47

数理解析研究所講究録
第2074巻 2018年 47-54

Resolvent expansion for the Schrödinger operator
on a graph with infinite rays
Kenichi ITO & Arne JENSEN $\dag er$*

In this article we report on the authors’ recent work [IJ3] on an expansion of
the resolvent for the Schrödinger operator on a graph with rays. We obtain precise
expressions for the first few coefficients of the expansion around the threshold 0
in terms only of the generalized eigenfunctions. This in particular justifies thenatural definition of threshold resonances for the generalized eigenfunctions solely
by the growth rate at infinity.

1

The free operator

In this section we define a graph with rays, and fix our free operator H_{0} on it. Here
we denote the set of vertices by G , and the set of edges by E_{G} , hence we consider
the graph (G, E_{G}) . We sometimes call it simply the graph G . The free operator
H_{0} is defined as a direct sum of the free Dirichlet Schrödinger operators on a finite
part and rays, being different from the graph Laplacian -\triangle c.
Let (K, E_{0}) be a connected, finite, undirected and simple graph, without loops
1 , . . . , N , be N copies of the discreteor multiple edges, and let (L_{ $\alpha$}, E_{ $\alpha$}) , $\alpha$half‐line, i.e.
=

L_{ $\alpha$}=\mathbb{N}=\{1 , 2, . .

E_{ $\alpha$}=\{\{n, n+1\}; n\in L_{ $\alpha$}\}.

We construct the graph (G, E_{G}) by jointing (L_{ $\alpha$}, E_{ $\alpha$}) to (K, E_{0}) at a vertex x_{ $\alpha$}\in K
for $\alpha$=1 , . . . , N :G=K\cup L_{1}\cup\cdots\cup L_{N},

E_{G}=E_{0}\cup E_{1}\cup\ldots E_{N}\cup\{\{x_{1}, 1^{(1)}\}, . . . , \{x_{N}, 1^{(N)}\}\}.
*

Department of Mathematics, Graduate School of Science, Kobe University, 1‐1, Rokkodai,
Nada‐ku, Kobe 657‐8501, Japan. \mathrm{E} ‐mail: ito‐ken@math.kobe‐u.ac.jp.
$\dag er$
Department of Mathematical Sciences, Aalborg University, Skjernvej 4\mathrm{A} , DK‐9220 Aalborg\emptyset , Denmark. \mathrm{E} ‐mail: matarne@math.aau.dk.

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Here we distinguished 1 of L_{ $\alpha$} by a superscript: 1 ( $\alpha$) \in L_{ $\alpha$} . Note that two different
half‐lines (L_{ $\alpha$}, E_{ $\alpha$}) and (L_{ $\beta$}, E_{ $\beta$}) , $\alpha$ \neq $\beta$ , could be jointed to the same vertex
x_{ $\alpha$}=x_{ $\beta$}\in K.
Let h_{0} be the free Dirichlet Schrödinger operators on K : For any functionu: K\rightarrow \mathbb{C} we define(h_{0}u)[x]=\displaystyle \sum_{\{x,y\}\in E_{0} (u[x]-u[y])+\sum_{ $\alpha$=1}^{N}s_{ $\alpha$}[x]u[x]wheres_{ $\alpha$}[x]=1 if x=x_{ $\alpha$}

ands_{ $\alpha$}[x]=for x\in K,

0 otherwise.

Note that the Dirichletboundary condition is considered being set on the boundaries 1^{( $\alpha$)}\in L_{ $\alpha$} outside
K . Similarly, for $\alpha$= 1 , . . . , N let h_{ $\alpha$} be the free Dirichlet Schrödinger operators
on L_{ $\alpha$} : For any function u : L_{ $\alpha$}\rightar ow \mathbb{C} we define(h_{ $\alpha$}u)[n]=\left\{ begin{ar y}{l 2u[1]-u[2]&\mathrm{f}\mathrm{o}\mathrm{r}n=1,\ 2u[n]-u[n+1]-u[n-1]&\mathrm{f}\mathrm{o}\mathrm{r}n\geq2. \end{ar y}\right.

Then we define the free operator H_{0} on G as a direct sum
H_{0}=h_{0}\oplus h_{1}\oplus\cdots\oplus h_{N} ,(1.1)

according to a direct sum decomposition

F(G)=F(K)\oplus F(L_{1})\oplus\cdots\oplus F(L_{N}),where F(X)=\{u:X\rightarrow \mathbb{C}\} denotes the set of all the functions on a space X.

