Splitting Quantum Graphs

We are interested particularly in sets of separated and self-adjoint boundary conditions $\Gamma=\{\alpha_{1},\alpha_{2},\beta_{1},\beta_{2}\}$, where these $\alpha_{i}$ and $\beta_{i}$ are $n\times n$ matrices satisfying the following:

$$\begin{split}\text{rank}(\begin{bmatrix}\alpha_{1}&\alpha_{2}\end{bmatrix})=\text{rank}(\begin{bmatrix}\beta_{1}&\beta_{2}\end{bmatrix})=n\,,\\ \alpha_{1}\alpha_{2}^{*}=\alpha_{2}\alpha_{1}^{*}\hskip 18.06749pt\text{and}\hskip 18.06749pt\beta_{1}\beta_{2}^{*}=\beta_{2}\beta_{1}^{*}\,,\end{split}$$

(1)

where the $*$ superscript represents the matrix adjoint. We say a function ${\vec{u}}$ satisfies $\Gamma$ boundary conditions if it satisfies the equations

$$\begin{split}\alpha_{1}{\vec{u}}(\vec{0})+\alpha_{2}{\vec{u}}^{\prime}(\vec{0})=\vec{0}\,,\\ \beta_{1}{\vec{u}}(\vec{\ell})+\beta_{2}{\vec{u}}^{\prime}(\vec{\ell})=\vec{0}\,.\end{split}$$

(2)

In this paper, ${\vec{u}}^{\prime}$ refers to a vector whose $i$th component is $u_{i}^{\prime}$, which is defined to be the derivative of the $i$th component $u_{i}$ of ${\vec{u}}$ with respect to the relevant spatial variable $x_{i}$.

We will follow the reasonable convention that the matrices $\beta_{1}$ and $\beta_{2}$ should both be diagonal. This comes from the natural assumption that the only interaction between functions should happen at shared vertices (in this case, only at the origin).

As an example, by choosing $\alpha_{1}$ and $\beta_{1}$ to be identity matrices while letting $\alpha_{2}$ and $\beta_{2}$ be zero matrices, $\Gamma$ would be equivalent to Dirichlet boundary conditions at all vertices. The Dirichlet trace would then be the column vector

$$\gamma_{D}{\vec{u}}=\begin{bmatrix}u_{1}(\ell_{1})&\ldots&u_{n}(\ell_{n})&u_{1}(0)&\ldots&u_{n}(0)\end{bmatrix}^{t}\,,$$

(4)

where $t$ represents the matrix (nonconjugating) transpose. One other notable trace that will show up often is the Neumann trace, defined as

$$\gamma_{N}{\vec{u}}=\begin{bmatrix}u_{1}^{\prime}(\ell_{1})&\ldots&u_{n}^{\prime}(\ell_{n})&-u_{1}^{\prime}(0)&\ldots&-u_{n}^{\prime}(0)\end{bmatrix}^{t}\,.$$

(5)

We are interested in solving eigenvalue problems of the form

$$\begin{cases}(H^{\Omega}-\lambda){\vec{u}}=\vec{0}\,,\\ \gamma_{\Gamma}{\vec{u}}=\vec{0}\,,\end{cases}$$

(6)

where ${\vec{u}}({\vec{x}},\lambda)$ is a function over $\Omega$, $H^{\Omega}$ is a differential expression defined as

$$H^{\Omega}=-\Delta+V:={\rm diag\,}(-\partial^{2}_{x_{i}}+V_{i}(x_{i})|1\leq i\leq n)\,,$$

(7)

where ${\rm diag\,}$ denotes a diagonal matrix whose $i$th diagonal entry is the $i$th argument specified. We assume that each potential function $V_{i}$ is in $L^{1}(0,\ell_{i})$.

Note that $\gamma_{\Gamma_{1}}u$ and $\gamma_{\Gamma_{2}}{\vec{u}}$ have a near identical definition to that in formula (3) for their respective $\alpha$ and $\beta$ matrices. However, in the case of $\gamma_{\Gamma_{1}}u$, the new vertex $s_{1}$ will take the place of the origin $0$, and in the case of $\gamma_{\Gamma_{2}}{\vec{u}}$ $s_{1}$ will take the place of $\ell_{j}$.

