Trace formulas for Schrödinger operators on star graphs
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Abstract
We derive trace formulas of the Buslaev–Faddeev type for quantum star graphs. One of the new ingredients is high energy asymptotics of the perturbation determinant.
1 Introduction
Besides being interesting mathematically, quantum graphs have many applications to chemistry and physics. Since the 1930s, they have been used to model structures ranging from nanotechnology and quantum wires to free electrons in organic molecules. Since quantum graphs have one dimensional edges, they are also used as simplified models for complicated behavior in physics, like quantum chaos and Anderson localization [1].
Using the perturbation determinant, we can define the limit amplitude a(k) and limit phase (or phase shift) \(\eta (k)\) for \(\zeta =k\in {\mathbb {R}}\). We set \(D(k)=:a(k)e^{i\eta (k)}\), with \(a(k)=D(k)\). As we will see later in (2.5), \(D(z)=1+O(\zeta ^{1})\) as \(\zeta \rightarrow \infty \), so we can choose \(\eta \) with the convention \(\eta (\infty )=0\). (We also require it to be continuous.) The names for a(k) and \(\eta (k)\) come from scattering theory, where they correspond to the amplitude shift and phase shift in the wavefunction as \(x\rightarrow \infty \).
The last thing we mention from scattering theory is zero energy resonances. The Schrödinger operator H has a zero energy resonance if there is a nontrivial bounded solution to \(\frac{d^2}{dx^2}\psi +V\psi =0\) that satisfies the continuity and Kirchhoff conditions. The multiplicity of the resonance is the dimension of the solution space. In physics, these resonances correspond to “halfbound states” or “metastable bound states”.
Our main result is the following trace formulas of the type in [2] relating \(\sum \lambda _j^{n}\) (for \(n\in \frac{1}{2}{\mathbb {N}}\)) to an expression involving the potential V.
Theorem 1
Remarks
 (i)
These trace formulas are analogous to the trace formulas for the halfline case, which can be found in [7, Sect. 4.6]. The differences are the values of the \(L_m\)’s and the possibility of having eigenvalues of multiplicity greater than one.
 (ii)
The additional requirement (1.7) in the case that \(\zeta =0\) is a resonance of multiplicity one comes from the hypotheses in [3, Prop. 4.5] for low energy asymptotics of the perturbation determinant.
We emphasize that while we are requiring each \(v_j\) to be smooth on \(e_j\), we do not impose restrictions on the values of \(v_j\) and its derivatives at the vertex. This raises the question of whether the vertex terms in the coefficients \(L_m\) disappear if V is smooth at the vertex. A natural notion of smoothness on \(\Gamma \) is that for any two distinct \(e_i\) and \(e_j\), the function on \({\mathbb {R}}\) obtained by combining \(v_i\) and \(v_j\) is smooth. This is easily seen to be equivalent to \(v_i^{(2k)}(0)=v_j^{(2k)}(0)\) for all \(1\le i,j\le n\) and \(v_i^{(2k+1)}(0)=0\) for all \(1\le i\le n\). We shall see that while for the first few coefficients \(L_m\) with m odd the vertex terms cancel, they do not for m even.
Corollary 1
 (1)
One integrates the perturbation determinant over a contour and takes limits in the form of the contour to obtain a family of identities.
 (2)
One analytically continues these identities and evaluates them at certain points.
In order to make this paper selfcontained we review necessary results from the literature in Sect. 2 and provide details for Steps 1 and 2 in Sect. 3. As we already mentioned, Sect. 4 contains the novel high energy asymptotics which lead, in particular, to the formulas for the coefficients \(L_m\) from (1.11). In Sect. 5, we use the star graph with \(n=2\) to recover some of the results for the whole real line that are given in [7, Sect. 5]. In contrast to [7, Sect. 5] we also obtain formulas if V is smooth away from a point, and we see explicitly the contribution to the trace formulas of the discontinuities of V and its derivatives at this point.
2 The perturbation determinant and other previous results

\(D(\zeta )\) is holomorphic in \(\mathfrak {I}\zeta >0\).

\(D(\zeta )\) has a zero in \(\zeta \) of order r if and only if \(\zeta ^2\) is an eigenvalue of multiplicity r of H.

