Trace formulas for Schr\"odinger operators on star graphs

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.


Introduction
A quantum graph is a triple (Γ, H, VC), where Γ is a metric graph with edges {e j }, H is a differential operator, and VC is a set of vertex conditions. The usual differential operator is the Schrödinger operator defined by (1.1) Hψ = − d 2 dx 2 ψ + V ψ, where ψ ∈ L 2 (Γ) := j L 2 (e j ) and V is a real-valued potential on Γ, with restriction to edge e j denoted by v j := V | ej . We assume throughout that V is sufficiently regular, for instance, in L 1 loc,unif on each edge. We can require continuity of ψ on Γ and impose Kirchoff boundary conditions on each vertex v, where the sum is over all edges e containing the vertex v, and the e+ indicates the derivative is taken in the outgoing direction. These conditions make H a self-adjoint operator on L 2 (Γ). 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].
One of the simplest examples of a quantum graph is a quantum star graph, which consists of a single vertex v and n ≥ 2 edges e j , 1 ≤ j ≤ n, each of which is identified with the half-line [0, ∞). Because of its similarity to n half-lines, many of the spectral theory and scattering theory results for the half-line case have analogues for star graphs. For example, Levinson's formula for star graphs and expressions for the spectral shift function and perturbation determinant for star graphs were proved in [3]. In this article, we will prove trace formulas of the Buslaev-Faddeev type for star graphs. We will first briefly review the scattering theory that is relevant to us here; see also [7, §4]. In the one dimensional case, the Schrödinger equation is just −u ′′ + V (x)u = zu, z = ζ 2 . Consider a half-line edge e j in the star graph with ej |v j (x)| dx < ∞.
Then there are two linearly independent solutions to the Schrödinger equation on e j , the regular solution ϕ j and the Jost solution θ j . The Jost solution is determined by the asymptotics, 4) and gives rise to the Jost function, ω j (ζ) := θ(0, ζ).
In the half-line case and under suitable conditions on the potential, the (modified) perturbation determinant D(ζ) (Section 2) is simply given by D(ζ) = ω(ζ) and is analytic in ζ for Im ζ > 0 and continuous up to Im ζ = 0 except possibly at ζ = 0. For the star graph, the perturbation determinant is more complicated, but an explicit formula in terms of the Jost functions and solutions is given in [3], which we will use in Section 2.
Using the perturbation determinant, we can define the limit amplitude a(k) and limit phase (or phase shift ) η(k) for ζ = k ∈ R. We set D(k) =: a(k)e iη(k) , with a(k) = |D(k)|. As we will see later in (2.5), D(z) = 1 + O(|ζ| −1 ) as |ζ| → ∞, so we can choose η with the convention η(∞) = 0. (We also require it to be continuous.) The names for a(k) and η(k) come from scattering theory, where they correspond to the amplitude shift and phase shift in the wavefunction as x → ∞.
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 − d 2 dx 2 ψ + V ψ = 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 "half-bound states" or "metastable bound states".
Our main result is the following trace formulas of the type in [2] relating |λ j | n (for n ∈ 1 2 N) to an expression involving the potential V .
Theorem 1 (trace formulas). Let Γ be a star graph with edges {e j }, 1 ≤ j ≤ n. Assume that and that for each j = 1, . . . , n and m ∈ N 0 := N ∪ {0}, If ζ = 0 is a resonance of multiplicity one, we also assume that Then letting r j be the multiplicity of the eigenvalue λ j , we have For m = 1, 2, . . ., The coefficients L m come from the asymptotic expansion log D(ζ) = ∞ m=1 L m (2iζ) −m , for |ζ| → ∞. The first few coefficients are given by (1.11) Remarks. 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 Γ is that for any two distinct e i and e j , the function on R obtained by combining v i and v j is smooth. This is easily seen to be equivalent to v 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.
Let us comment the outline of this paper and on our approach to Theorem 1. There is a wellknown strategy to obtain trace formulas which goes back to [2] (see, for instance, [7] for a textbook presentation) and which we will follow here. It consists in the following steps: (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. A key step both in Steps 1 and 2 is to find a representation of the perturbation formula in terms of Jost solutions. This allows to prove bounds on the perturbation determinant which are necessary both to control the limit of the contour in Step 1 and to perform the analytic continuation in Step 2. In order to carry out this program we will rely on [3], which contains a useful formula for the perturbation determinant on a star graph, see Proposition 1, and gives the low energy asymptotics of the perturbation determinant, see Proposition 2. Using these results we can carry out Step 1 in a rather straightforward manner. For Step 2, however, we also need high energy asymptotics for the perturbation determinant, and that is our main technical result in this paper. While they also rely on the representation formula from [3] they require an inductive procedure and careful remainder estimates. We provide those in Section 4 and in Appendix A.
In order to make this paper self-contained we review necessary results from the literature in Section 2 and provide details for Steps 1 and 2 in Section 3. As we already mentioned, Section 4 contains the novel high energy asymptotics which lead, in particular, to the formulas for the coefficients L m from (1.11). In Section 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, §5]. In contrast to [7, §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.