We will make heavy use of the Evans function, a powerful and popular tool in the stability theory of traveling waves [1, 4, 6, 8, 9]. In general, Evans functions are not uniquely defined. To avoid this issue, we specify a unique construction of an Evans function $E^{\Omega}_{\Gamma}$ associated with a given operator $H^{\Omega}_{\Gamma}$ in Definition 2.2. Thus, for the remainder of the paper, we will simply refer to the Evans function of an operator, always assuming that it is constructed by Definition 2.2.

If $\Omega$ is split at some some non-vertex point $x_{j}=s_{1}$ into regions $\Omega_{1}$ and $\Omega_{2}$ as discussed in Remark 1.3, then a one-sided map is defined to be a Dirichlet-to-Neumann map associated with $s_{1}$. In particular, we specify two such maps given our splitting.

In this paper we develop methods for finding the spectra of Schrödinger differential operators on quantum star graphs. In particular, we wish to be able to split a quantum star graph into smaller subgraphs, and then find the spectrum of the original problem by examining the spectra of the related subgraph problems. This is particularly helpful for dealing with operators with localized potentials, since it allows us to isolate the zero potential and nonzero potential portions from one another, resulting in simpler subproblems.

In Section 2, we find a general formula for $R_{\lambda}^{\Gamma}{\vec{v}}$, the resolvent of $H^{\Omega}_{\Gamma}$ applied to an arbitrary $L^{2}(\Omega)$ function ${\vec{v}}$.

In Section 3, we express ${\vec{u}}_{\Gamma,i}$ (a solution to an inhomogeneous $\Gamma$ boundary value problem) in terms of $R^{\Gamma}_{\lambda}{\vec{v}}$. Such functions are the building blocks of the two-sided maps. This expression (Theorem 3.4) involves three key projection matrices: the Dirichlet, Neumann, and Robin projections. The various properties of these matrices allow us to algebraically apply boundary conditions to solutions in a general way, without relying on any knowledge of the specific type of condition (beyond being representable by the $\alpha$ and $\beta$ matrices discussed earlier).

In Section 4, we prove the main results of this paper. They relate the eigenvalues of $H^{\Omega}_{\Gamma}$ to those of the smaller operators that come from splitting $\Omega$ as discussed in Remark 1.3. We state them here:

A similar result has already been proven for the interval case in [7].

Since zeros of Evans functions locate eigenvalues, Theorem 1.5 can be used to easily prove anther one of the main results.

In [2], a similar eigenvalue counting theorem has already been proven for the case of dividing multidimensional domains by hypersurfaces and defining two-sided maps with respect to these hypersurfaces.

We will also demonstrate how repeated applications of Theorem 1.5 can be used to split a quantum graph into three subdomains over which the eigenvalues can be counted separately, which is useful for more general barrier and well problems.

We can also construct new Dirichlet-to-Neumann maps $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$, now associated at two vertices $s_{1}$ and $s_{2}$, and sum them to make another two-sided map. The construction of $\mathcal{M}_{1}$ and $\mathcal{M}_{2}$ is different for splittings on the same wire vs splittings on different wires. The specific constructions are detailed in Definitions 4.5 and 4.8. In either case, we can utilize the determinant of the two-sided map $|(\mathcal{M}_{1}+\mathcal{M}_{2})(\lambda)|$ in the following theorem:

Just as with Theorem 1.5, the equality in Theorem 1.10 is one of meromorphic functions, so the desired behavior is preserved. Also, we again obtain a corresponding eigenvalue counting theorem.

Finally, in Section 5, we illustrate some applications of the main theorems to quantum star graphs with potential barriers and wells.

While it is possible to construct Dirichlet-to-Neumann maps column-by-column by solving several boundary-value problems, this method does not do much to illuminate the processes by which the zeros and poles containing spectral information appear in the map. Provided that $\lambda\in\rho(H^{\Omega}_{\Gamma})$, this map-making consists of finding functions ${\vec{u}}_{\Gamma,i}\in H^{2}(\Omega)$ which serve as solutions to the problem:

$$\begin{cases}(H^{\Omega}-\lambda){\vec{u}}_{\Gamma,i}=\vec{0}\,,\\ \gamma_{\Gamma}{\vec{u}}_{\Gamma,i}=\vec{e}_{i},\end{cases}$$

(16)

where $\vec{e}_{i}$ is the $i$th standard basis vector for ${\mathbb{C}}^{2n}$, then applying the desired trace to such a solution. Note that we will have $\vec{e}_{i}$ representing standard basis vectors for ${\mathbb{C}}^{n}$ and ${\mathbb{C}}^{2n}$ interchangeably, since the dimension will be clear from context. Instead of solving for ${\vec{u}}_{\Gamma,i}$ by its defining boundary-value problem, we will construct it using the resolvent of $H^{\Omega}_{\Gamma}$.