In the definition (1.1) interactions between K and L_{ $\alpha$} are absent, and the free
operator H_{0} does not coincide with the graph Laplacian -\triangle c defined as(-\displaystyle \triangle_{G}u)[x]=\sum_{\{x,y\}\in E_{G} (u[x]-u[y]).

In fact, we can write-\displaystyle\trianglec=H_{0}+J, =-\sum_{$\alpha$=1}^{N}(|\mathcal{S}_{$\alpha$}\rangle\langlef_{$\alpha$}|+f_{$\alpha$}\rangle\{s_{$\alpha$}|)

,

(1.2)

where f_{ $\alpha$}[x] =1 if x= 1^{( $\alpha$)} and f_{ $\alpha$}[x] =0 otherwise. The operator H_{0} is actuallysimpler and more useful than -\triangle c , since it does not have a zero eigenvalue ora zero resonance, and the asymptotic expansion of its resolvent around 0 does
not have a singular part. This fact effectively simplifies the expansion procedure
for the perturbed resolvent, and enables us to obtain more precise expressions forthe coefficients than those in [IJ1]. The interaction J is a special case of general
perturbations considered in Assumption 2.1, see Proposition 2.2. Hence the graphLaplacian -\triangle c can be treated as a perturbation of the free operator H_{0}.

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The perturbed operator

In this section we introduce our class of perturbations. We also provide a simple
classification of threshold types in terms of the growth rate of the generalized
eigenfunctions. This classification will be justified by our main results presented
in Section 3.

Set for s\in \mathbb{R}\mathcal{L}^{s}=\ell^{1,s}(G)= (\ell^{1}(K))\oplus(\ell^{1,s}(L_{1}))\oplus\cdots\oplus(\ell^{1,s}(L_{N}))(\mathcal{L}^{s})^{*}=\ell^{\infty,-s}(G)= (\ell^{\infty}(K))\oplus(\ell^{1,s}(L_{1}))\oplus\cdots\oplus(\ell^{1,s}(L_{N})),,where for $\alpha$=1 , . . . , N\displaystyle \el ^{1,s}(L_{ $\alpha$})=\{x:L_{ $\alpha$}\rightar ow \mathb {C}; \sum_{n\in L_{ $\alpha$} (1+n^{2})^{s/2}|x[n]| <\infty\},\displaystyle \el ^{\infty,-s}(L_{ $\alpha$})=\{x:L_{ $\alpha$}\rightar ow \mathb {C}; \sup_{n\in L_{ $\alpha$} (1+n^{2})^{-s/2}|x[n]|<\infty\}.
We consider the following class of perturbations, cf. [JN1, IJ1, IJ2].
Assumption 2.1. Assume that V\in \mathcal{B}(\mathcal{H}) is self‐adjoint, and that there exist an
injective operator v \in \mathcal{B}(\mathcal{K}, \mathcal{L}^{ $\beta$}) with $\beta$ \geq 1 and a self‐adjoint unitary operator
U\in \mathcal{B}(\mathcal{K}) , both defined on some abstract Hilbert space \mathcal{K} , such that

V=vUv^{*}\in \mathcal{B}( \mathcal{L}^{ $\beta$})^{*}, \mathcal{L}^{ $\beta$}).

We note that V is compact on \mathcal{H} under Assumption 2.1. Let us provide acriterion for Assumption 2.1 in terms of weighted \ell^{2} ‐spaces. We use the standard
weighted space notation such as \ell^{2,s}(G) , s\in \mathbb{R}.
Proposition 2.2. Assume that V \in \mathcal{B}(\mathcal{H}) is self‐adjoint, and that it extends toan operator in \mathcal{B}(\ell^{2,- $\beta$-1/2- $\epsilon$}(G), \ell^{2, $\beta$+1/2+ $\epsilon$}(G) for some $\beta$\geq 1 and $\epsilon$ > 0 . Then
Vsati \mathcal{S}fies Assumption 2.1 for the \mathcal{S}ame $\beta$.