\(D^{1}(\zeta )\frac{d}{d\zeta ^2}D(\zeta )={\text {tr}}(R_0(\zeta ^2)R(\zeta ^2))\).
Remark
The perturbation determinant is sometimes defined with an argument of \(z=\zeta ^2\). Then the definition is \(D(z):=\det (\mathbbm {1}+\sqrt{V}R_0(z)\sqrt{V})\) for \(z\in \rho (H_0)\), and the last property listed above takes on a nicer form. We will however continue to use the definition with \(\zeta \).
Proposition 1
The importance of this proposition is that it connects a spectral theoretic object, namely \(D(\zeta )\), with ODE objects, namely the \(\theta _j\)’s. Results of this type go back to [6]; see also [5].
Remark
Proposition 2
3 Adapting results from the halfline case
Trace formula derivations for the halfline case can be found in [7, Sect. 4.6]. We will follow the same method here, but with some adaptations to ensure the results hold for star graphs. We will assume we know the coefficients \(L_m\) in the asymptotic expansion of \(\log D(\zeta )\), which will result in proving Theorem 1 except for the formulas (1.11) for the \(L_m\)’s. (The expressions for the \(L_m\)’s will be derived in Sect. 4.)
Lemma 1
Proof
Lemma 2
Proof (of most of Theorem 1)
4 High energy asymptotic expansions
Lemma 3
We fix a branch of the logarithm using \(\frac{K(\zeta )}{in\zeta }=1+O(\zeta ^{1})\) and requiring \(\log (\frac{K(\zeta )}{in\zeta })\rightarrow 0\) as \(\zeta \rightarrow \infty \). Integrating, we get the following:
Corollary 2
5 Reduction to the real line
A star graph with only \(n=2\) edges can be identified with the whole real line. So using results for star graphs, we can prove things about scattering on the real line. The real line case with v smooth is handled directly in [7, Sect. 5], but we will recover the results for the real line with v smooth away from a point by setting \(n=2\) in the star graph case. A potential v on \({\mathbb {R}}\) is viewed as a pair of potentials \(v_1(x):={v}(x)\) and \(v_2(x):={v}(x), x\ge 0\) on the \(n=2\) star graph. Using results for star graphs from [3] along with Theorem 1, we can show:
Corollary 3
 (i)The perturbation determinant is$$\begin{aligned} D(\zeta )&=\frac{1}{2i\zeta }\left[ \theta _1'(0,\zeta )\theta _2(0,\zeta )+\theta _2'(0,\zeta )\theta _1(0,\zeta )\right] \end{aligned}$$(5.1)where W is the Wronskian and \(\theta _1,\theta _2\) are the Jost solutions on the two halfline edges.$$\begin{aligned}&=\frac{1}{2i\zeta }W(\theta _2(\cdot ,\zeta ),\theta _1(\cdot ,\zeta ))=:m(\zeta ), \end{aligned}$$(5.2)
 (ii)We have the trace formula$$\begin{aligned} {\text {tr}}(R_0(z)R(z))=\frac{\frac{d}{d\zeta ^2}D(\zeta )}{D(\zeta )}=\frac{\dot{m}(\zeta )}{2\zeta m(\zeta )},\quad z=\zeta ^2. \end{aligned}$$(5.3)
 (iii)(low energy asymptotics) Assume \(\int _{\mathbb {R}}(1+x^2)v(x)\,dx<\infty \), and let \(W(\zeta ):=W(\theta _2(\cdot ,\zeta ),\theta _1(\cdot ,\zeta ))\). If \(W(0)=0\), then \(\theta _1(x,0)=\alpha \theta _2(x,0)\) for some \(\alpha \in {\mathbb {R}}{\setminus }\{0\}\), and in this case,$$\begin{aligned} W(\zeta )=i(\alpha +\alpha ^{1})\zeta +O(\zeta ^2),\quad \zeta \rightarrow 0. \end{aligned}$$(5.4)
 (iv)
Assume again \(\int _{\mathbb {R}}(1+x^2)v(x)\,dx<\infty \). The Schrödinger operator H has a zero energy resonance of order one if \(W(0)=0\), and no zero energy resonance if \(W(0)\ne 0\).
 (v)(Levinson’s formula) Suppose \(\int _{\mathbb {R}}(1+x)v(x)\,dx<\infty \), and let N be the number of negative eigenvalues of H and \(m\in \{0,1\}\) be the multiplicity of the resonance at \(\zeta =0\). If \(m=1\), also suppose that \(\int _{{\mathbb {R}}}(1+x^2)v(x)\,dx<\infty \). Then$$\begin{aligned} \eta (\infty )\eta (0)=\pi \left( N+\frac{m1}{2}\right) . \end{aligned}$$(5.5)
 (vi)(trace formulas) Suppose that \(\int _{\mathbb {R}}(1+x)v(x)\,dx<\infty \) and (1.6) for \(j=1,2\) hold. Then we get the trace formulas in Theorem 1. In particular, if \(v\in C^\infty ({\mathbb {R}})\), the first few \(L_m\)’s are$$\begin{aligned} L_1= & {} \int _{\infty }^\infty v(x)\,dx,\quad L_2=0,\nonumber \\ L_3= & {} \int _{\infty }^{\infty }v^2(x)\,dx,\quad L_4=0,\nonumber \\ L_5= & {} \int _{\infty }^\infty (v'(x)^2+2v^3(x)). \end{aligned}$$(5.6)
Remarks
 (i)
In (vi), using Theorem 1 allows for a potential v that is discontinuous at \(x=0\). We emphasize that in this case the trace formulas contain additional contributions from the discontinuities of v and its derivatives at 0.
 (ii)
 (iii)
The Jost functions \(\theta _1,\theta _2\) extend easily from \([0,\infty )\) to \({\mathbb {R}}\). This follows from the existence proof of Jost solutions on the halfline given in e.g. [7, Sect. 4.1], and ensures that (5.2) makes sense. The \(\theta _1\) here agrees with the Jost solution \(\theta _1\) described in [7, Sect. 5.1], but because we identify \(e_2\) with \([0,\infty )\) rather than \((\infty ,0]\), the x argument in \(\theta _2\) must be negated to match the \(\theta _2\) in [7, Sect. 5.1].
Proof
(i) and (ii) follow immediately from Proposition 1. (iii) follows from the proof of Proposition 2 (low energy asymptotics) found in [3], though some of the steps are simplified in the case \(n=2\). The fact \(\alpha \in {\mathbb {R}}\) comes from \(\theta _j(x,0)=\overline{\theta _j(x,0)}\). (iv) follows from Proposition 2. (v) follows from the result for star graphs proved in [3]. For (vi), because \(v\in C^\infty ({\mathbb {R}})\), we require \(v_1^{(2k)}(x)=v_2^{(2k)}(x)\) and \(v_1^{(2k+1)}(x)=v_2^{(2k+1)}(x)\) for all \(k\in {\mathbb {N}}_0\). Then we just compute \(L_m, 1\le m\le 5\) via (1.11). \(\square \)
Footnotes
 1.
This is just the number of compositions of m. Also, since there are no powers of \((2i\zeta )^{1}\), we actually have far fewer elements to sum over, but the \(2^{m1}\) bound will be sufficient.
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