For $\lambda\in\rho(H^{\Omega}_{\Gamma})$, a key property of the resolvent $R_{\lambda}^{\Gamma}$ of $H^{\Omega}_{\Gamma}$ is that, for $\lambda\in\rho(H^{\Omega}_{\Gamma})$, it acts as the inverse operator of $H^{\Omega}_{\Gamma}-\lambda$. That is,

$$R_{\lambda}^{\Gamma}(H^{\Omega}_{\Gamma}-\lambda){\vec{v}}=(H^{\Omega}_{\Gamma}-\lambda)R_{\lambda}^{\Gamma}{\vec{v}}={\vec{v}},$$

(17)

for any function ${\vec{v}}$ on $\Omega$.

Equation (17) actually gives us a way to find $R_{\lambda}^{\Gamma}{\vec{v}}$ for an arbitrary $L^{2}(\Omega)$ function ${\vec{v}}$ by solving the inhomogeneous boundary-value problem:

$$\begin{cases}(H^{\Omega}_{\Gamma}-\lambda){\vec{y}}={\vec{v}}\,,\\ \gamma_{\Gamma}{\vec{y}}=\vec{0}.\end{cases}$$

(18)

The unique solution ${\vec{y}}$ to this problem must be the resolvent. We will solve for $R_{\lambda}^{\Gamma}{\vec{v}}$ by adding together a complementary and particular solution. Normally, we could generate a particular solution via the method of Variation of Parameters, but since each component of a function on $\Omega$ has a different spatial variable, we need to make some slight adjustments to the method.

We will start by obtaining a complementary solution. First, suppose $\lambda\in\rho(H^{\Omega}_{\Gamma})$. Next, let the entries of $\alpha_{1}$ and $\alpha_{2}$ be named as follows:

$$\alpha_{1}=\begin{bmatrix}a_{1,1}&\ldots&a_{1,n}\\ \vdots&\ddots&\vdots\\ a_{n,1}&\ldots&a_{n,n}\end{bmatrix}\hskip 36.135pt\alpha_{2}=\begin{bmatrix}b_{1,1}&\ldots&b_{1,n}\\ \vdots&\ddots&\vdots\\ b_{n,1}&\ldots&b_{n,n}\end{bmatrix}\,,$$

(19)

and similarly let the nonzero diagonal entries of $\beta_{1}$ and $\beta_{2}$ be named

$$\beta_{1}={\rm diag\,}(g_{1},...,g_{n})\hskip 36.135pt\beta_{2}={\rm diag\,}(h_{1},...,h_{n})\,.$$

(20)

Consider the matrix $\begin{bmatrix}-\alpha_{2}^{*}\\ \alpha_{1}^{*}\end{bmatrix}$. We can use each one of the columns as initial conditions for complementary homogeneous problems to get solutions ${\vec{y}}_{i}$ which solve:

$$\begin{cases}(H^{\Omega}-\lambda){\vec{y}}_{i}=\vec{0}\,,\\ y_{i,j}(0)=-\overline{b_{i,j}}\text{ for $1\leq j\leq n$}\,,\\ y^{\prime}_{i,j}(0)=\overline{a_{i,j}}\text{ for $1\leq j\leq n$}\,,\end{cases}$$

(21)

where $y_{i,j}$ is the $j$th component of ${\vec{y}}_{i}$ and the overline represents complex conjugation. Observe that each of these $n$ solutions satisfies $\Gamma$ boundary conditions at the origin. We denote the set of all ${\vec{y}}_{i}$ as $\mathcal{Y}$.

Similarly, we can define another $n$ solutions by using the columns of $\begin{bmatrix}-\beta_{2}^{*}\\ \beta_{1}^{*}\end{bmatrix}$ as initial conditions for homogeneous problems to generate solutions ${\vec{z}}_{i}$ which satisfy:

$$\begin{cases}(H^{\Omega}-\lambda){\vec{z}}_{i}=\vec{0}\,,\\ {\vec{z}}_{i}(\vec{\ell})=-\overline{h}_{i}\vec{e}_{i}\,,\\ {\vec{z}}_{i}^{\prime}(\vec{\ell})=\overline{g_{i}}\vec{e}_{i}\,.\end{cases}$$

(22)

First, note that these solutions ${\vec{z}}_{i}$ satisfy $\Gamma$ boundary conditions at the outer endpoints of the quantum graph. Also, we have that $z_{i,j}=0$ for all $j\neq i$ due to the zero conditions. We denote the set of all ${\vec{z}}_{i}$ as $\mathcal{Z}$.