By this criterion we can see that the interaction J from (1.2) satisfies Assump‐tion 2.1. For another criterion for Assumption 2.1 we refer to [IJ1, Appendix \mathrm{B} ].
Under Assumption 2.1 we let

H=H_{0}+V,

and consider the solutions to the zero eigen‐equation H $\Psi$=0 in the largest space
where it can be defined. Define the generalized zero eigenspace \overline{\mathcal{E} as\overline{\mathcal{E} =\{ $\Psi$\in(\mathcal{L}^{ $\beta$})^{*}; H $\Psi$=0\}.

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Let \mathrm{n}^{( $\alpha$)} \in(\mathcal{L}^{1})^{*}, 1^{( $\alpha$)}\in(\mathcal{L}^{0})^{*} be the functions defined as\mathrm{n}^{( $\alpha$)}[x]=\{mfor x=m\in L_{ $\alpha$},0

for x\in G\backslash L_{ $\alpha$},1^{( $\alpha$)}[x]=\{1

for x\in L_{ $\alpha$},0

for x\in G\backslash L_{ $\alpha$},respectively, and abbreviate the spaces spanned by these functions as\mathbb{C}\mathrm{n}=\mathbb{C}\mathrm{n}^{(1)}\oplus\cdots\oplus \mathbb{C}\mathrm{n}^{(N)}, \mathbb{C}1=\mathbb{C}1^{(1)}\oplus\cdots\oplus \mathbb{C}1^{(N)}.
We can show that under Assumption 2.1 with $\beta$\geq 1 the generalized eigenfunctions
have specific asymptotics:\overline{\mathcal{E} \subset \mathb {C}\mathrm{n}\oplus \mathb {C}1\oplus \mathcal{L}^{ $\beta$-2}
With this asymptotics we consider the following subspaces:\mathcal{E}=\overline{\mathcal{E} \cap(\mathbb{C}1\oplus \mathcal{L}^{ $\beta$-2}) , \mathrm{E}=\overline{\mathcal{E} \cap \mathcal{L}^{ $\beta$-2}
A function in \overline{\mathcal{E} \backslash\mathcal{E} should be called a non‐resonance eigenfunction, one in \mathcal{E}\backslash \mathrm{E}a resonance eigenfunction, and one in \mathrm{E} a bound eigenfunction, but we shall often
call them generalized eigenfunction\mathcal{S} or simply eigenfunctions.

Let us introduce the same classification of threshold as in [IJ1, Definition 1.6].
Definition 2.3. The threshold z=0 is said to be

1. a regular point, if \mathcal{E}=\mathrm{E}=\{0\} ;2. an exceptional point of the first kind, if \mathcal{E}\rightar ow\supset \mathrm{E}=\{0\} ;3. an exceptional point of the second kind, if \mathcal{E}=\mathrm{E}\rightar ow\supset\{0\} ;4. an exceptional point of the third kind, if \mathcal{E}\rightar ow\supset \mathrm{E}\rightar ow\supset\{0\}.
It should be noted here that there is a dimensional relation:\dim(\overline{\mathcal{E}}/\mathcal{E})+\dim(\mathcal{E}/\mathrm{E})=N, 0\leq\dim \mathrm{E}<\infty,
the former of which reflects a certain topological stability of the non‐decaying
eigenspace under small perturbations.
We can also show that for any $\Psi$_{1}\in\overline{\mathcal{E} and $\Psi$_{2}\in \mathcal{E} , if we let
NN$\Psi$_{1} - \displaystyle \sum c_{ $\alpha$}^{(1)}\mathrm{n}^{( $\alpha$)} \in \mathbb{C}1 \oplus \mathcal{L}^{ $\beta$-2}, $\Psi$_{2} - \sum c_{ $\alpha$}^{(2)}1^{( $\alpha$)} \in \mathcal{L}^{ $\beta$-2},$\alpha$=1$\alpha$=1

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then these coefficients are orthogonal:\displaystle\sum_{$\alpha$=1}^{N}\overline{c}_ $\alpha$}^{(2)}c_{$\alpha$}^{(1)}=0.
By this fact it would be natural to introduce orthogonality in \overline{\mathcal{E} in terms of theasymptotics, and accordingly define the generalized orthogonal projections. We use\rangle to denote the duality between between \mathcal{L}^{8} and (\mathcal{L}^{S})^{*} If $\beta$ \geq 2 then \{ $\Phi$, $\Psi$\}
is defined for $\Phi$ \in \mathrm{E} and $\Psi$ \in \mathcal{E} . If we only assume $\beta$\geq 1 then we must assume$\Phi$\cdot $\Psi$ \in \mathcal{L}^{0} to justify the notation \{ $\Phi$, $\Psi$\rangle . Here ( $\Phi$\cdot $\Psi$)[n]$\Psi$[n] $\Phi$[n], n \in G , isthe pointwise product.
=