The perturbation determinant and other previous results
The free Schrödinger operator is just the operator H 0 := − d dx 2 , i.e. there is no potential V . The corresponding resolvent will be denoted by R 0 (z) := (H 0 −z) −1 , while the resolvent for H will be denoted by R(z) := (H − z) −1 . The perturbation determinant (PD), introduced by Krein in 1953, produces a holomorphic function on the resolvent set ρ(H 0 ) that is determined by the pair of operators H, H 0 . In our case, we will actually need to look at the modified perturbation determinant (see [8,7] for details). It is closely related to the spectral shift function, which has applications to many areas, including spectral theory, scattering theory, and trace formulas [4]. The modified perturbation determinant, which we will just call the perturbation determinant, is given by It is shown in [3] that this is well-defined in our case. Some useful facts about the perturbation determinant that can be found in e.g. [8,7] are as follows.
• D(ζ) has a zero in ζ of order r if and only if ζ 2 is an eigenvalue of multiplicity r of H.
Remark. The perturbation determinant is sometimes defined with an argument of z = ζ 2 . Then the definition is D(z) := det(½ + √ V R 0 (z) |V |) for z ∈ ρ(H 0 ), and the last property listed above takes on a nicer form. We will however continue to use the definition with ζ.
Proposition 1 (formula for the PD, [3]). If the potential V is a real-valued function on Γ such that (1.5) holds, then for ζ 2 = z ∈ ρ(H 0 ), The importance of this proposition is that it connects a spectral theoretic object, namely D(ζ), with ODE objects, namely the θ j 's. Results of this type go back to [6]; see also [5].
Proposition 1 also yields the leading order of the high energy asymptotics of D(ζ). In fact, from asymptotics for the half-line, ω j (ζ) = θ j (0, ζ) = 1 + O(|ζ| −1 ) and θ ′ (0, ζ) = iζ + O(1) as |ζ| → ∞, we have which is the same limiting behavior as in the half-line case. To get the coefficients L m in Theorem 1, we need a full asymptotic expansion for K(ζ), which we will compute in Section 4.