Due to the rank conditions of the $\alpha$ and $\beta$ matrices imposed in equation (1), we have that $\mathcal{Y}$ is a linearly independent set and $\mathcal{Z}$ is a linearly independent set. Additionally, it can bee seen that the combined set $\mathcal{Y}\cup\mathcal{Z}$ is linearly dependent if and only if $\lambda\in\sigma(H^{\Omega}_{\Gamma})$. Thus, since $\lambda\in\rho(L_{\Gamma})$, we must have that $\mathcal{Y}\cup\mathcal{Z}$ forms a linearly independent set of $2n$ functions.

This definition also extends to the operators defined over subgraphs like $H^{\Omega_{1}}_{\Gamma_{1}}$ and $H^{\Omega_{2}}_{\Gamma_{2}}$; we simply alter the initial value problems defining $Y$ and $Z$ to incorporate $s_{1}$ as reflected in the new boundary conditions.

We wish to emulate Variation of Parameters so as to obtain a particular solution from our complementary solution set. We cannot do normal Variation of Parameters on the whole set of complementary solutions since each solution has multiple spatial variables, but we can do Variation of Parameters for each edge. So, we construct a particular solution ${\vec{y}}_{p}$ component by component. The $j$th component of the particular solution ${\vec{y}}_{p}$ can be built by looking at all the $y_{i,j}$ and $z_{i,j}$ (we call such components subvectors), finding a linearly independent pair, and using Variation of Parameters on them to find a particular solution to the one-dimensional second order problem:

$$(-\frac{\partial^{2}}{\partial x_{i}^{2}}+V_{i}-\lambda)u=v_{i}\hskip 54.2025ptx_{i}\in(0,\ell_{i})\,.$$

(25)

This proof shows that our componentwise construction of ${\vec{y}}_{p}$ is well defined for all $\lambda\in\rho(H^{\Omega}_{\Gamma})$. We also see in the proof that the choice of each $y_{\tau_{j},j}$ is local with respect to $\lambda$. Thus, for specific $\lambda$ we can use Variation of Parameters to define the $i$th entry of ${\vec{y}}_{p}$ as

	$\displaystyle({\vec{y}}_{p})_{i}(x_{i},\lambda):=$	$\displaystyle-\frac{1}{D_{i}(\lambda)}(y_{\tau_{i},i}(x_{i},\lambda)\int_{x_{i}}^{\ell_{i}}v_{i}(t_{i})z_{i,i}(t_{i},\lambda)\,dt_{i}$
		$\displaystyle+z_{i,i}(x_{i},\lambda)\int_{0}^{x_{i}}v_{i}(t_{i})y_{\tau_{i},i}(t_{i},\lambda)\,dt_{i})\,,$		(26)

where $D_{i}:=W_{0}(y_{\tau_{i},i},z_{i,i})$ is a Wronskian for solutions on the $i$th wire.

Let $S_{0}=\begin{bmatrix}Y&Z\end{bmatrix}$. Then our general solution (which is $R_{\lambda}^{\Gamma}{\vec{v}})$ will be of the form

$${\vec{u}}=S_{0}{\vec{c}}+{\vec{y}}_{p}\,,$$

(27)

where ${\vec{c}}$ is a vector of scalars in ${\mathbb{C}}^{2n}$. To make sure ${\vec{u}}$ satisfies the $\Gamma$ boundary conditions, we will need to adjust the components of ${\vec{c}}$ so that

$$\gamma_{\Gamma}(S_{0}{\vec{c}})=-\gamma_{\Gamma}{\vec{y}}_{p}\,.$$

(28)

By the linearity of $\gamma_{\Gamma}$, we can modify equation (28) by taking the trace of every column of $S_{0}$, arranging these in a matrix $\tilde{S}_{0}$, and then multiplying this matrix by ${\vec{c}}$. We define $\tilde{S}_{0}$ as

