Elliptic differential operators on infinite graphs with general conditions on vertices

ABSTRACT

Let Γ be an unbounded metric-oriented graph embedded in ${\mathbb{R}}^n$, $\mathcal{E}$, $\mathcal{V}$ be countable sets of edges $\mathfrak{e}\in \mathcal{E}$, and vertices $v\in \mathcal{V}$ of Γ. The graph Γ is equipped with a differential equation (1) $Au\left(x\right)=\sum _{j=0}^r{a}_j\left(x\right)u^{\left(j\right)}\left(x\right)=f\left(x\right),x\in \mathrm{\Gamma }\setminus \mathcal{V}$(1) with piece-wise smooth coefficients $a_j,$ $j=,0,1,\dots ,r$, and general conditions at the vertices $v\in \mathcal{V}$ (2) $\begin{array}{rl}{B}_{k}u\left(v\right)& =\sum _{j=0}^{m_k}{b}_{j,k}\left(v\right)u_{{\mathcal{E}}_v}^{\left(j\right)}\left(v\right)={\varphi }_k\left(v\right)\in {\mathbb{C}}^{d\left(v\right)},\\ & \times v\in \mathcal{V},k=1,\dots ,m,\end{array}$(2) where $d\left(v\right)$ is the number of edges incident to v, $b_{j,k}\left(v\right)$ are $d\left(v\right)\times d\left(v\right)$ complex matrices, $u_{{\mathcal{E}}_v}^{\left(j\right)}\left(v\right)=\left(u_1^{\left(j\right)}\right(v),\dots ,u_{d\left(v\right)}^{\left(j\right)}(v\left)\right)\in {\mathbb{C}}^{d\left(v\right)},$ $u_i^{\left(j\right)}\left(v\right),i=1,\dots ,d\left(v\right)$ are limit values at the vertex $v\in \mathcal{V}$ of the derivatives $u_i^{\left(j\right)}\left(x\right)$ taken along the edges ${\mathfrak{e}}_i\in {\mathcal{E}}_v$ according to their orientation. We associate with equation (1) and the vertex conditions (2) an operator $\mathfrak{A}\mathfrak{=}\left(A,B_1,\dots ,B_m\right)$ acting from the Sobolev space $H^s\left(\mathrm{\Gamma }\right)$ to the space $H^{s-r}\left(\mathrm{\Gamma }\right)\oplus {\mathcal{L}}^2({\mathcal{V}\mathcal{)}}^m$ where ${\mathcal{L}}^2({\mathcal{V}\mathcal{)}\mathcal{=}\oplus }_{v\in \mathcal{V}}{\mathbb{C}}^{d\left(v\right)}$. We study the smoothness, and exponential behavior at infinity of solutions of equation $\mathfrak{A}u=\left(f,{\varphi }_1,\dots ,{\varphi }_m\right)$, and for periodic graphs we obtain the necessary and sufficient conditions of the Fredholmness of $\mathfrak{A}$ and a description of the essential spectrum of the realization of $\mathfrak{A}$ in $L^2\left(\mathrm{\Gamma }\right).$ We give applications of these results to the Schrödinger operators on periodic graphs with general conditions at the vertices.

KEYWORDS:

• Quantum graphs
• a priori estimate
• Fredholm theory
• essential spectrum
• limit operators

MATHEMATICS SUBJECT CLASSIFICATION (2010):

• 34B45
• 47A10

Acknowledgments

The author is grateful to the referee for valuable comments that contribute to the improvement of the work.

Disclosure statement

No potential conflict of interest was reported by the author.