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.

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 {vj}j=1J, infinite leads {ek}k=1K, K > 0, and M finite edges {em}m=K+1M+K. We assume that each finite edge, em, has two distinct vertices as its boundary (a non-restrictive no-loop condition) and we write vem for these two vertices v. An infinite lead has one vertex. The set of (at most two) common vertices of em and e is denoted by eme and we denote by emv the set of all edges having v as a vertex.

The finite edges are assigned length ℓm, K + 1 ≤ mM + K and we put ℓk = ∞, 1 ≤ kK, for the infinite edges. To obtain a quantum graph we define a Hilbert space, given by

L2m=1K+ML2([0,m]),L2u=(u1,,uM+K),umL2([0,m]).
(1.1)

We then consider the simplest quantum graph Hamiltonian which is an unbounded operator P on L2 defined by (Pu)m=x2um with

D(P)={u:umH2([0,m]),um(v)=u(v),veme,emvνum(v)=0}.

Here ∂ν denotes the outward pointing normal at the boundary of ev,

umH2([0,m]),νum(0)=um(0),νum(m)=um(m).

The space Dloc(P) is defined by replacing H2 by Hloc2 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.

FIG. 1.

A graph given by a cycle {ek}k=K+12K connected to K infinite leads {ek}k=1K at K vertices: vk, eK+keK+k−1 = vk, e2KeK+1 = v1, ekeK+k = vk. The lengths of finite edges are given by ℓk(t) = e−2ak(t)k, 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 Imλ̈=λ2k=1Kȧu,ek(λ)2, where u is the normalized eigenfunction corresponding to u and ek(λ) is the generalized eigenfunction normalized in the kth lead—see (1.2).

FIG. 1.

A graph given by a cycle {ek}k=K+12K connected to K infinite leads {ek}k=1K at K vertices: vk, eK+keK+k−1 = vk, e2KeK+1 = v1, ekeK+k = vk. The lengths of finite edges are given by ℓk(t) = e−2ak(t)k, 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 Imλ̈=λ2k=1Kȧu,ek(λ)2, where u is the normalized eigenfunction corresponding to u and ek(λ) is the generalized eigenfunction normalized in the kth lead—see (1.2).

Close modal

When K > 0 then the projection on the continuous spectrum of P is given in terms of generalized eigenfunctions ek(λ), 1 ≤ kK, which for λ ∉ Specpp(P) are characterized as follows:

ek(λ)Dloc(P),(Pλ2)ek(λ)=0,emk(λ,x)=δmkeiλx+smk(λ)eiλx,1mK.
(1.2)

The family λek(λ)Dloc(P) 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 ≤ mM + K,

m(t)=eam(t)m,am(0)=0,
(1.3)

and the corresponding family of operators, P(t). We consider a(t) as a function which is constant on the edges and denote

ȧta(0),(ȧu)m(x)=ȧmum(x).

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 ↦ (Pλ2)−1 to ℂ as an operator Lcomp2Lloc2 (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).

Theorem 1.
Suppose thatλ2 > 0 is a simple eigenvalue ofP = P(0) anduis the corresponding normalized eigenfunction. Then fortt0there exists a smooth functiontλ(t) such thatλ(t) ∈ Res(P(t)) and
Imλ̈=k=1KFk2,Fkλȧu,ek(λ)+λ1vemv14ȧm(3νum(v)ek(λ,v)¯u(v)νemk(λ,v)¯),
(1.4)
where 〈•, •〉 denotes the inner product on (1.1).

The proof is given in Section III and that section is concluded with two examples: the first gives graphs and eigenvalues for which Fk=λȧu,ek(λ)—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.

FIG. 2.

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 λ̃(t)=λ+tλ̇+i2t2Imλ̈, 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.

FIG. 2.

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 λ̃(t)=λ+tλ̇+i2t2Imλ̈, 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.

Close modal
FIG. 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 Imλ̈=14λ2γ̇u,e(λ)L2(X)2—see Theorem 3.

FIG. 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 Imλ̈=14λ2γ̇u,e(λ)L2(X)2—see Theorem 3.

Close modal
FIG. 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 tanλ+2tanλ2=0, 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 λ(t)λ(0)+tλ̇+i2t2Imλ̈ is not as accurate as in Fig. 2 since now Reλ̈0. The main point is to illustrate the appearance of the boundary contributions to Fk in (1.4).

