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–Kuchment^{1} 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 Letter^{11} 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, *e _{m}*, has two distinct vertices as its boundary (a non-restrictive no-loop condition) and we write

*v*∈

*e*for these two vertices

_{m}*v*. An infinite lead has one vertex. The set of (at most two) common vertices of

*e*and

_{m}*e*

_{ℓ}is denoted by

*e*∩

_{m}*e*

_{ℓ}and we denote by

*e*∋

_{m}*v*the set of all edges having

*v*as a vertex.

The finite edges are assigned length ℓ_{m}, *K* + 1 ≤ *m* ≤ *M* + *K* and we put ℓ_{k} = ∞, 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 *L*^{2} defined by $(Pu)m=\u2212\u2202x2um$ with

Here ∂_{ν} denotes the outward pointing normal at the boundary of *e _{v}*,

The space $Dloc(P)$ is defined by replacing *H*^{2} by $Hloc2$ when ℓ_{m} = ∞.

Quantum graphs with infinite leads fit neatly into the general abstract framework of *black box* scattering^{15} and hence we can quote general results Ref. 8 [Chapter 4] in spectral and scattering theory.

When *K* > 0 then the projection on the continuous spectrum of *P* is given in terms of generalized eigenfunctions *e ^{k}*(

*λ*), 1 ≤

*k*≤

*K*, which for

*λ*∉ Spec

_{pp}(

*P*) are characterized as follows:

The family $\lambda \u21a6ek(\lambda )\u2208Dloc(P)$ extends holomorphically to a neighbourhood of ℝ and that defines *e ^{k}*(

*λ*) for all

*λ*. We will in fact be interested in

*λ*∈ Spec

_{pp}(

*P*). The functions

*e*parametrize the continuous spectrum of

^{k}*P*—see Ref. 8 [Sec. 4.4] and (3.13) below.

We now consider a family of quantum graphs obtained by varying the lengths ℓ_{m}, *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* − *λ*^{2})^{−1} to ℂ as an operator $Lcomp2\u2192Lloc2$ (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*).

*Suppose that*

*λ*

^{2}> 0

*is a simple eigenvalue of*

*P*=

*P*(0)

*and*

*u*

*is the corresponding normalized eigenfunction. Then for*$t\u2264t0$

*there exists a smooth function*

*t*↦

*λ*(

*t*)

*such that*

*λ*(

*t*) ∈ Res(

*P*(

*t*))

*and*

*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 $Fk=\lambda \u3008a\u0307u,ek(\lambda )\u3009$—see Figures 1 and 2. The second example gives a graph and an eigenvalue for which the boundary terms in the formula for *F _{k}* are needed—see Fig. 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–Nenciu^{5} 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 *E*_{0} 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, ∂_{E}ℙ(*E*) = ∫_{𝒜}*e*(*E*, *a*) ⊗ *e*(*E*, *a*)^{∗}*dμ*(*a*), $a\u2208A$, this expression becomes

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*(*λ*_{0}, *ct*)∩Res(*P*(*t*)) ≠ 0̸ for some *c* and for $t\u2264t0$, where the constants *c* and *t*_{0} depend on *λ*_{0} and *P*(*t*). (Here and below $D(\lambda 0,r):={\lambda :\lambda \u2212\lambda 0<r}$.) 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 $P(t)=\u2212\u2202x2+tV(x)$, where $V\u2208Cc\u221e(R)$ and *t* → 0; infinitely many resonances for *t* ≠ 0^{22} disappear and *P*(0) has only one resonance at 0.) Also, the result is not uniform if we vary *λ*_{0} or the lengths of the edges.