Definition 2.4. We call a subset \{$\Psi$_{ $\gamma$}\}_{ $\gamma$}\subset \mathcal{E} a resonance basis, if the set \{[$\Psi$_{ $\gamma$}]\}_{ $\gamma$}of representatives forms a basis in \mathcal{E}/\mathrm{E} . It is said to be orthonormal, if

1. for any $\gamma$ and $\Psi$\in \mathrm{E} one has \overline{ $\Psi$}\cdot$\Psi$_{ $\gamma$}\in \mathcal{L}^{0}\mathrm{a}\mathrm{n}\mathrm{d}\{ $\Psi,\ \Psi$_{ $\gamma$}\}=0 ;2. there exists an orthonormal system \{c^{( $\gamma$)}\}_{ $\gamma$}\subset \mathbb{C}^{N} such that for any $\gamma$$\Psi$_{ \gam a$}-\displaystle\sum_{$\alpha$=1}^{N}c_{$\alpha$}^{($\gam a$)}1^{($\alpha$)}\in\mathcal{L}^ $\beta$-2}
The orthogonal resonance projection \mathcal{P} is defined as\displayst le\mathcal{P}=\sum_{$\gam a$}| \Psi$_{$\gam a$}\ langle$\Psi$_{$\gam a$}|.
Definition 2.5. We call a basis \{$\Psi$_{ $\gamma$}\}_{ $\gamma$} \subset \mathrm{E} a bound basis to distinguish it froma resonance basis. It is said to be orthonormal, if for any $\gamma$ and $\gamma$' one has\overline{ $\Psi$}_{$\gamma$'}\cdot$\Psi$_{ $\gamma$}\in \mathcal{L}^{0} and\{$\Psi$_{$\gamma$'}, $\Psi$_{ $\gamma$}\rangle=$\delta$_{ $\gamma \gamma$'}.

The orthogonal bound projection \mathrm{P} is defined as\displayst le\mathrm{P}=\sum_{$\gam a$}| \Psi$_{$\gam a$}\ langle$\Psi$_{$\gam a$}|.
We remark that the above orthogonal projections \mathcal{P} and \mathrm{P} are independent ofchoice of orthonormal bases.

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3

Main results

In this section we present the main theorems of [IJ3] classifying the resolvent
expansions according to threshold types given in Definition 2.3. In the statements
below we have to impose different assumptions on the parameter $\beta$ depending on
threshold types. For simplicity we state the results only for integer values of $\beta$,
but an extension to general $\beta$ is straightforward.
We setR( $\kappa$)=(H+$\kappa$^{2})^{-1} for -$\kappa$^{2}\not\in $\sigma$(H) ,\mathcal{B}^{s}=\mathcal{B}(\mathcal{L}^{s}, (\mathcal{L}^{S})^{*}) .

Theorem 3.1. Assume that the threshold 0 is a regular point, and that Assump‐tion 2.1 is fulfilled for some integer $\beta$\geq 2 . ThenR($\kap a$)=\displayst le\sum_{j=0}^{$\beta$-2}$\kap a$^{j}G_{j}+\mathcal{O}($\kap a$^{$\beta$-1})

in \mathcal{B}^{ $\beta$-2}

with G_{j} \in \mathcal{B}^{j+1} for j even, and G_{j} \in \mathcal{B}^{j} for j odd. The coefficients G_{j} can becomputed explicitly. In particular,
G_{-2}=\mathrm{P}=0, G_{-1}=\mathcal{P}=0.

Theorem 3.2. Assume that the threshold 0 i\mathcal{S} an exceptional point of the firstkind, and that Assumption 2.1 i\mathcal{S} fulfilled for some integer $\beta$\geq 3 . ThenR($\kap a$)=\displaystyle\sum_{j=-1}^{$\beta$-4}$\kap a$^{j}G_{j}+\mathcal{O}($\kap a$^{$\beta$-3})

in \mathcal{B}^{ $\beta$-1}

with G_{j} \in \mathcal{B}^{j+3} for j even, and G_{j} \in \mathcal{B}^{j+2} for j odd. The coefficients G_{j} can becomputed explicitly. In particular,
G_{-2}=\mathrm{P}=0, G_{-1}=\mathcal{P}\neq 0.

