# On the Essential Spectrum of Quantum Graphs

On the Essential Spectrum of Quantum Graphs Let $$\Gamma \subset \mathbb {R}^{n}$$ Γ ⊂ R n be a graph periodic with respect to the action of a group $$\mathbb {G}$$ G isomorphic to $$\mathbb {Z}^{m},1\le m\le n.$$ Z m , 1 ≤ m ≤ n . We consider a one-dimensional Schrödinger operator \begin{aligned} S_{q}u(x)=\left( -\frac{d^{2}}{dx^{2}}+q(x)\right) u(x),u\in C_{0}^{\infty }(\Gamma \backslash \mathcal {V)},q\in L^{\infty }(\Gamma ) \end{aligned} S q u ( x ) = - d 2 d x 2 + q ( x ) u ( x ) , u ∈ C 0 ∞ ( Γ \ V ) , q ∈ L ∞ ( Γ ) defined on the edges of the graph $$\Gamma$$ Γ , where $$\mathcal {V}$$ V is the set of the vertices of $$\Gamma$$ Γ . The operator $$S_{q}$$ S q is extended to a closed unbounded operator $$\mathcal {H}_{q}$$ H q in $$L^{2}(\Gamma )$$ L 2 ( Γ ) with domain $$\tilde{H} ^{2}(\Gamma )$$ H ~ 2 ( Γ ) consisting of functions u belonging to the Sobolev space $$H^{2}(e)$$ H 2 ( e ) on the edges e of the graph $$\Gamma$$ Γ and satisfying the Kirchhoff–Neumann conditions at the vertices of $$\Gamma .$$ Γ . For the unbounded operator $$\mathcal {H}_{q}$$ H q we introduce a family $$Lim (\mathcal {H}_{q})$$ L i m ( H q ) of limit operators $$\mathcal {H}_{q}^{g}$$ H q g defined by the sequences $$\mathbb {G\ni }g_{m}\rightarrow \infty$$ G ∋ g m → ∞ and prove that \begin{aligned} sp_{ess}\mathcal {H}_{q}= {\displaystyle \bigcup \limits _{\mathcal {H}_{q}^{g}\in Lim(\mathcal {H}_{q})}} sp\mathcal {H}_{q}^{g}. \end{aligned} s p e s s H q = ⋃ H q g ∈ L i m ( H q ) s p H q g . We apply this result to the calculation of the essential spectra of self-adjoint Schrödinger operators with periodic potentials perturbed by terms slowly oscillating at infinity. We show that such perturbations significantly change the structure of the spectrum of Schrödinger operators with periodic potentials. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Integral Equations and Operator Theory Springer Journals

# On the Essential Spectrum of Quantum Graphs

, Volume 88 (3) – Jul 6, 2017
24 pages

/lp/springer_journal/on-the-essential-spectrum-of-quantum-graphs-RejA5AlERp
Publisher
Springer International Publishing
Subject
Mathematics; Analysis
ISSN
0378-620X
eISSN
1420-8989
D.O.I.
10.1007/s00020-017-2386-6
Publisher site
See Article on Publisher Site

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. ### Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. ### Organize your research It’s easy to organize your research with our built-in tools. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. ### Monthly Plan • Read unlimited articles • Personalized recommendations • No expiration • Print 20 pages per month • 20% off on PDF purchases • Organize your research • Get updates on your journals and topic searches$49/month

14-day Free Trial

Best Deal — 39% off

### Annual Plan

• All the features of the Professional Plan, but for 39% off!
• Billed annually
• No expiration
• For the normal price of 10 articles elsewhere, you get one full year of unlimited access to articles.

$588$360/year

billed annually

14-day Free Trial