The next theorem adapts the method of Tang–Zworski^{20} and Stefanov^{17} (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,* ℓ_{m}*, have the property that* $\u2113m\u2208L$*,* *K* + 1 ≤ *m* ≤ *M* + *K, where* $L$ *is a fixed compact subset of the open half-line.*

*Then for any*$L\u22d0(0,\u221e)$

*,*

*I*⋐(0, ∞)

*,*

*R*> 0

*, and*

*γ*< 1

*there exists*ε

_{0}> 0

*such that*

*implies*

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 ℓ_{j} = ℓ_{j}(*t*) ∈ *C*^{1}(ℝ), and that *λ*_{0} > 0 is an eigenvalue of *P*(0). Then for any *γ* < 1 there exists *t*_{0} such that for $t\u2264t0$,

To apply Theorem 2 we need to construct an approximate mode of *P*(*t*) using the eigenfunction of *P*(0). Thus, let *u*^{0} be a normalized eigenfunction of *P*(0) with eigenvalue *λ*_{0}; in particular $uk0\u22610$, 1 ≤ *k* ≤ *K*. Choose *χ _{j}* ∈

*C*

^{∞}(ℝ),

*j*= 1, 2, such that

*χ*≥ 0,

_{j}*χ*

_{0}+

*χ*

_{1}= 1,

*χ*(

_{j}*s*) = 1 near $j\u2212s<13$ and define $um\u2212(x)\u2254\chi 0(x/\u2113m)um0(x)$ and $um+(x)\u2254\chi 1(x/\u2113m)um0(x)$,

*u*

^{0}=

*u*

^{+}+

*u*

^{−}.

*P*(

*t*),

*u*=

*u*(

*t*) needed in (1.6),

*t*small enough $suppum\u2212\u2282[0,23)\u2113m(0)\u2282[0,\u2113m(t))$ and $suppum+\u2282(23,1]\u2113m(0)\u2282(\delta m(t),1]\u2113m(0)$. Hence the values of

*u*(

_{m}*t*) and ∂

_{ν}

*u*(

_{m}*t*) at the vertices are the same as those of $um0$ and $um(t)\u2208D(P(t))$. Also, since $(\u2212\u2202x2\u2212\lambda 02)um0=0$ and $\chi 0(k)=\u2212\chi 1(k)$, (and putting ℓ

_{m}= ℓ

_{m}(0))

*γ*and

*t*

_{0}) that (1.8) holds.□

*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*(*λ*_{0},*C*_{0}ε), where*C*_{0}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.By adapting Stefanov’s methods

^{17}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 *L*^{2}(*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 *E _{j}* → ∞.) The eigenvalues

*E*> 0 are

_{j}*embedded*in the continuous spectrum. In addition the resolvent

*R*(

*λ*) ≔ (

*P*−

*λ*

^{2})

^{−1}:

*L*

^{2}→

*L*

^{2}, Im

*λ*> 0, has a meromorphic continuation to

*λ*∈ ℂ as an operator $R(\lambda ):Cc\u221e(X)\u2192C\u221e(X)$. Its poles are called

*scattering resonances*. Under generic perturbation of the metric in

*X*

_{0}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) −

*λ*

^{2})

*u*= 0

*,*$uL2=1$

*.*

*Then there exists a smooth function*

*t*↦

*λ*(

*t*)

*,*$t<t0$

*, such that*

*λ*(0) =

*λ,*

*λ*(

*t*)

*is a scattering resonance of*

*P*(

*t*)

*and*

*where*

*e*(

*λ*,

*x*)

*is given in (2.3),*$f\u0307\u2254\u2202tf|t=0$

*and*

*L*

^{2}(∂

*X*)

*is defined using the metric induced by*

*g.*

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), (*X _{k}*,

*g*|

_{Xk}) ≃ ([

*a*, ∞) × ℝ/ℓ

_{k}_{k}ℤ,

*dr*

^{2}+

*e*

^{−2r}

*dθ*

^{2}, 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* = *λ*^{2} and to write 〈•, •〉 for the *L*^{2}(*X*, *d*vol_{g}) inner product and 〈•, •〉_{L2(∂X)} for the inner product on *L*^{2}(∂) with the measure induced by the metric *g*.

*P*(

*t*) −

*λ*

^{2})

^{−1}(see Proposition 4.3 below for a general argument). Let

*t*↦

*u*(

*t*),

*u*(0) =

*u*denote a smooth family of resonant states

*u*(

*t*) is

*outgoing*—see Ref. 8 [Sec. 4.4].

