We present a Fermi golden rule giving rates of decay of states obtained by perturbing embedded eigenvalues of a quantum graph. To illustrate the procedure in a notationally simpler setting, we first describe a Fermi golden rule for boundary value problems on surfaces with constant curvature cusps. We also provide a resonance existence result which is uniform on compact sets of energies and metric graphs. The results are illustrated by numerical experiments.
I. INTRODUCTION AND STATEMENT OF RESULTS
Quantum graphs are a useful model for spectral properties of complex systems. The complexity is captured by the graph but analytic aspects remain one dimensional and hence relatively simple. We refer to the monograph by Berkolaiko–Kuchment1 for references to the rich literature on the subject.
In this note we are interested in graphs with infinite leads and consequently with continuous spectra. We study dissolution of embedded eigenvalues into the continuum and existence of resonances close to the continuum. Our motivation comes from a recent Physical Review Letter11 by Gnutzmann–Schanz–Smilansky and from a mathematical study by Exner–Lipovský.10
We consider an oriented graph with vertices , infinite leads , K > 0, and M finite edges . We assume that each finite edge, em, has two distinct vertices as its boundary (a non-restrictive no-loop condition) and we write v ∈ em for these two vertices v. An infinite lead has one vertex. The set of (at most two) common vertices of em and eℓ is denoted by em∩eℓ and we denote by em∋v the set of all edges having v as a vertex.
The finite edges are assigned length ℓm, K + 1 ≤ m ≤ M + K and we put ℓk = ∞, 1 ≤ k ≤ K, for the infinite edges. To obtain a quantum graph we define a Hilbert space, given by
We then consider the simplest quantum graph Hamiltonian which is an unbounded operator P on L2 defined by with
Here ∂ν denotes the outward pointing normal at the boundary of ev,
The space is defined by replacing H2 by when ℓm = ∞.
Quantum graphs with infinite leads fit neatly into the general abstract framework of black box scattering15 and hence we can quote general results Ref. 8 [Chapter 4] in spectral and scattering theory.
A graph given by a cycle connected to K infinite leads at K vertices: vk, eK+k∩eK+k−1 = vk, e2K∩eK+1 = v1, ek∩eK+k = vk. The lengths of finite edges are given by ℓk(t) = e−2ak(t)ℓk, K + 1 ≤ k ≤ 2K. If ℓk(0)’s are rationally related then P(0) has eigenvalues, λ(0), embedded in the continuous spectrum. If λ(0) is simple then λ(0) belongs to a smooth family of resonances, λ(t), Im λ(t) ≤ 0. Theorem 1 and Example 1 in Section III show that in this case , where u is the normalized eigenfunction corresponding to u and ek(λ) is the generalized eigenfunction normalized in the kth lead—see (1.2).
A graph given by a cycle connected to K infinite leads at K vertices: vk, eK+k∩eK+k−1 = vk, e2K∩eK+1 = v1, ek∩eK+k = vk. The lengths of finite edges are given by ℓk(t) = e−2ak(t)ℓk, K + 1 ≤ k ≤ 2K. If ℓk(0)’s are rationally related then P(0) has eigenvalues, λ(0), embedded in the continuous spectrum. If λ(0) is simple then λ(0) belongs to a smooth family of resonances, λ(t), Im λ(t) ≤ 0. Theorem 1 and Example 1 in Section III show that in this case , where u is the normalized eigenfunction corresponding to u and ek(λ) is the generalized eigenfunction normalized in the kth lead—see (1.2).
When K > 0 then the projection on the continuous spectrum of P is given in terms of generalized eigenfunctions ek(λ), 1 ≤ k ≤ K, which for λ ∉ Specpp(P) are characterized as follows:
The family extends holomorphically to a neighbourhood of ℝ and that defines ek(λ) for all λ. We will in fact be interested in λ ∈ Specpp(P). The functions ek parametrize the continuous spectrum of P—see Ref. 8 [Sec. 4.4] and (3.13) below.
We now consider a family of quantum graphs obtained by varying the lengths ℓm, K + 1 ≤ m ≤ M + K,
and the corresponding family of operators, P(t). We consider a(t) as a function which is constant on the edges and denote
The works, Refs. 10 and 11, considered the case in which P(0) has embedded eigenvalues and investigated the resonances of the deformed family P(t) converging to these eigenvalues as t → 0. Here we present a Fermi golden rule type formula (see Section II for references to related mathematical work) which gives an infinitesimal condition for the disappearance of an embedded eigenvalue. It becomes a resonance of P and one can calculate the infinitesimal rate of decay. Resonances are defined as poles of the meromorphic continuation of λ ↦ (P − λ2)−1 to ℂ as an operator (see Refs. 9 and 8 [Sec. 4.2] and for a self-contained general argument Proposition 4.1). We denote the set of resonances of P by Res(P).