FIG. 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 tanλ+2tanλ2=0, 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 λ(t)λ(0)+tλ̇+i2t2Imλ̈ is not as accurate as in Fig. 2 since now Reλ̈0. The main point is to illustrate the appearance of the boundary contributions to Fk in (1.4).

Close modal

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

2t2Γ(0)=πEu,VP(E0)Vu.

If the continuous spectrum has a nice parametrization by generalized eigenfunctions, ∂Eℙ(E) = ∫𝒜e(E, a) ⊗ e(E, a)(a), aA, this expression becomes

t2Γ(0)=πAVu,e(E,a)2dμ(a).

In the case of quantum graphs A 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 tt0, where the constants c and t0 depend on λ0 and P(t). (Here and below D(λ0,r):={λ:λλ0<r}.) 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 P(t)=x2+tV(x), where VCc(R) 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

HRm=1KL2([0,R])m=K+1K+ML2([0,m]).
(1.5)

Theorem 2.

Suppose thatPis defined above and the lengths,m, have the property thatmL,K + 1 ≤ mM + K, whereLis a fixed compact subset of the open half-line.

Then for anyL(0,),I⋐(0, ∞),R > 0, andγ < 1 there exists ε0 > 0 such that
uHRD(P),λ0Isuch thatuL2=1,(Pλ02)u=ε<ε0
(1.6)
implies
Res(P)D(λ0,εγ).
(1.7)

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 tt0,

Res(P(t))D(λ0,tγ).
(1.8)

Proof.

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 uk00, 1 ≤ kK. Choose χjC(ℝ), j = 1, 2, such that χj ≥ 0, χ0 + χ1 = 1, χj(s) = 1 near js<13 and define um(x)χ0(x/m)um0(x) and um+(x)χ1(x/m)um0(x), u0 = u+ + u.

We now define a quasimode for P(t), u = u(t) needed in (1.6),
um(t)=um(x)+um+(xδm(t)),δm(t)m(t)m(0).
For t small enough suppum[0,23)m(0)[0,m(t)) and suppum+(23,1]m(0)(δm(t),1]m(0). Hence the values of um(t) and ∂νum(t) at the vertices are the same as those of um0 and um(t)D(P(t)). Also, since (x2λ02)um0=0 and χ0(k)=χ1(k), (and putting ℓm = ℓm(0))
[(P(t)λ02)u(t)]m=m2(χ0((xδm(t))/m)um0(xδm(t))χ0(x/m)um0(x))+2m1(χ0((xδm(t))/m)xum0(xδm(t))χ0(x/m)xum0(x)).
We note that all the terms are supported in (13δm(t),23+δm(t))m(0) and elementary estimates show that (P(t)λ02)u(t)Ct. For instance,
χ0(x)(um0(xδm(t))um0(x))Cδm(t)maxx12m(0)16+|δm(t)xum0(x)Cδm(t)(x2u0mL2((14,34)m(0))+u0mL2((14,34)m(0)))C(λ02+1)t.
From (1.7) we conclude (after decreasing γ and t0) that (1.8) holds.□

Remarks.

  1. 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 L,R, 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.

  2. 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.

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)

X=X1X0,X0=X1X,XX1=,(X1,g|X1)([a,)r×(R/Z)θ,dr2+e2rdθ2).
(2.1)

We consider the following family of unbounded operators on L2(X):

P(t)=Δg14,D(P(t))={uH2(X):νu|X=γ(t)u|X},
(2.2)

where tγ(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:

Spec(P)=Specpp(P)Specac(P),Specac(P)=[0,),
Specpp(P)={Ej}j=0J,14E0<E1E2,0J+.

(When J = + ∞ then Ej → ∞.) The eigenvalues Ej > 0 are embedded in the continuous spectrum. In addition the resolvent R(λ) ≔ (Pλ2)−1 : L2L2, Im λ > 0, has a meromorphic continuation to λ ∈ ℂ as an operator R(λ):Cc(X)C(X). 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:

(Pλ2)e(λ,x)=0,10e(λ,x)|X1dθ=er2eiλr+s(λ)eiλr,(R(λ)R(λ))f=i2λe(λ,x)f,e(λ,),λR,fCc(X),
(2.3)

see Ref. 8 [Theorem 4.20]. With these preliminaries in place we can now prove

Theorem 3.

