The Point Spectrum Of Periodic Quantum Trees

Abstract

We study the point spectrum of a periodic quantum tree equipped with a Schrödinger type differential operator with delta-type vertex conditions, using subsets of the compact graph that generates the tree. We prove analogs of existing discrete results concerning the eigenvalues of such operators (see Aomoto, 1991 and see Banks, Garza-Vargas and Mukherjee, 2022). In particular, we define the density of states measure and find the measure of eigenvalues of the periodic tree. While most results carry over from the discrete case, a notable difference between the continuum and discrete cases is that a \textbf{regular} quantum periodic tree may have eigenvalues. We prove that after an arbitrarily small adjustment of edge lengths, the point spectrum of the universal cover of a compact quantum graph, with at least one cycle and the standard Schrödinger operator, is empty.

Figures (5)

• Figure 1: The universal cover from Example \ref{['example_regular']}. Depicted is the neighborhood of the vertex $r$. The set $D_{+}$ is shown in blue, and the set $D_{-}$ in red. All sinusoidal functions represent the eigenfunction on each edge, parameterized from the endpoint closer to $r$ to the endpoint farther from $r$.
• Figure 2: Graph 1 is a quantum graph $\Gamma = (V, E, \ell, \mathcal{H}_{\Gamma})$ where $\alpha_{\upsilon} < \infty$ for each $\upsilon \in V$, and Graph 2 is its derived graph. For every $\upsilon \in V$, we separated between the principal vertex and the shadow vertices. Each vertex $\upsilon$ has $\deg(\upsilon)$ shadow vertices, each corresponding to a different edge incident to $\upsilon$ in $\Gamma$. Every shadow vertex is connected to the principal vertex of the neighboring vertex to which it was connected in $\Gamma$, and to the shadow vertex corresponding to the same edge with the opposite orientation. The principal vertices are also connected to one another if they were connected in $\Gamma$, and each principal vertex is connected to all of its corresponding shadow vertices.
• Figure 3: A quantum graph $\Gamma$, in which the Q-Aomoto set corresponding to $\lambda=\pi^{2}$ is highlighted. The graph is equipped with the standard Schrödinger operator and Kirchhoff boundary conditions at all vertices. Edge lengths are indicated along the edges. The vertex $D$ together with its incident edge, and the vertex $C$ together with its incident edges, form two connected components of the first type described in Observation \ref{['Observation_in_Q-Aomoto']}. The edge $(A,B)$ forms a connected component of the second type. In this example, $\partial X_{\pi^{2}}(\Gamma)=\{A,B\}$.
• Figure 4: The quantum graph from Figure \ref{['fig:Q-Aomoto_Example']} and its associated discrete graph corresponding to the eigenvalue $\lambda = \pi^2$. The Q-Aomoto set for $\lambda$ is highlighted. In the discrete graph below, the vertices corresponding to the connected components of the Q-Aomoto set are also highlighted. Note that the edge $\left(A, B\right)$, representing a Q-Aomoto component of the second type described in Observation \ref{['Observation_in_Q-Aomoto']}, is represented as a vertex in the discrete model.
• Figure 5: The top graph contains a loop $L$ connected to the rest of the graph via a vertex $u$ of degree $5$. In the bottom graph, the vertex $u$ is split into five distinct vertices of degree $1$, each retaining one of the incident edges. This operation separates $L$ from the rest of the graph, and the remaining part of the graph is also partitioned into two connected components, denoted by $\Gamma_{1}^{L}$ and $\Gamma_{2}^{L}$.

Theorems & Definitions (30)

• Remark 1.2
• Example 1.3
• Theorem 1.4
• Definition 1.5
• Theorem 1.6
• Definition 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10
• Definition 1.11