The proof is given in Section III and that section is concluded with two examples: the first gives graphs and eigenvalues for which —see Figures 1 and 2. The second example gives a graph and an eigenvalue for which the boundary terms in the formula for Fk are needed—see Fig. 4.
A simple graph with embedded eigenvalues, M = K = 2. Solid lines and dashed lines indicate the trajectory of λ(t) and of the second order approximation , respectively. (The colour coding indicates the parameter t shown in the colour bar.) We approximate the real part linearly using (3.11) and the imaginary quadratically using (1.4). The four cases are (a) ℓ3(t) = 1 − t, ℓ4(t) = 1 − t, (b) ℓ3(t) = 1 − t, ℓ4(t) = 1, (c) ℓ3(t) = 1 − t, ℓ4(t) = 1 + t, (d) ℓ3(t) = 1 − t, ℓ4(t) = 1 + 2t.
A simple graph with embedded eigenvalues, M = K = 2. Solid lines and dashed lines indicate the trajectory of λ(t) and of the second order approximation , respectively. (The colour coding indicates the parameter t shown in the colour bar.) We approximate the real part linearly using (3.11) and the imaginary quadratically using (1.4). The four cases are (a) ℓ3(t) = 1 − t, ℓ4(t) = 1 − t, (b) ℓ3(t) = 1 − t, ℓ4(t) = 1, (c) ℓ3(t) = 1 − t, ℓ4(t) = 1 + t, (d) ℓ3(t) = 1 − t, ℓ4(t) = 1 + 2t.
A surface with one cusp end and a boundary. Suppose we consider a family of boundary conditions for the Laplacian −Δ: ∂νw = γ(t) w at ∂X. The Laplacian has a continuous spectrum with a family of generalized eigenfunctions e(λ) ∈ C∞(X)—see (2.3). Suppose that for t = 0, λ2 is a simple embedded eigenvalue of −Δ with the boundary condition ∂νw = γ(0) w, with the normalized eigenfunction given by u. Then λ = λ(0) belongs to a smooth family of resonances of Laplacians with boundary condition ∂νw = γ(t) w, and —see Theorem 3.
A surface with one cusp end and a boundary. Suppose we consider a family of boundary conditions for the Laplacian −Δ: ∂νw = γ(t) w at ∂X. The Laplacian has a continuous spectrum with a family of generalized eigenfunctions e(λ) ∈ C∞(X)—see (2.3). Suppose that for t = 0, λ2 is a simple embedded eigenvalue of −Δ with the boundary condition ∂νw = γ(0) w, with the normalized eigenfunction given by u. Then λ = λ(0) belongs to a smooth family of resonances of Laplacians with boundary condition ∂νw = γ(t) w, and —see Theorem 3.
The graph from example 2: In this case boundary terms in our Fermi golden rule appear at some embedded eigenvalues such as λ0 which is the smallest solution of , λ0 ≈ 1.9106. We consider the following variation of length: ℓ3 = 1 − t, ℓ4 = 1 + t, ℓ5 = 1 − t, ℓ6 = 1 + t, and (a) ℓ7 = 1, (b) ℓ7 = 1 + t/2, (c) ℓ7 = 1 + t. As pointed out by the referee, the approximation is not as accurate as in Fig. 2 since now . The main point is to illustrate the appearance of the boundary contributions to Fk in (1.4).
The graph from example 2: In this case boundary terms in our Fermi golden rule appear at some embedded eigenvalues such as λ0 which is the smallest solution of , λ0 ≈ 1.9106. We consider the following variation of length: ℓ3 = 1 − t, ℓ4 = 1 + t, ℓ5 = 1 − t, ℓ6 = 1 + t, and (a) ℓ7 = 1, (b) ℓ7 = 1 + t/2, (c) ℓ7 = 1 + t. As pointed out by the referee, the approximation is not as accurate as in Fig. 2 since now . The main point is to illustrate the appearance of the boundary contributions to Fk in (1.4).
