nonempty boundary. In general, the resulting Ja... |

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Citation Context ...and almost periodic systems. The most popular examples are the Harper and related almost Mathieu equation dating back to [13, 14, 15]; for more recent results and an extensive bibliography see, e.g., =-=[16, 17, 18, 19, 20, 21]-=-. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples have also been studied [22, 23]. In case when \Gamma is a line with an array of point interact... |

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Citation Context ...and almost periodic systems. The most popular examples are the Harper and related almost Mathieu equation dating back to [13, 14, 15]; for more recent results and an extensive bibliography see, e.g., =-=[16, 17, 18, 19, 20, 21]-=-. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples have also been studied [22, 23]. In case when \Gamma is a line with an array of point interact... |

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Citation Context ...ion Schrodinger operators on L 2 (\Gamma) , where \Gamma is a graph, were introduced into quantum mechanics long time ago [1]. In recent years we have wittnessed of a renewed interest to them --- see =-=[2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12] and other-=-, often nonrigorous, studies quoted in these papers --- motivated mostly by the fact that they provide a natural, if idealized, model of semiconductor "quantum wire" structures. Jacobi matri... |

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Citation Context ...lot of attention in the last decade, in particular, as a laboratory for random and almost periodic systems. The most popular examples are the Harper and related almost Mathieu equation dating back to =-=[13, 14, 15]-=-; for more recent results and an extensive bibliography see, e.g., [16, 17, 18, 19, 20, 21]. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples hav... |

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Citation Context ...ree δ coupling, αj = 0 , is added at each extra link. Vertices with just two neighbors are also useful, if we want to amend the potentials Vjn with a family of point interactions — for an example see =-=[35]-=-. 3. If (i)–(iv) are relaxed, the implications (b), (c) may still hold with ℓ2 (II) replaced by a weighted ℓ2 space. If Γ is equipped with a global metric, one can establish a relation between the exp... |

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Citation Context ... operator in question is a Kronig--Penney--type Hamiltonian, there is a bijective correspondence --- dubbed the French connection by B. Simon [16] --- between such systems and certain Jacobi matrices =-=[18, 24, 25, 26, 27, 28]-=-. The aim of this paper is to show that the same duality can be established for a wide class of Schrodinger operators on graphs, including the case of a nonempty boundary. In general, the resulting Ja... |

7 |
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Citation Context ... coupling, ff j = 0 , is added at each extra link. Vertices with just two neighbors are also useful, if we want to amend the potentials V jn with a family of point interactions --- for an example see =-=[35]-=-. 3. If (i)--(iv) are relaxed, the implications (b), (c) may still hold with ` 2 (I I ) replaced by a weighted ` 2 space. If \Gamma is equipped with a global metric, one can establish a relation betwe... |

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Citation Context ...lot of attention in the last decade, in particular, as a laboratory for random and almost periodic systems. The most popular examples are the Harper and related almost Mathieu equation dating back to =-=[13, 14, 15]-=-; for more recent results and an extensive bibliography see, e.g., [16, 17, 18, 19, 20, 21]. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples hav... |

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Citation Context ...sults and an extensive bibliography see, e.g., [16, 17, 18, 19, 20, 21]. The underlying lattices are mostly periodic of dimension one or two; however, more complicated examples have also been studied =-=[22, 23]-=-. In case when \Gamma is a line with an array of point interactions, i.e., the Schrodinger operator in question is a Kronig--Penney--type Hamiltonian, there is a bijective correspondence --- dubbed th... |

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Citation Context ... operator in question is a Kronig--Penney--type Hamiltonian, there is a bijective correspondence --- dubbed the French connection by B. Simon [16] --- between such systems and certain Jacobi matrices =-=[18, 24, 25, 26, 27, 28]-=-. The aim of this paper is to show that the same duality can be established for a wide class of Schrodinger operators on graphs, including the case of a nonempty boundary. In general, the resulting Ja... |

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Citation Context ...are below bounded, one can express for k 2 2 ae(H C ) " ae(H 0 ) the resolvent difference (H C \Gamma k 2 ) \Gamma1 \Gamma (H 0 \Gamma k 2 ) \Gamma1 , where H 0 = H D ff ; H N fi , by Krein's for=-=mula [32, 33, 34]-=-; this expression becomes singular under the condition (3.7) or (3.10), respectively. 2. If two vertices are joined by more than a single link, the theorem can be applied if a vertex with the free ffi... |

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Citation Context ...are below bounded, one can express for k 2 2 ae(H C ) " ae(H 0 ) the resolvent difference (H C \Gamma k 2 ) \Gamma1 \Gamma (H 0 \Gamma k 2 ) \Gamma1 , where H 0 = H D ff ; H N fi , by Krein's for=-=mula [32, 33, 34]-=-; this expression becomes singular under the condition (3.7) or (3.10), respectively. 2. If two vertices are joined by more than a single link, the theorem can be applied if a vertex with the free ffi... |

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Citation Context ...ng is ideal, ff j = 0 or fi j = 0 , the loose ends correspond to the Dirichlet condition, ! j = 0 , and the "tooth" lengths are ` j := jjj` , we get thus an equation reminiscent of the Maryl=-=and model [17, 36]-=- with a fixed coupling strength and an additional periodic modulation of the potential. Acknowledgments. The author is grateful for the hospitality extended to him in the Institute of Mathematics, Uni... |

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