Suppose that the operatorsP(t) are defined by (2.2) and thatλ > 0 is a simple eigenvalue ofP(0) and (P(0) − λ2) u = 0,uL2=1.

Then there exists a smooth functiontλ(t),t<t0, such thatλ(0) = λ,λ(t) is a scattering resonance ofP(t) and
Imλ̈=14λ2γ̇u,eL2(X)2,e(x)=e(λ,x),
(2.4)
wheree(λ, x) is given in (2.3),ḟtf|t=0andL2(∂X) is defined using the metric induced byg.

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−2r2, 1 ≤ kK. In that case the generalized eigenfunction is normalized using

1m0mek(λ,x)|Xmdθ=er2δkmeiλr+skm(λ)eiλr.

The Fermi golden rule for the boundary value problem (2.2) is given by

Imλ̈=14λ2k=1Kγ̇u,ekL2(X)2,ek(x)=ek(λ,x).
(2.5)

Proof.

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.

We first define the following orthogonal projection:
1rRu10u|X1{rR}dθ,1rR:L2(X)L2([R,),erdr),R>a,1rRI1rR,HR1rRL2(X).
(2.6)
The smoothness of scattering resonances arising from a smooth perturbation of a simple resonance follows from smooth dependence of the continuation of (P(t) − λ2)−1 (see Proposition 4.3 below for a general argument). Let tu(t), u(0) = u denote a smooth family of resonant states
(P(t)z(t))u(t)=0,10u(t)|X1dθ=a(t)er2eiλ(t)r,a(0)=0,Imλ(t)0,λ(0)2=z(0).
(2.7)
The second equation in (2.7) means that u(t) is outgoing—see Ref. 8 [Sec. 4.4].
The self-adjointness of P(t) and integration by parts for the zero mode in the cusp show that for u = u(t) and P = P(t),
0=Im(Pz)u,1rRu=Imr(1rRu)(R)1rRu¯(R)Imz1rRuL2(X)2.
(2.8)
(See Ref. 8 (4.4.17) for a detailed presentation in the general black box setting.) Since Imż=0 (as Im z(t) ≤ 0, see also (2.12) below) and since 1rRu(0) = 0, we have, at t = 0, Imz̈=2Imr(1rRu̇)(R)1rRu̇¯. We would like to argue as in (2.8) but in reverse. However, as u̇ will not typically be in D(P) we now obtain the boundary terms
Imz̈=2Im(Pz)u̇,1rRu̇+2Imνu̇,u̇L2(X).
(2.9)
We now need an expression for u̇. Since (P(t) − z(t)) u(t) = 0, ∂νu|X = γu|X, we have (at t = 0),
(Pz)u̇=żu,νu̇|X=γ̇u|X.
(2.10)
In addition, differentiation of the second condition in (2.7) shows that u̇ is outgoing.
Without loss of generality we can assume that u = u(0) is real valued. Choose gC̄(X,R) (real valued, compactly supported, and smooth up to the boundary) such that νg|X=γ̇u|X. We claim that
żu(Pz)g,u=0.
(2.11)
In fact, Green’s formula shows that the left hand side of (2.11) is equal to ż+Xγ̇u2. On the other hand, using the fact that 1rRu(0) = u(0),
0=ddt(P(t)z(t))u(t),1rRu(t)|t=0=żu(Pz)u̇,u=ż+Xνu̇u=ż+Xγ̇u2.
(2.12)
In view of (2.11), vg+R(λ)(żu(Pz)g), λ2 = z, λ > 0, is well defined, outgoing (see (2.7)), and solves the boundary value problem (2.10) satisfied by u̇. Since the eigenvalue at z is simple that means that u̇v is a multiple of u (see Ref. 8 [Theorem 4.18] though in this one dimensional case this is particularly simple). Hence
u̇=αu+g+R(λ)(żu(Pz)g).
(2.13)
With this formula in place we return to (2.9). First we note that the first term on the right hand side vanishes
Im(Pz)u̇,1rRu̇=Imżu,u̇=żImu,αu+g+R(λ)(żu(Pz)g)=żImα+żImu,R(λ)(żu(Pz)g)=żImα.
(2.14)
Here we used the fact that u and g were chosen to be real. The last identity followed from (2.11). To analyse the second term on the right hand side of (2.9) we recall some properties of the Schwartz kernel of the resolvent
R(λ)(x,y)=R(λ)(y,x)=R(λ¯)(x,y)¯,λC.
(2.15)
(The first property follows from considering λ = ik, k ≫ 1, and using the fact that Pu¯=Pū, and the second from considering Im λ ≫ 1, z = λ2, and noting that ((Pz)1)=(Pz̄)1.) Using (2.9), (2.10), (2.14), (2.13), (2.15), (2.12), and the fact that u and g are real, we now see that (with λ=z>0)
Imz̈=2żImα+2Imγ̇u,u̇L2(X)=2żImα+2Imαγ̇u,u+2Imγ̇u,[R(λ)(żu(Pz)g)]|XL2(X)=1iγ̇u,[(R(λ)R(λ))(żu(Pz)g)]|XL2(X).
(2.16)
Since (R(λ) − R(−λ)) u = 0 we have now to use (2.3) to see that
[(R(λ)R(λ))(żu(Pz)g)]|X=i2λe(λ)|XXe(λ)¯(Pz)g=i2λe(λ)|XX(νe¯(λ)gνge(λ)¯)=i2λe(λ)|Xγ̇u,eL2(X).
Inserting this into (2.16) gives (2.4) completing the proof.□