*P*(

*t*) and integration by parts for the zero mode in the cusp show that for

*u*=

*u*(

*t*) and

*P*=

*P*(

*t*),

*z*(

*t*) ≤ 0, see also (2.12) below) and since 1

_{r≥R}

*u*(0) = 0, we have, at

*t*= 0, $Im\u2009z\u0308=\u22122Im\u2009\u2202r(1r\u2265Ru\u0307)(R)1r\u2265Ru\u0307\xaf$. We would like to argue as in (2.8) but in reverse. However, as $u\u0307$ will not typically be in $D(P)$ we now obtain the boundary terms

*P*(

*t*) −

*z*(

*t*))

*u*(

*t*) = 0, ∂

_{ν}

*u*|

_{∂X}=

*γu*|

_{∂X}, we have (at

*t*= 0),

*u*=

*u*(0) is real valued. Choose $g\u2208C\u0304\u221e(X,R)$ (real valued, compactly supported, and smooth up to the boundary) such that $\u2202\nu g|\u2202X=\gamma \u0307u|\u2202X$. We claim that

_{r≤R}

*u*(0) =

*u*(0),

*λ*

^{2}=

*z*,

*λ*> 0, is well defined, outgoing (see (2.7)), and solves the boundary value problem (2.10) satisfied by $u\u0307$. Since the eigenvalue at

*z*is simple that means that $u\u0307\u2212v$ is a multiple of

*u*(see Ref. 8 [Theorem 4.18] though in this one dimensional case this is particularly simple). Hence

*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

*λ*=

*ik*,

*k*≫ 1, and using the fact that $Pu\xaf=Pu\u0304$, and the second from considering Im

*λ*≫ 1,

*z*=

*λ*

^{2}, and noting that $((P\u2212z)\u22121)\u2217=(P\u2212z\u0304)\u22121$.) 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 $\lambda =z>0$)

*R*(

*λ*) −

*R*(−

*λ*))

*u*= 0 we have now to use (2.3) to see that

## 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 $H2\u2254\u2a01m=1M+KH2([0,\u2113m])$. Then for *u*, *v* ∈ *H*^{2}, $(\u2202xku)m\u2254\u2202xkum$,

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 $P\u02dc(t)$ (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

Let $P\u02dc(t)$ be defined in $Lt2$ by $(P\u02dc(t)u)m=\u2212\u2202x2um$,

That is just the family of Neumann Laplace operators on the graph with the lengths *e*^{−aj(t)}ℓ_{j}.

On *L*^{2} we define a new family of operators: $P(t)\u2254U(t)P\u02dc(t)U(t)\u2217$. It is explicitly given by $[P(t)u]m=\u2212e2am(t)\u2202x2um$,

Using Proposition 4.3 from Sec. IV we see that for small *t* there exists a smooth family $t\u21a6u(t)\u2208Hloc2$ such that

We defined $HR$ by (1.5) and denote by 1_{x≤R} the orthogonal projection $L2\u2192HR$.

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

We recall that *e _{m}*, 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 $u\u0307$ at *t* = 0,

Hence,

We used here the fact that *u*(*v*) ≔ *u _{m}*(

*v*) does not depend on

*m*. The second condition can be formulated as $u\u0307m(v)=w(v)\u221212a\u0307m(v)u(v)$, where

*w*≔ ∂

_{t}(

*e*

^{a(t)/2}

*u*(

*t*))|

_{t=0}is continuous on the graph.

