Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Pankrashkin, Konstantin

doi:10.1007/s11005-006-0088-0

Cited by 84 publications

(99 citation statements)

References 36 publications

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“…The proof, however, contains an element which allows one to bypass the need for symmetry (lemma 4.4) which is required in the proof of theorem 4.2, and thus gives the possibility to generalize this result to non-Neumann graphs and graphs with potentials. It was shown in [54] that theorem 2.3 holds also for δ-type conditions and for some electric potentials, if the the inverse Hill discriminant is used instead of the arccos. Therefore, one might try to repeat the proof above, replacing the arccos with the corresponding inverse Hill discriminant.…”

Section: Discretized Versions Of a Metric Graphmentioning

confidence: 99%

The nodal count {0,1,2,3,…} implies the graph is a tree

Band

2014

Phil. Trans. R. Soc. A.

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Sturm's oscillation theorem states that the n th eigenfunction of a Sturm–Liouville operator on the interval has n −1 zeros (nodes) (Sturm 1836 J. Math. Pures Appl. 1 , 106–186; 373–444). This result was generalized for all metric tree graphs (Pokornyĭ et al. 1996 Mat. Zametki 60 , 468–470 ( doi:10.1007/BF02320380 ); Schapotschnikow 2006 Waves Random Complex Media 16 , 167–178 ( doi:10.1080/1745530600702535 )) and an analogous theorem was proved for discrete tree graphs (Berkolaiko 2007 Commun. Math. Phys. 278 , 803–819 ( doi:10.1007/S00220-007-0391-3 ); Dhar & Ramaswamy 1985 Phys. Rev. Lett. 54 , 1346–1349 ( doi:10.1103/PhysRevLett.54.1346 ); Fiedler 1975 Czechoslovak Math. J. 25 , 607–618). We prove the converse theorems for both discrete and metric graphs. Namely if for all n , the n th eigenfunction of the graph has n −1 zeros, then the graph is a tree. Our proofs use a recently obtained connection between the graph's nodal count and the magnetic stability of its eigenvalues (Berkolaiko 2013 Anal. PDE 6 , 1213–1233 ( doi:10.2140/apde.2013.6.1213 ); Berkolaiko & Weyand 2014 Phil. Trans. R. Soc. A 372 , 20120522 ( doi:10.1098/rsta.2012.0522 ); Colin de Verdière 2013 Anal. PDE 6 , 1235–1242 ( doi:10.2140/apde.2013.6.1235 )). In the course of the proof, we show that it is not possible for all (or even almost all, in the metric case) the eigenvalues to exhibit a diamagnetic behaviour. In addition, we develop a notion of ‘discretized’ versions of a metric graph and prove that their nodal counts are related to those of the metric graph.

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Section: Discretized Versions Of a Metric Graphmentioning

confidence: 99%

The nodal count {0,1,2,3,…} implies the graph is a tree

Band

2014

Phil. Trans. R. Soc. A.

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“…This relation between quantum and combinatorial graph operators is well known and has been exploited many times (e.g., [1,5,8,34,35,39,50]). …”

Section: Then λ Is In the Spectrum Of The Graphene Hamiltonian H If Amentioning

confidence: 99%

“…This derivation was triggered by the one done in [36] for the photonic crystal case, as well by [2,3], albeit the presented computation is simpler and more convenient for our purpose than the one in [36]. It reflects the known idea (e.g., [1,5,34,35,50]) that spectral analysis of quantum graph Hamiltonians (at least on graphs with all edges of equal lengths) splits into two essentially unrelated parts: analysis on a single edge, and then spectral analysis on the combinatorial graph, the former being independent on the graph structure, and the latter independent on the potential.…”

Section: Introductionmentioning

confidence: 97%

On the Spectra of Carbon Nano-Structures

Kuchment

Post

2007

Commun. Math. Phys.

157

218

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“…Magnetic Hamiltonians on quantum graphs were studied e.g. in [Pan06,GG08,BW14,Ber13,EM15,Exn97,Kur11,EL11]. In this review, attention is paid to magnetic quantum graphs in Section 11.…”

Section: Introductionmentioning

confidence: 99%

New isotope technologies in environmental physics

Povinec¹,

Betti²,

Jull³

et al. 2008

Acta Physica Slovaca. Reviews and Tutorials

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In the current review, we study the model of quantum graphs. We focus mainly on the resonance properties of quantum graphs. We define resolvent and scattering resonances and show their equivalence. We present various results on the asymptotics of the number of resolvent resonances in both non-magnetic and magnetic quantum graphs and find bounds on the coefficient by the leading term of the asymptotics. We explain methods how to find the spectral and resonance condition. Most of the notions and theorems are illustrated in examples. We show how to find resonances numerically and, in a simple example, we find trajectories of resonances in the complex plane. We discuss Fermi's golden rule for quantum graphs and distribution of the mean intensity for the topological resonances.

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Spectra of Schrödinger Operators on Equilateral Quantum Graphs

Cited by 84 publications

References 36 publications

The nodal count {0,1,2,3,…} implies the graph is a tree

The nodal count {0,1,2,3,…} implies the graph is a tree

On the Spectra of Carbon Nano-Structures

New isotope technologies in environmental physics

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