We follow the same strategy as in the proof of Theorem 3 but with some notational complexity due to the graph structure.

Let H2m=1M+KH2([0,m]). Then for u, vH2, (xku)mxkum,

x2f,gL2=xf,xgL2vemvνfm(v)ḡm(v)=f,x2gL2+vemvfm(v)νḡm(v)νfm(v)ḡm(v).
(3.1)

We note here that the sum over vertices can be written as a sum over edges

vemvνfm(v)ḡm(v)=m=1M+Kvemνfm(v)ḡm(v).
(3.2)

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 P˜(t) (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

Lt2m=1M+KL2([0,eam(t)m]),L2L02,U(t):Lt2L2,
[U(t)u]m(y)eam(t)/2um(eam(t)y),U(t)1=U(t).

Let P˜(t) be defined in Lt2 by (P˜(t)u)m=x2um,

D(P˜(t))={u:umH2([0,eaj(t)m]),um(v)=u(v),veme,emvνum(v)=0}.

That is just the family of Neumann Laplace operators on the graph with the lengths eaj(t)j.

On L2 we define a new family of operators: P(t)U(t)P˜(t)U(t). It is explicitly given by [P(t)u]m=e2am(t)x2um,

(3.3)
D(P(t))={uH2:eam(t)/2um(v)=ea(t)/2u(v),veme,
emve3am(t)/2νum(v)=0}.

Using Proposition 4.3 from Sec. IV we see that for small t there exists a smooth family tu(t)Hloc2 such that

(P(t)z(t))u(t)=0,uk(t,x)=a(t)eiλ(t)x,1kK,Imλ(t)0,λ(0)2=z,λ(0)>0.
(3.4)

We defined HR by (1.5) and denote by 1xR the orthogonal projection L2HR.

Writing P = P(t), u = u(t), z = z(t) we see, as in (2.8), that

0=Im(Pz)u,1xRu=Imm=1Kxum(R)ūm(R)ImzuHR2.
(3.5)

We recall that em, 1 ≤ mK are the infinite edges with unique boundaries. Hence, using (3.1), at t = 0,

Imz̈=2Imm=1Kxu̇m(R)u̇¯m(R)=2Im(Pz)u̇,1xRu̇+2Imvemvνu̇m(v)u̇¯m(v).
(3.6)

We now look at the equation satisfied by u̇ at t = 0,

ddt(P(t)z(t))u(t)=2ȧ(x2u)żu+(Pz)u̇=(2ȧzż)u+(Pz)u̇.
(3.7)

Hence,

(x2z)u̇m=(ż2zȧm)um,emvνu̇m(v)=32emvȧmνum(v),u̇m(v)u̇(v)=12(ȧȧm)u(v),veme.
(3.8)

We used here the fact that u(v) ≔ um(v) does not depend on m. The second condition can be formulated as u̇m(v)=w(v)12ȧm(v)u(v), where w ≔ ∂t(ea(t)/2u(t))|t=0 is continuous on the graph.