To find an expression for $u\u0307$ (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 $\u2211em\u220bv\u2202\nu um(v)=0$.) 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

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* = *λ*^{2} and *e ^{k}* =

*e*(

^{k}*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

Inserting (3.14) and (3.15) into (3.6), using (3.11) and $Im\u2009z\u0308=2\lambda Im\u2009\lambda \u0308$ gives (1.4). □

*M*bonds and

*K*leads. Suppose that an embedded eigenvalue

*λ*is simple and satisfies

*u*(

_{m}*x*) =

*C*sin(

_{m}*λx*) where

*e*and a lead are meeting at a vertex. Since the graph is connected,

_{m}*u*(

_{m}*x*) =

*C*sin(

_{m}*λx*) for

*K*+ 1 ≤

*m*≤

*M*+

*K*. Let $nm=\lambda \u2113m\pi $ and let

*u*(0) =

_{m}*u*(ℓ

_{m}_{m}) = 0 and

*F*in (1.4) to

_{k}*M*= 5,

*K*= 2 and four vertices: see in Fig. 4. Let ℓ

_{m}(0) = 1, 3 ≤

*m*≤ 7. Then the sequence of embedded eigenvalues

*λ*is given as

*S*

_{1}∪

*S*

_{2}where

*λ*∈

*S*

_{1}, then (3.16) is satisfied. If

*λ*∈

*S*

_{2}, however, we have (with

*v*

_{1}and

*v*

_{2}corresponding to

*x*= 0 for

*e*

_{3},

*e*

_{6}and

*e*

_{4},

*e*

_{5}respectively, and

*v*

_{4}to

*x*= 0 for

*e*

_{7})

*C*> 0 is the normalization constant. Note 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 Carleman^{3} 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–Pushnitski^{6} and Davies–Exner–Lipovský^{7} 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* Ω_{1}⋐Ω_{2}⋐ℂ*, where* Ω_{j} *are open sets.*

*Then there exist constants*

*C*

_{1}

*depending only on*Ω

_{2}

*and*$L$

*, and*

*C*

_{2}

*depending on*Ω

_{1}, Ω

_{2}

*,*

*R*

*and*$L$

*such that*

*where the elements of*Res(

*P*)

*are included according to their multiplicities.*

*analytically*to all of ℂ,

_{r≤R}

*R*(

*λ*)1

_{r≤R}we follow the general argument of Ref. 15 (see also Ref. 8 [Secs. 4.2 and 4.3]). For that we choose $\chi j\u2208Cc\u221e$,

*j*= 0, …, 3 to be equal to 1 on all edges and to satisfy

*k*= 1, …,

*K*. For

*λ*

_{0}with Im

*λ*

_{0}> 0, we define

*λ*

_{0}=

*e*

^{πi/4}

*μ*,

*μ*≫ 1. Then

*K*(

*λ*,

*λ*

_{0})

*χ*

_{3}is compact, and

*λ*↦ (

*I*+

*K*

_{0}(

*λ*,

*λ*

_{0})

*χ*

_{3})

^{−1}is a meromorphic family of operators. We now put

*I*+

*K*(

*λ*,

*λ*

_{0}))

^{−1}. (See Ref. 8 [Sec. 4.2] and in particular Ref. 8 (4.2.19).)

*K*(

*λ*,

*λ*

_{0}) is of trace class for

*λ*∈ ℂ and that for any compact subset Ω⋐ℂ there exists a constant

*C*

_{3}depending only on Ω, $L$, and

*λ*

_{0}such that

*R*on each infinite lead. Let $P\u02dcmin,P\u02dcmax$ be the same operators but on metric graphs where all the length $\u2113j\u2208L$,

*K*+ 1 ≤

*j*≤

*K*+

*M*were replaced by $\u2113min\u2254minL$ and $\u2113max\u2254maxL$, respectively. These operators have discrete spectra and the ordered eigenvalues of these operators satisfy

*Suppose that the unbounded operator*$P\u02dck(t):H\rho \u2192H\rho $

*,*

*ρ*> 0

*, with edge length given by*

*and*

*If*0 =

*μ*

_{0}(

*t*) ≤

*μ*

_{1}(

*t*) ≤

*μ*

_{2}(

*t*)⋯,

*is the ordered sequence of eigenvalues of*

*P*(

_{k}*t*)

*, then*

*μ*(

_{p}*t*)

*is a non-decreasing function of*

*t.*

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 *μ ^{n}*(

*t*) ∈ ℝ, $un(t)\u2208D(P(s))$, such that

*μ*(

^{n}*s*) =

*μ*, and for small

*t*−

*s*,

*P*(

*t*)

*u*(

^{n}*t*) =

*μ*(

^{n}*t*)

*u*(

^{n}*t*), and ${un(t)}n=1N$ is an orthonormal setting spanning $1P(t)\u2212\mu \u2264\epsilon L2$, for ε > 0 small enough. The lemma follows from showing that ∂

_{t}

*μ*(

^{n}*s*) ≥ 0 for any

*n*.