Let us compare (1.4) to the Fermi golden rule in more standard settings of mathematical physics as presented in Reed–Simon 14 [Sec. XII.6, Notes to Chapter XII] (see also Cornean–Jensen–Nenciu5 for references to more recent advances). In that case we have a Hamiltonian H(t) = H + tV such that H has a simple eigenvalue at E0 embedded in the continuous spectrum of H(0) with the normalized eigenfunction u. Let ℙ(E) = 1(−∞,E)∖{E0}(H) be a modified spectral projection. Then for small t the operator H(t) has a family of resonances with imaginary parts given by Γ(t)/2 where the width Γ(t) function satisfies
If the continuous spectrum has a nice parametrization by generalized eigenfunctions, ∂Eℙ(E) = ∫𝒜e(E, a) ⊗ e(E, a)∗dμ(a), , this expression becomes
In the case of quantum graphs is a discrete set—see (3.13) below—and the formula is close to our formula (1.4). In Ref. 8 [Theorem 4.22] a general formula for black box perturbations is given and it applies verbatim to perturbations of quantum graph Hamiltonians when the domain of the perturbation does not change. The difference here lies in the fact that the domain changes and that produces additional boundary terms. (We present the results in the simplest case of Kirchhoff boundary conditions.) To explain the method in a similar but notationally simpler setting, we first prove the Fermi golden rule in scattering on surfaces with cusp ends and boundaries. There the change in the domain comes from the change in boundary conditions.
Formula (1.4) gives a condition for the existence of a resonance with a nontrivial imaginary part (decay rate) near an embedded eigenvalue of the unperturbed operator: D(λ0, ct)∩Res(P(t)) ≠ 0̸ for some c and for , where the constants c and t0 depend on λ0 and P(t). (Here and below .) However, it is difficult to estimate the speed with which the resonance λ(t) moves—that is already visible in comparing Fig. 2 with Fig. 4. (A striking example is given by , where and t → 0; infinitely many resonances for t ≠ 022 disappear and P(0) has only one resonance at 0.) Also, the result is not uniform if we vary λ0 or the lengths of the edges.
The next theorem adapts the method of Tang–Zworski20 and Stefanov17 (see also Ref. 8 [Sec. 7.3]) to obtain existence of resonances near any approximate eigenvalue and in particular near an embedded eigenvalue—see the example following the statement. In particular this applies to the resonances studied in Refs. 10 and 11. The method applies however to very general Hamiltonians—for semiclassical operators on graphs the general black box results of Refs. 20 and 17 apply verbatim. The point here is that the constants are uniform even though the dependence on t is slightly weaker.
To formulate the result we define
Suppose that P is defined above and the lengths, ℓm, have the property that , K + 1 ≤ m ≤ M + K, where is a fixed compact subset of the open half-line.
As a simple application of Theorem 1 related to Theorem 2 we present the following.
Example. Suppose that P(t) is the family of operators defined by choosing ℓj = ℓj(t) ∈ C1(ℝ), and that λ0 > 0 is an eigenvalue of P(0). Then for any γ < 1 there exists t0 such that for ,
To apply Theorem 2 we need to construct an approximate mode of P(t) using the eigenfunction of P(0). Thus, let u0 be a normalized eigenfunction of P(0) with eigenvalue λ0; in particular , 1 ≤ k ≤ K. Choose χj ∈ C∞(ℝ), j = 1, 2, such that χj ≥ 0, χ0 + χ1 = 1, χj(s) = 1 near and define and , u0 = u+ + u−.
Remarks.
A slightly sharper statement than (1.7) can already be obtained from the proof in Section IV. It is possible that in fact Res(P)∩D(λ0, C0ε), where C0 depends on , and δ. That is suggested by the fact that the converse to this stronger conclusion is valid—see Proposition 4.5. This improvement would require finer complex analytic arguments. It is interesting to ask if methods more specific to quantum graphs, in place of our general methods, could produce this improvement.
By adapting Stefanov’s methods17 one can strengthen the conclusion by adding a statement about multiplicities (see also Ref. 8 [Exercise 7.1]) but again we opted for a simple presentation.
II. A FERMI GOLDEN RULE FOR BOUNDARY VALUE PROBLEMS: SURFACES WITH CUSPS
To illustrate the Fermi golden rule in the setting of boundary value problems we consider surfaces, X, with cusps of constant negative curvature. The key point is that the domain of the operator changes and the general results such as Ref. 14 [Theorem XII.24] or Ref. 8 [Theorem 4.22] do not apply.