To find an expression for u̇ (similar to (2.13)) we first find

gm=1KCc([0,))m=K+1M+KC([0,m]),

such that

emvνgm(v)=32emvȧmνum(v),gm(v)g(v)=12(ȧȧm)u(v).
(3.9)

We can assume without loss of generality that both g and u are real valued.

In analogy to (2.11) we claim that

(ż2zȧ)u(Pz)g,u=0.
(3.10)

In fact, using (3.1), (3.7), and (3.8) we obtain

0=ddt(P(t)z(t))u(t),1xRu(t)|t=0=żu2zȧu(Pz)u̇,u=ż2zȧu,u+vemv(νu̇m(v)u(v)u̇m(v)νum(v))=ż2zȧu,u+vemv(32ȧmνum(v)u(v)(w(v)12ȧmu(v))νum(v))=ż2zȧu,uvemvȧmνum(v)u(v).
(3.11)

(We used the continuity of u and the Neumann condition emvνum(v)=0.) 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

u̇=αu+g+R(λ)(żu2zȧu(Pz)g).
(3.12)

With this in place we return to (3.6). The first term on the right hand side is

2Im(Pz)u̇,1xRu̇=2Imżu2zȧu,u̇=2Imżu2zȧu,αu+g+R(λ)(żu2zȧu(Pz)g)=2Imα(ż2zȧu,u)4zImȧu,R(λ)(żu2zȧu(Pz)g).

(We used here the simplifying assumption that g and u are real valued.)

As in (2.16) we conclude that

4zImȧu,R(λ)(żu+2zȧu+(Pz)g)=2ziȧu,[(R(λ)R(λ))](2zȧu+(Pz)g).

Now, as in (2.3), Ref. 8 [Theorem 4.20] shows that

(R(λ)R(λ))f=i2λk=1Kek(λ,x)f,ek(λ,),λR,fHR,
(3.13)

which means that (with z = λ2 and ek = ek(λ))

2ziȧu,[(R(λ)R(λ)](2zȧu+(Pz)g)=λk=1Kȧu,ek2λ2ȧu+(Pz)g,ek¯=2λ3k=1Kȧu,ek2λk=1Kȧu,ekek,(Pz)g.

The second term on the right hand side is now rewritten using (3.1) and the boundary conditions (3.9),

λk=1Kȧu,ekvemv(νemk(v)gm(v)νgm(v)ek(v))=λk=1Kȧu,ekvemv12ȧm(νemk(v)u(v)+3νum(v)ek(v)).

We conclude that

2Im(Pz)u̇,1xRu̇=2Imα(ż2zȧu,u)2λ3k=1Kȧu,ek22λk=1Kȧu,ekvemv14ȧm(3νum(v)ek(v)νemk(v)u(v)).
(3.14)

A similar analysis of the second term on the right hand side of (2.9) shows that

2Imvemvνu̇m(v)u̇¯m(v)=Imα2vemvȧmνum(v)u(v)2λ1k=1Kvemv14ȧm(νemk(v)u(v)3νum(v)ek(v))22λk=1Kȧu,ek¯vemv14ȧm(3νum(v)e¯k(v)νe¯mk(v)u(v)).
(3.15)

Inserting (3.14) and (3.15) into (3.6), using (3.11) and Imz̈=2λImλ̈ gives (1.4). □

Example 1.
Consider a connected graph with M bonds and K leads. Suppose that an embedded eigenvalue λ is simple and satisfies
λmπZ,m=K+1,,M+K.
(3.16)
Then
Imλ̈=k=1Kλȧu,ek(λ)2.
(3.17)

Proof.
um(x) = Cmsin(λx) where em and a lead are meeting at a vertex. Since the graph is connected, um(x) = Cmsin(λx) for K + 1 ≤ mM + K. Let nm=λmπ and let
emk(λ,x)=Amksin(λx)+Bmkcos(λx).
Then um(0) = um(ℓm) = 0 and
νum(0)=(1)nm+1νum(m),emk(λ,m)=(1)nmemk(λ,0).
We can use this and (3.2) to reduce Fk in (1.4) to
Fk=λȧu,ek(λ)+λ1m=K+1M+K34ȧm(νum(0)ek(λ,0)¯+νum(m)ek(λ,m)¯)=λȧu,ek(λ).
Theorem 1 then gives (3.17). □