*s*= 0. We can then use the same calculation as in (3.11) with

*z*=

*μ*(0),

^{n}*a*(

_{m}*t*) =

*δ*, and

_{km}t*u*=

*u*(0). That gives

^{n}*a*,

*b*∈ ℝ, a calculation shows that

Inequality (4.6) follows from the lemma as we can change the length of the edges in succession. The Weyl law for $P\u02dc$ (see Ref. 1), and the fact that $P\u02dc\chi 3=P\chi 3$ (where *χ*_{3} denotes the multiplication operator), now shows that for any operator $A:L2\u2192D(P)$,

where the constant *C*_{4} depends only on $L$. From this we deduce (4.5) and $\u2022=\u2022L2\u2192L2$. For instance,

Here we used the facts that *χ*_{2} ≡ 1 on the support of *χ*_{1}, hence [*P*, *χ*_{2}]*χ*_{1} = 0, and that [*P*, [*P*, *χ*_{2}]][*P*, *χ*_{2}] are second and first order operators, respectively, and that *R*(*λ*_{0}) maps *L*^{2} to $D(P)$. The other terms in *K*(*λ*, *λ*_{0}) 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 $\Omega 3={\lambda :\lambda \u2212\lambda 0<R$, where *R* is large enough so that Ω_{2} ⊂ Ω_{3}. It follows that for a constant *C*_{3} depending only on Ω_{3} and $L$, and hence only on Ω_{2},

we have

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

we obtain

that is

where *C*_{4} depends only on *λ*_{0} 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*, *λ*_{0})) in Ω_{3}. That proves the first bound in (4.1).

We can write

where *g*(*λ*) is holomorphic in Ω_{3}. From the upper bound (4.7) and the lower bound (4.8) we conclude that $g(\lambda )\u2264C5$ in a smaller disc containing Ω_{2}, with *C*_{5} 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* ∈ Ω_{1} and *C*_{7} depending only on Ω_{j}’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),

That is the only property used in the proof of

*Let*

*P*(

*t*)

*be the family of unbounded operators on*

*L*

^{2}

*(of a fixed metric graph) defined by (3.3). Let*

*R*(

*λ*,

*t*)

*be the resolvent of*

*P*(

*t*)

*meromorphically continued to*ℂ

*. Suppose that*

*γ*

*is a smooth Jordan curve such that*

*R*(

*λ*,

*t*)

*has no poles on*

*γ*

*for*$t<t0$

*. Then for*$\chi j\u2208Cc\u221e$

*,*

*j*= 1, 2

*,*

*In particular, if*

*λ*

_{0}

*is a simple pole of*

*R*(

*λ*, 0)

*then there exist smooth families*

*t*↦

*λ*(

*t*)

*and*$t\u21a6u(t)\u2208Dloc(P(t))$

*such that*

*λ*(0) =

*λ*

_{0}

*,*

*λ*(

*t*) ∈ Res(

*P*(

*t*))

*, and*

*u*(

*t*)

*is a resonant state of*

*P*(

*t*)

*corresponding to*

*λ*(

*t*)