Adapting results from the half-line case
Trace formula derivations for the half-line case can be found in [7, §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(ζ), 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 Section 4.)  (Figure 1.) There are two main differences for a star graph: D may have zeros that are not simple, and integration over the contour C ε must use different low energy asymptotics for D(ζ). where iκ j = i|λ j | 1/2 is a zero of D(z) of order r j . Integration by parts along a semicircle C + r of radius r yields, If r = R, this integral goes to zero for Re s < 1 2 since D(ζ) = 1 + O(|ζ| −1 ); this is the same as in the half-line case. For r = ε, recall from Proposition 2 that D(ζ) = cζ m−1 (1 + o(1)) as ζ → 0, and c = 0, so |ε 2s log D(ε)| = | log(cε m−1 (1 + o(1)))|ε 2 Re s ≤ (| log ε m−1 | + | log |c|| + o(1) + π)ε 2 Re s → 0 as ε → 0, m = 0, 1, 2, . . ., since Re s > 0. Similarly, r j e πis κ 2s j .
Integrating by parts and using log D(ζ) = log a(k) + iη(k) along with (2.2) yields the desired formula.
We can analytically continue F and G to Re s > 0 just as in the half-line case. Suppose we have the asymptotic expansion The function G(s) is analytic everywhere except half-integer points m + 1 2 , m = 0, 1, 2, . . ., where it has simple poles with residues (−1) m 2 −2m−2 L 2m+1 . If m ≥ 1 and m − 1 2 < Re s < m + 1 2 , then Proof Using Lemma 2 yields the trace formulas in Theorem 1.

High energy asymptotic expansions
In this section, we will compute the coefficients in the asymptotic series expansion, To this end, assume (1.6) holds for each j = 1, . . . , n and m ∈ N 0 . Using the formula for the PD (2.1), we have which is permissible since each separated term is 1 + O(|ζ| −1 ) and hence has argument close to zero as |ζ| → ∞. From [7, §4], there is the following asymptotic series for the logarithm of each Jost function ω j (ζ), Also, these g m (x) occur in the asymptotic expansion Now it remains to find an asymptotic expansion for log K(ζ) inζ . By (4.3), To take the logarithm, we use the same method as in the half-line case, which is to find the asymptotic expansion for the logarithmic derivative, and then integrate. First we do long division: are asymptotic expansions, then we have the asymptotic expansion, We fix a branch of the logarithm using K(ζ) inζ = 1+O(|ζ| −1 ) and requiring log K(ζ) inζ → 0 as |ζ| → ∞. Integrating, we get the following: Corollary 2 (logarithm expansion). Let A(x, ζ), B(x, ζ) be as above, and suppose we have  We also define C 1 := C 1 (0) = a 1 (0), and for m ≥ 2, if a m (x) → 0 as x → ∞, In our case, A(x, ζ) = K(x,ζ) inζ , so a 1 (x) ≡ 0, a m (x) := 2 n g [1] m−1 (x) + · · · + g [n] m−1 (x) , and we get the asymptotic series (We show in Appendix A that the functions A(x, ζ) and B(x, ζ) in our case do indeed satisfy the necessary hypotheses to apply the corollary.) The first few C m 's (other than C 1 = 0) are Equation (4.2) becomes, m . Now using (4.8) and the definition of the ℓ m 's, we get (1.11).

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, §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 R is viewed as a pair of potentials v 1 (x) := v(x) and v 2 (x) := v(−x), x ≥ 0 on the n = 2 star graph. Using results for star graphs from [3] along with Theorem 1, we can show: Corollary 3 (real line). Consider the Schrödinger operator H = − d 2 dx 2 + v on the real line and suppose R |v(x)| dx < ∞. Then the following hold.
(i) The perturbation determinant is where W is the Wronskian and θ 1 , θ 2 are the Jost solutions on the two half-line edges.
(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. 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 α ∈ R comes from θ j (x, 0) = θ j (x, 0). (iv) follows from Proposition 2. (v) follows from the result for star graphs proved in [3].  Here we show that the asymptotic expansion for log K(ζ) inζ derived in Section 4 is valid, in particular, that Corollary 2 applies to our specific A(x, ζ) and B(x, ζ). Ignoring x, it is easy to verify that B(x, ζ) has a valid asymptotic series in terms of powers (2iζ) −m , but to integrate, we need error bounds involving x as well. Recall that A(x, ζ) = K(x, ζ) inζ = 1 + m−1 (x). From properties of the g m 's which can be found in e.g. [7, §4.4], it follows that for all x ≥ 0 and Im ζ ≥ 0, |ζ| ≥ c > 0, The maximum possible binomial coefficient on any A M (x, ζ) j is Summing over all compositions of m and then over all m, we get the error bound