Example 2.
Let us consider a graph with M = 5, K = 2 and four vertices: see in Fig. 4. Let ℓm(0) = 1, 3 ≤ m ≤ 7. Then the sequence of embedded eigenvalues λ is given as S1S2 where
S1=πZ,S2=λ:tanλ+2tanλ2=0,λπ2Z.
If λS1, then (3.16) is satisfied. If λS2, however, we have (with v1 and v2 corresponding to x = 0 for e3, e6 and e4, e5 respectively, and v4 to x = 0 for e7)
u3(x)=Csin(λx),u4(x)=Csin(λx),u5(x)=Csin(λx),
u6(x)=Csin(λx),u7(x)=Csinλsinλ2sinλx12,
where C > 0 is the normalization constant. Note that
u(v3)=Csinλ0,u(v4)=Csinλ0.
So we do not have the simple formula (3.17) in this case.

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.

Proposition 4.1.

Suppose thatPsatisfies the assumptions of Theorem 2 and Ω1⋐Ω2⋐ℂ, where Ωjare open sets.

Then there exist constantsC1depending only on Ω2andL, andC2depending on Ω1, Ω2,RandLsuch that
Res(P)Ω2C1,1rRR(λ)1rRL2L2C2ζRes(P)Ω2λζ1,λΩ1,
(4.1)
where the elements of Res(P) are included according to their multiplicities.

Proof.
Let R0(λ):k=1KLcomp2(ek)Hloc2H0,loc1(ek) be defined as the diagonal operator acting on each component as R00(λ), the Dirichlet resolvent on Lcomp2([0,)) continued analytically to all of ℂ,
R00(λ)f(x)=0eiλ(x+y)eiλxy2iλf(y)dy.
To describe 1rRR(λ)1rR we follow the general argument of Ref. 15 (see also Ref. 8 [Secs. 4.2 and 4.3]). For that we choose χjCc, j = 0, …, 3 to be equal to 1 on all edges and to satisfy
χj|ekCc([0,2R)),χ0|ek(x)=1,xR,χj|ek(x)=1,xsuppχj1|ek,
for k = 1, …, K. For λ0 with Im λ0 > 0, we define
Q(λ,λ0)(1χ0)R0(λ)(1χ1)+χ2R(λ0)χ1,Q(λ,λ0):Lcomp2Dloc(P).
Then
(Pλ2)Q(λ,λ0)=I+K(λ,λ0),
K0(λ,λ0)[P,χ0]R0(λ)(1χ1)+(λ02λ2)χ2R(λ0)χ1+[P,χ2]R(λ0)χ1.
We now choose λ0 = eπi/4μ, μ ≫ 1. Then
I+K0(λ0,λ0)andI+K0(λ0,λ0)χ3are invertible onL2,
(4.2)
K(λ, λ0) χ3 is compact, and
R(λ)=Q(λ,λ0)(I+K0(λ,λ0)χ3)1(IK0(λ,λ0)(1χ3)),
(4.3)
where λ ↦ (I + K0(λ, λ0) χ3)−1 is a meromorphic family of operators. We now put
K(λ,λ0)K0(λ,λ0)χ3
and conclude that
1rRR(λ)1rR=1rRQ(λ,λ0)χ3(I+K(λ,λ0))11rR,
(4.4)
and the set of resonances is given by the poles of (I + K(λ, λ0))−1. (See Ref. 8 [Sec. 4.2] and in particular Ref. 8 (4.2.19).)
We now claim that K(λ, λ0) is of trace class for λ ∈ ℂ and that for any compact subset Ω⋐ℂ there exists a constant C3 depending only on Ω, L, and λ0 such that
K(λ,λ0)trC3.
(4.5)
To see this, let P˜ be the operator of H3R where we put, say the Neumann boundary condition at 3R on each infinite lead. Let P˜min,P˜max be the same operators but on metric graphs where all the length jL, K + 1 ≤ jK + M were replaced by minminL and maxmaxL, respectively. These operators have discrete spectra and the ordered eigenvalues of these operators satisfy
λp(P˜max)λp(P˜)λp(P˜min).
(4.6)
This is a consequence of the following lemma.