*.*

The proof of (4.10) under the condition (4.9) follows from (4.4) and the definitions of *Q*(*λ*, *λ*_{0}) and *K*(*λ*, *λ*_{0}). 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* ∈ *L*^{2} and define *u*(*t*) ≔ *R*(*λ*_{0}, *t*) *f* ∈ *L*^{2}. Formally, $u\u0307\u2254\u2202tu(t)$ satisfies (3.8) with $z\u0307=0$ and $z=\lambda 02$. We can find a smooth family *g*(*t*) ∈ *L*^{2} satisfying (3.9) with *u* = *u*(*t*). We then have ∂_{t}*u*(*t*) = *g* + *R*(*λ*_{0}, *t*)(−2∂_{t}*a*(*t*) *u*(*t*) − *G*(*t*)), where $Gm\u2254(\u2212e\u22122a(t)\u2202x2\u2212\lambda 02)gm(t)$. By considering difference quotients, a similar argument shows that *u*(*t*) ∈ *L*^{2} is differentiable. The argument can be iterated showing that *u*(*t*) ∈ *C*^{∞}((−*t*_{0}, *t*_{0}), *L*^{2}) and that proves (4.9).□

We now give

*δ*≪

*ρ*≪ 1 to be chosen,

*ρ*, and

*δ*, Ω(

*ρ*,

*δ*) +

*D*(0,

*δ*)⋐Ω

_{1}⋐Ω

_{2}we apply Proposition 4.1 to see that for

*λ*> 0 and the fact that

*λ*

_{0}∈

*I*⋐(0, ∞) give

To derive a contradiction we use the following simple lemma:

*Suppose that*

*f*(

*z*)

*is holomorphic in a neighbourhood of*Ω ≔ [ −

*ρ*,

*ρ*] +

*i*[ −

*δ*

_{−},

*δ*

_{+}]

*,*

*δ*

_{±}> 0

*. Suppose that, for*

*M*> 1

*,*

*M*

_{±}> 0

*, and*0 <

*δ*

_{+}≤

*δ*

_{−}< 1

*,*

*and that*$\rho 2>(1+2logM)\delta \u22122$

*. Then*

*m*

_{±}= log

*M*

_{±},

*m*= log

*M*> 0,

*z*=

*x*+

*iy*, and

*z*= ±

*δ*

_{±}, $Rez\u2264\rho $, $u(z)\u2264\delta \u22122K\u22641$ since we assumed $\rho 2>(1+2m)\delta \u22122$). When $Rez=\rho $ then $u(z)\u22642m\u2212K(\rho 2\u2212\delta \u22122)\u22640$. The maximum principle for subharmonic functions shows that $logf(0)\u2212\theta m+\u2212(1\u2212\theta )m\u2212\u22641$ and that concludes the proof.□

We apply this lemma to *f*(*z*) ≔ 〈1_{r≤R}*R*(*z* + *λ*_{0})1_{r≤R}*φ*, *ψ*〉, *φ*, *ψ* ∈ *L*^{2}, with *M*_{+} = *C*_{3}/*δ*_{+}, *M* = *M*_{−} = *C*_{2}*δ*^{−C1}. If we show that

we obtain a contradiction to (1.6) by putting $\phi =(P\u2212\lambda 02)u$ and *ψ* = *u* and using the support property of *u* (the outgoing resolvent is the right inverse of $P\u2212\lambda 02$ on compactly supported function):

For *γ* < 1 choose *γ* < *γ*_{1} < *γ*_{2} < *γ*_{3} < 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}

*Suppose that*

*P*

*satisfies the assumptions of Theorem 2 and let*

*R*> 0,

*δ*> 0

*. There exists a constant*

*C*

_{0}

*depending only on*

*R*,

*δ, and*$L$

*such that for any*0 < ε <

*δ*/2

*,*

*λ*is a resonance of

*P*with $\lambda \u2212\lambda 0<\epsilon $ and let

*v*be the corresponding resonant state. Then in each infinite lead,

*v*(

_{m}*x*) =

*a*

_{m}e^{iλx}, 1 ≤

*m*≤

*K*. As in (3.5),

*λ*≠ 0, it follows that $\u2211m=1Kam2=2Im\u2009\lambda vH02\u22642\epsilon vH02$.

*r*<

*R*/2 and $\chi \u2208Cc\u221e([0,2))$ is equal to 1 on [0, 1]. We then define $u\u02dc\u2208HR\u2229DP$ by

## 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.