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  1. 4 Acta Physica Polonica A

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  1. 1 2017

  2. 1 2013

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  1. 4 Kurasov P.

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Surgery of Graphs: M-Function and Spectral Gap

100%
Kurasov P.
Acta Physica Polonica A
|
2017
|
vol. 132
|
issue 6
1666-1671
EN
We discuss behaviour of the spectral gap for quantum graphs when two metric graphs are glued together. It appears that precise answer to this question can be given using a natural generalisation of the Titchmarsh-Weyl M-functions.
2
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Inverse Problems for Quantum Graphs: Recent Developments and Perspectives

100%
Kurasov P.
Acta Physica Polonica A
|
2011
|
vol. 120
|
issue 6A
A-132-A-141
EN
An introduction into the area of inverse problems for the Schrödinger operators on metric graphs is given. The case of metric finite trees is treated in detail with the focus on matching conditions. For graphs with loops we show that for almost all matching conditions the potential on the loop is not determined uniquely by the Titchmarsh-Weyl function. The class of all admissible potentials is characterized.
3
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On the Inverse Problem for Quantum Graphs with One Cycle

100%
Kurasov P.
Acta Physica Polonica A
|
2009
|
vol. 116
|
issue 5
765-771
EN
Quantum graphs having one cycle are considered. It is shown that if the cycle contains at least three vertices, then the potential on the graph can be uniquely reconstructed from the corresponding Titchmarsh-Weyl function (Dirichlet-to-Neumann map) associated with graph's boundary, provided certain non-resonant conditions are satisfied.
4
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On the Spectral Gap for Laplacians on Metric Graphs

100%
Kurasov P.
Acta Physica Polonica A
|
2013
|
vol. 124
|
issue 6
1060-1062
EN
We discuss lower and upper estimates for the spectral gap of the Laplace operator on a finite compact connected metric graph. It is shown that the best lower estimate is given by the spectral gap for the interval with the same total length as the original graph. An explicit upper estimate is given by generalizing Cheeger's approach developed originally for Riemannian manifolds.
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