Lemma 4.2.
Suppose that the unbounded operatorP˜k(t):HρHρ,ρ > 0, with edge length given by
k(t)=ρ,1kK,m(t)=eδmktm,K+1mM+K,
and
D(P˜k(t))={u:umH2([0,m(t)]),um(v)=u(v),veme,emvνum(v)=0}.
If 0 = μ0(t) ≤ μ1(t) ≤ μ2(t)⋯, is the ordered sequence of eigenvalues ofPk(t), thenμp(t) is a non-decreasing function oft.

Proof.

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, un(t)D(P(s)), such that μn(s) = μ, and for small ts, P(t) un(t) = μn(t) un(t), and {un(t)}n=1N is an orthonormal setting spanning 1P(t)μεL2, for ε > 0 small enough. The lemma follows from showing that ∂tμn(s) ≥ 0 for any n.

Without loss of generality we can assume that s = 0. We can then use the same calculation as in (3.11) with z = μn(0), am(t) = δkmt, and u = un(0). That gives
μp(0)=2μp(0)u,uL2(ek)+vekνuk(v)uk(v).
Since uk(x)=asinμpx+bcosμpx, for some a, b ∈ ℝ, a calculation shows that
μp(0)=μp(0)k(a2+b2)0,
completing the proof.□

Inequality (4.6) follows from the lemma as we can change the length of the edges in succession. The Weyl law for P˜ (see Ref. 1), and the fact that P˜χ3=Pχ3 (where χ3 denotes the multiplication operator), now shows that for any operator A:L2D(P),

χ3Aχ3trC4Pχ3Aχ3+C4χ3Aχ3,

where the constant C4 depends only on L. From this we deduce (4.5) and =L2L2. For instance,

[P,χ2]R(λ0)χ1trC4P[P,χ2]R(λ0)χ1+C4[P,χ2]R(λ0)χ1=C4[P,[P,χ2]]R(λ0)χ1+C4(1+λ02)[P,χ2]R(λ0)χ1C5.

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 D(P). 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 Ω3={λ:λλ0<R, where R is large enough so that Ω2 ⊂ Ω3. It follows that for a constant C3 depending only on Ω3 and L, and hence only on Ω2,

we have

det(I+K(λ,λ0))eC3.
(4.7)

(For basic facts about determinants see, for instance, Ref. 8 [Sec. B.5].) Writing

(I+K(λ0,λ0))1=(I(I+K(λ0,λ0))1K(λ0,λ0)),

we obtain

det(I+K(λ0,λ0))1=det(I+K(λ0,λ0)1exp(I+K(λ0,λ0))1K(λ0,λ0)treC4,

that is

det(I+K(λ0,λ0))eC4,
(4.8)

where C4 depends only on λ0 and L. 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

det(I+K(λ0,λ))=eg(λ)ζRes(P)Ω3(λζ),

where g(λ) is holomorphic in Ω3. From the upper bound (4.7) and the lower bound (4.8) we conclude that g(λ)C5 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

det(I+K(λ0,λ))eC6ζRes(P)Ω2λζ,λΩ1.

To deduce the second bound in (4.1) from this we use the inequality

(I+A)1det(I+A)det(I+A)

which gives

1rRR(λ)1rR=1rRQ(λ,λ0)χ3(I+K(λ,λ0))11rR1rRQ(λ,λ0)χ3det(I+|K(λ,λ0))det(I+K(λ,λ0))1C7eK(λ,λ0)trζRes(P)Ω2λζ1,

for λ ∈ Ω1 and C7 depending only on Ωj’s L, 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),

(P(t)λ02)1C((t0,t0);L(L2,L2)),Imλ0>0.
(4.9)

That is the only property used in the proof of

Proposition 4.3.
LetP(t) be the family of unbounded operators onL2(of a fixed metric graph) defined by (3.3). LetR(λ, t) be the resolvent ofP(t) meromorphically continued to. Suppose thatγis a smooth Jordan curve such thatR(λ, t) has no poles onγfort<t0. Then forχjCc,j = 1, 2,
γχ1R(ζ,t)χ2dζC((t0,t0);L(L2,L2)).
(4.10)
In particular, ifλ0is a simple pole ofR(λ, 0) then there exist smooth familiestλ(t) andtu(t)Dloc(P(t))such thatλ(0) = λ0,λ(t) ∈ Res(P(t)), andu(t) is a resonant state ofP(t) corresponding toλ(t).

Proof.