Thus we assume that (X, g) is a Riemannian surface with a smooth boundary and a decomposition (see Fig. 3)
We consider the following family of unbounded operators on L2(X):
where t ↦ γ(t) ∈ C∞(∂X) is a smooth family of functions on ∂X and ∂ν is the outward pointing normal derivative. The spectrum of the operator P has the following well known decomposition:
(When J = + ∞ then Ej → ∞.) The eigenvalues Ej > 0 are embedded in the continuous spectrum. In addition the resolvent R(λ) ≔ (P − λ2)−1 : L2 → L2, Im λ > 0, has a meromorphic continuation to λ ∈ ℂ as an operator . Its poles are called scattering resonances. Under generic perturbation of the metric in X0 all embedded eigenvalues become resonances. For proofs of these well known facts see Ref. 4 and also Ref. 8 [Sec. 4.1 (Example 3), Sec. 4.2 (Example 3), and Sec. 4.4.2] for a presentation from the point of view of black box scattering.15
The generalized eigenfunctions, e(λ, x), describing the projection onto the continuous spectrum have the following properties:
see Ref. 8 [Theorem 4.20]. With these preliminaries in place we can now prove
Suppose that the operators P(t) are defined by (2.2) and that λ > 0 is a simple eigenvalue of P(0) and (P(0) − λ2) u = 0, .
1. In the case of scattering on constant curvature surfaces with cusps the Fermi golden rule was explicitly stated by Phillips–Sarnak—see Ref. 13 and for a recent discussion.12 For a presentation from the black box point of view see Ref. 8 [Sec. 4.4.2].
2. The proof generalizes immediately to the case of several cusps (which is analogous to a quantum graph with several leads), (Xk, g|Xk) ≃ ([ak, ∞) × ℝ/ℓkℤ, dr2 + e−2rdθ2, 1 ≤ k ≤ K. In that case the generalized eigenfunction is normalized using
The Fermi golden rule for the boundary value problem (2.2) is given by
For notational simplicity we assume that γ(0) ≡ 0, that is that P(0) is the Neumann Laplacian on X. We will also omit the parameter t when that is not likely to cause confusion. It is also convenient to use z = λ2 and to write 〈•, •〉 for the L2(X, dvolg) inner product and 〈•, •〉L2(∂X) for the inner product on L2(∂) with the measure induced by the metric g.
III. PROOF OF THEOREM 1
We follow the same strategy as in the proof of Theorem 3 but with some notational complexity due to the graph structure.
Let . Then for u, v ∈ H2, ,
We note here that the sum over vertices can be written as a sum over edges
Just as in Section II the domain of the deformed operators will change but we make a modification which will keep the Hilbert space on which (we change the notation from Section I and will use P(t) for a unitarily equivalent operator) acts fixed by changing the lengths in (1.3). For that let
Let be defined in by ,
That is just the family of Neumann Laplace operators on the graph with the lengths e−aj(t)ℓj.
On L2 we define a new family of operators: . It is explicitly given by ,
Using Proposition 4.3 from Sec. IV we see that for small t there exists a smooth family such that
We defined by (1.5) and denote by 1x≤R the orthogonal projection .
Writing P = P(t), u = u(t), z = z(t) we see, as in (2.8), that
We recall that em, 1 ≤ m ≤ K are the infinite edges with unique boundaries. Hence, using (3.1), at t = 0,
We now look at the equation satisfied by at t = 0,
Hence,
We used here the fact that u(v) ≔ um(v) does not depend on m. The second condition can be formulated as , where w ≔ ∂t(ea(t)/2u(t))|t=0 is continuous on the graph.
To find an expression for (similar to (2.13)) we first find
such that
We can assume without loss of generality that both g and u are real valued.
In analogy to (2.11) we claim that
(We used the continuity of u and the Neumann condition .) Since g and u satisfy the same boundary conditions, (3.8) and (3.9), (3.10) follows from (3.11).
As in the derivation of (2.13) we now see that for some α ∈ ℂ we have
With this in place we return to (3.6). The first term on the right hand side is
(We used here the simplifying assumption that g and u are real valued.)
As in (2.16) we conclude that
which means that (with z = λ2 and ek = ek(λ))
The second term on the right hand side is now rewritten using (3.1) and the boundary conditions (3.9),
We conclude that
A similar analysis of the second term on the right hand side of (2.9) shows that
IV. PROOF OF THEOREM 2
The proof adapts to the setting of quantum graphs and of quasimodes u satisfying (1.6) the arguments of Ref. 20. They have origins in the classical work of Carleman3 on completeness of eigenfunctions for classes of non-self-adjoint operators, see also Refs. 17 and 19.