	$\displaystyle\tilde{S}_{0}(\lambda)$	$\displaystyle=\begin{bmatrix}\beta_{1}S_{0}(\vec{\ell},\lambda)+\beta_{2}S_{0}^{\prime}(\vec{\ell},\lambda)\\ \alpha_{1}S_{0}(\vec{0},\lambda)+\alpha_{2}S_{0}^{\prime}(\vec{0},\lambda)\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}\beta_{1}Y(\vec{\ell},\lambda)+\beta_{2}Y^{\prime}(\vec{\ell},\lambda)&\beta_{1}Z(\vec{\ell},\lambda)+\beta_{2}Z^{\prime}(\vec{\ell},\lambda)\\ \alpha_{1}Y(\vec{0},\lambda)+\alpha_{2}Y^{\prime}(\vec{0},\lambda)&\alpha_{1}Z(\vec{0},\lambda)+\alpha_{2}Z^{\prime}(\vec{0},\lambda)\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}\beta_{1}Y(\vec{\ell},\lambda)+\beta_{2}Y^{\prime}(\vec{\ell},\lambda)&0\\ 0&\alpha_{1}Z(\vec{0},\lambda)+\alpha_{2}Z^{\prime}(\vec{0},\lambda)\end{bmatrix}\,.$		(29)

The zero blocks in (2.4) appear since $Y$ and $Z$ are defined to satisfy $\alpha$ and $\beta$ conditions, respectively. We can rewrite equation $\eqref{eq:15}$ as

$$\tilde{S}_{0}(\lambda){\vec{c}}=-\gamma_{\Gamma}{\vec{y}}_{p}\,,$$

(30)

Due to the integral bounds we chose, we find that

$$\gamma_{\Gamma}{\vec{y}}_{p}=\begin{bmatrix}\vec{0}\\ \alpha_{1}{\vec{y}}_{p}(\vec{0},\lambda)+\alpha_{2}{\vec{y}}_{p}^{\prime}(\vec{0},\lambda)\end{bmatrix}\,.$$

(31)

Equations (30) and (31) imply that we can always choose $c_{1}=c_{2}=...=c_{n}=0$, and thus we can determine $c_{n+i}$ for $1\leq i\leq n$ by solving the smaller system:

$$C(\lambda)\begin{bmatrix}c_{n+1}\\ \vdots\\ c_{2n}\end{bmatrix}=-\alpha_{1}{\vec{y}}_{p}(\vec{0},\lambda)-\alpha_{2}{\vec{y}}_{p}^{\prime}(\vec{0},\lambda)\,,$$

(32)

where $C(\lambda)$ is given by:

$$C(\lambda)=\alpha_{1}Z(\vec{0},\lambda)+\alpha_{2}Z^{\prime}(\vec{0},\lambda)\,.$$

(33)

That is, $C(\lambda)$ is the lower right block of $\tilde{S}_{0}$. We can find an exact formula for $C(\lambda)$. First, we give names to the various ${\vec{z}}_{i}$ at the origin:

	$\displaystyle-s_{i}(\lambda)$	$\displaystyle=z_{i,i}(0,\lambda)\,,$
	$\displaystyle r_{i}(\lambda)$	$\displaystyle=z_{i,i}^{\prime}(0,\lambda)\,.$		(34)

Then from equation (33), $C(\lambda)$ is given by

	$\displaystyle C(\lambda)$	$\displaystyle=\begin{bmatrix}\alpha_{1}&\alpha_{2}\end{bmatrix}\begin{bmatrix}{\rm diag\,}(-s_{1}(\lambda),...,-s_{n}(\lambda))\\ {\rm diag\,}(r_{1}(\lambda),...,r_{n}(\lambda))\end{bmatrix}$
		$\displaystyle=\begin{bmatrix}\begin{vmatrix}-s_{1}(\lambda)&-b_{1,1}\\ r_{1}(\lambda)&a_{1,1}\end{vmatrix}&\ldots&\begin{vmatrix}-s_{n}(\lambda)&-b_{1,n}\\ r_{n}(\lambda)&a_{1,n}\end{vmatrix}\\ \vdots&\ddots&\vdots\\ \begin{vmatrix}-s_{1}(\lambda)&-b_{n,1}\\ r_{1}(\lambda)&a_{n,1}\end{vmatrix}&\ldots&\begin{vmatrix}-s_{n}(\lambda)&-b_{n,n}\\ r_{n}(\lambda)&a_{n,n}\end{vmatrix}\end{bmatrix}\,.$		(35)

If $C(\lambda)$ is invertible then we can solve for the coefficients $c_{n+1},...,c_{2n}$ by inversion in equation (32). The invertibility is guaranteed by the following Lemma.