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 fL2 and define u(t) ≔ R(λ0, t) fL2. Formally, u̇tu(t) satisfies (3.8) with ż=0 and z=λ02. 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 Gm(e2a(t)x2λ02)gm(t). 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

Proof of Theorem 2.
We proceed by contradiction by assuming that, for 0 < δρ ≪ 1 to be chosen,
Res(P)(Ω(ρ,δ)+D(0,δ))=,Ω(ρ,δ)[λ0ρ,λ0+ρ]i[0,δ]
does not contain any resonances. Choosing pre-compact open sets, independent of ε, ρ, and δ, Ω(ρ, δ) + D(0, δ)⋐Ω1⋐Ω2 we apply Proposition 4.1 to see that for
1rRR(λ)1rRC2δC1,λΩ(ρ,δ).
(4.11)
On the other hand, the resolvent estimate in the physical half-plane Im λ > 0 and the fact that λ0I⋐(0, ∞) give
1rRR(λ)1rRC3/Imλ,Imλ>0,ReλReλ0<ρ.
(4.12)

To derive a contradiction we use the following simple lemma:

Lemma 4.4.
Suppose thatf(z) is holomorphic in a neighbourhood of Ω ≔ [ − ρ, ρ] + i[ − δ, δ+],δ± > 0. Suppose that, forM > 1,M± > 0, and 0 < δ+δ < 1,
f(z)M±,Imz=±δ+,Rezρ,f(z)M,zΩ,
(4.13)
and thatρ2>(1+2logM)δ2. Then
f(0)eM+θM1θ,θδδ++δ.
(4.14)

Proof.
We consider the following subharmonic function defined in a neighbourhood of Ω. To define it we put m± = logM±, m = logM > 0, z = x + iy, and
u(z)logf(x+iy)δm++δ+m+y(m+m)δ++δKx2+Ky2,
where K2m/(ρ2δ2). Then for Im z = ± δ±, Rezρ, u(z)δ2K1 since we assumed ρ2>(1+2m)δ2). When Rez=ρ then u(z)2mK(ρ2δ2)0. The maximum principle for subharmonic functions shows that logf(0)θm+(1θ)m1 and that concludes the proof.□

We apply this lemma to f(z) ≔ 〈1rRR(z + λ0)1rRφ, ψ〉, φ, ψL2, with M+ = C3/δ+, M = M = C2δC1. If we show that

f(0)1εφψ,
(4.15)

we obtain a contradiction to (1.6) by putting φ=(Pλ02)u and ψ = u and using the support property of u (the outgoing resolvent is the right inverse of Pλ02 on compactly supported function):

1=R(λ0)(Pλ02)u,u=1rRR(λ0)1rR(Pλ02)u,u1εε1.

For γ < 1 choose γ < γ1 < γ2 < γ3 < 1 and put

ρ=εγ1,δ=εγ2,δ+=εγ3.

Then (4.14) implies (4.15) and that completes the proof.□

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 

Proposition 4.5.
Suppose thatPsatisfies the assumptions of Theorem 2 and letR > 0, δ > 0. There exists a constantC0depending only onR, δ, andLsuch that for any 0 < ε < δ/2,
D(λ0,ε)Res(P),λ0>δuHRDP,u=1,(Pλ02)uC0ε(λ0+ε).
(4.16)

Proof.
Suppose that λ is a resonance of P with λλ0<ε and let v be the corresponding resonant state. Then in each infinite lead, vm(x) = ameiλx, 1 ≤ mK. As in (3.5),
Im(λ2)vH02=Imm=1Kxvm(0)v̄m(0)=Imm=1Kiλam2=Reλm=1Kam2.
Since Reλ ≠ 0, it follows that m=1Kam2=2ImλvH022εvH02.
Suppose r < R/2 and χCc([0,2)) is equal to 1 on [0, 1]. We then define u˜HRDP by
u˜m(x)χ(x/r)vm(x),1mKvm(x)K+1mK+M.
Now,
u˜2=vH02+m=1Kam2Re2Imλxχ(x/r)2dx=vH02(1+O(εre2εr)),
and hence,
(Pλ02)u˜2=λ2λ022u˜2+[P,χ(/r)]u˜2(2ε(λ0+ε))2u˜2+Cm=1Kam2(r2+(λ0+ε)2)e2εrCr,δε2(λ0+ε)2vH02.
We conclude that we can take uũ/ũ as the quasimode.□

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.

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