We start with general results which are a version of the arguments of Ref. 8 [Sec. 7.2]. In particular they apply without modification to quantum graphs with general Hamiltonians and general boundary conditions. We note that for metric graphs considered here much more precise estimates are obtained by Davies–Pushnitski6 and Davies–Exner–Lipovský7 but since we want uniformity we present an argument illustrating the black box point of view.
Suppose that P satisfies the assumptions of Theorem 2 and Ω1⋐Ω2⋐ℂ, where Ωj are open sets.
From Ref. 2 [Theorem 3.10] we know that if μ is an eigenvalue of P(s) of multiplicity N then we can choose analytic functions μn(t) ∈ ℝ, , such that μn(s) = μ, and for small t − s, P(t) un(t) = μn(t) un(t), and is an orthonormal setting spanning , for ε > 0 small enough. The lemma follows from showing that ∂tμn(s) ≥ 0 for any n.
Inequality (4.6) follows from the lemma as we can change the length of the edges in succession. The Weyl law for (see Ref. 1), and the fact that (where χ3 denotes the multiplication operator), now shows that for any operator ,
where the constant C4 depends only on . From this we deduce (4.5) and . For instance,
Here we used the facts that χ2 ≡ 1 on the support of χ1, hence [P, χ2]χ1 = 0, and that [P, [P, χ2]][P, χ2] are second and first order operators, respectively, and that R(λ0) maps L2 to . The other terms in K(λ, λ0) are estimated similarly and that gives (4.5). (Finer estimates for large λ are possible—see Ref. 8 [Secs. 4.3 and 7.2] and 20—but we concentrate here on uniformity near a given energy.)
Now, let , where R is large enough so that Ω2 ⊂ Ω3. It follows that for a constant C3 depending only on Ω3 and , and hence only on Ω2,
we have
(For basic facts about determinants see, for instance, Ref. 8 [Sec. B.5].) Writing
we obtain
that is
where C4 depends only on λ0 and . The Jensen formula (see, for instance, Ref. 21 [Sec. 3.61]) then gives a bound on the number of zeros of det(I + K(λ, λ0)) in Ω3. That proves the first bound in (4.1).
We can write
where g(λ) is holomorphic in Ω3. From the upper bound (4.7) and the lower bound (4.8) we conclude that in a smaller disc containing Ω2, with C5 depending only on the previous constants. (For instance, we can use the Borel–Carathéodory inequality—see Ref. 21 [Sec. 5.5].) Hence
To deduce the second bound in (4.1) from this we use the inequality
which gives
for λ ∈ Ω1 and C7 depending only on Ωj’s , and R. This completes the proof.□
Before proving Theorem 2 we will use the construction of the meromorphic continuation in the proof of Proposition 4.1 to give a general condition for smoothness of a family of resonances (see also Ref. 16),
That is the only property used in the proof of
The proof of (4.10) under the condition (4.9) follows from (4.4) and the definitions of Q(λ, λ0) and K(λ, λ0). From that the conclusion about the deformation of a simple resonance is immediate—see Ref. 8 [Theorems 4.7 and 4.9].
It remains to establish (4.9). Suppose f ∈ L2 and define u(t) ≔ R(λ0, t) f ∈ L2. Formally, satisfies (3.8) with and . We can find a smooth family g(t) ∈ L2 satisfying (3.9) with u = u(t). We then have ∂tu(t) = g + R(λ0, t)(−2∂ta(t) u(t) − G(t)), where . By considering difference quotients, a similar argument shows that u(t) ∈ L2 is differentiable. The argument can be iterated showing that u(t) ∈ C∞((−t0, t0), L2) and that proves (4.9).□
We now give
To derive a contradiction we use the following simple lemma:
We apply this lemma to f(z) ≔ 〈1r≤RR(z + λ0)1r≤Rφ, ψ〉, φ, ψ ∈ L2, with M+ = C3/δ+, M = M− = C2δ−C1. If we show that
we obtain a contradiction to (1.6) by putting and ψ = u and using the support property of u (the outgoing resolvent is the right inverse of on compactly supported function):
For γ < 1 choose γ < γ1 < γ2 < γ3 < 1 and put
For completeness we also include the following proposition which would be a converse to Theorem 2 for γ = 1. The more subtle higher dimensional version in the semiclassical setting was given by Stefanov.18
Acknowledgments
We are grateful for the support of National Science Foundation under the Grant No. DMS-1500852. We would also like to thank Semyon Dyatlov for helpful discussions and assistance with figures and the two anonymous referees whose comments and careful reading led to many improvements.