Theorem 3.3. Assume that the threshold 0 i_{\mathcal{S} an exceptional point of the second
kind, and that A_{\mathcal{S} sumption 2 .1 is fulfilled for \mathcal{S}ome integer $\beta$\geq 4 . ThenR($\kap a$)=\displaystyle\sum_{j=-2}^{$\beta$-6}$\kap a$^{j}G_{j}+\mathcal{O}($\kap a$^{$\beta$-5})

in \mathcal{B}^{ $\beta$-2}

with G_{j} \in \mathcal{B}^{j+3} for j even, and G_{j} \in \mathcal{B}^{j+2} for j odd. The coefficients G_{j} can becomputed explicitly. In particular,
G_{-2}=\mathrm{P}\neq 0, G_{-1}=\mathcal{P}=0.

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Theorem 3.4. A_{\mathcal{S}\mathcal{S} ume that the thre \mathcal{S}hold0 is an exceptional point of the thirdkind, and that A_{\mathcal{S}\mathcal{S} umption 2 .1 is fulfilled for \mathcal{S}ome integer $\beta$\geq 4 . ThenR($\kap a$)=\displaystyle\sum_{j=-2}^{$\beta$-6}$\kap a$^{j}G_{j}+\mathcal{O}($\kap a$^{$\beta$-5})

in \mathcal{B}^{ $\beta$-2}

with G_{j} \in \mathcal{B}^{j+3} for j even, and G_{j} \in \mathcal{B}^{j+2} for j odd. The coefficients G_{j} can becomputed explicitly. In particular,
G_{-2}=\mathrm{P}\neq 0, G_{-1}=\mathcal{P}\neq 0.
Theorems 3. 1-3.4 justify the classification of threshold types only by the growthproperties of eigenfunctions:
Corollary 3.5. The thre \mathcal{S}hold type determines and is determined by the coefficients
G_{-2} and G_{-1} from Theorem \mathcal{S}3.1-3.4.

We can also compute the coefficients G_{0} and G_{1} . They can be considered aspart of the main results of [IJ3]. However, their expressions are very long, and weomit them in this article, see [IJ3, Appendix \mathrm{B} ]. These results are generalizationsof [IJ1] on the discrete full line \mathb {Z} and [IJ2] on the discrete half‐line \mathb {N} . The
strategy for proofs is also similar to [IJ1, IJ2], implementing the expansion schemeof [JN1, JN2] in its full generality. However, due to our choice of the free operator
the expansion procedure gets simplified.
Acknowledgement \mathrm{s}KI was supported by JSPS KAKENHI Grant Number 17\mathrm{K}05325 . The authors
were partially supported by the Danish Council for Independent Research | Natural
Sciences, Grant \mathrm{D} $\Gamma$ \mathrm{F}-4181-00042.

References[IJ1] K. Ito and A. Jensen, A complete classification of threshold properties
for one‐dimensional discrete Schrödinger operators. Rev. Math. Phys. 27(2015), no. 1, 1550002, 45 pp.

[IJ2] K. Ito and A. Jensen, Resolvent expansions for the Schrödinger operator on
the discrete half‐line. J. Math. Phys. 58 (2017), no. 5, 052101, 24 pp.
[IJ3] K. Ito and A. Jensen, Resolvent expansion for the Schrodinger operator on
a graph with infinite rays. Preprint 2017.
https: // arxiv. \mathrm{o}\mathrm{r}\mathrm{g}/\mathrm{a}\mathrm{b}\mathrm{s}/1712.01592

54[JN1] A. Jensen and G. Nenciu, A unified approach to resolvent expansions atthresholds. Rev. Math. Phys., 13 (2001), 717‐754.
[JN2] A. Jensen and G. Nenciu, Erratum: “A unified approach to resolvent expan‐
sions at thresholds”’ [Rev. Math. Phys. 13(6), 717‐754 (2001)]. Rev. Math.
Phys. 16(5), 675‐677 (2004).