Resonances near thresholds in slightly Twisted Waveguides

# Resonances near thresholds in slightly Twisted Waveguides

Vincent Bruneau Université de Bordeaux, IMB, UMR 5251, 33405 TALENCE cedex, France Pablo Miranda Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173. Santiago, Chile.  and  Nicolas Popoff Université de Bordeaux, IMB, UMR 5251, 33405 TALENCE cedex, France
###### Abstract.

We consider the Dirichlet Laplacian in a straight three dimensional waveguide with non-rotationally invariant cross section, perturbed by a twisting of small amplitude. It is well known that such a perturbation does not create eigenvalues below the essential spectrum. However, around the bottom of the spectrum, we provide a meromorphic extension of the weighted resolvent of the perturbed operator, and show the existence of exactly one resonance near this point. Moreover, we obtain the asymptotic behavior of this resonance as the size of the twisting goes to 0. We also extend the analysis to the upper eigenvalues of the transversal problem, showing that the number of resonances is bounded by the multiplicity of the eigenvalue and obtaining the corresponding asymptotic behavior.

AMS 2000 Mathematics Subject Classification: 35J10, 81Q10, 35P20.
Keywords: Twisted waveguide, Dirichlet Laplacian, Resonances near thresholds.

## 1. Introduction

Let be a bounded domain in with Lipschitz boundary. Set and . Define as the Laplacian in with Dirichlet boundary conditions. Consider (the Laplacian in with Dirichlet boundary conditions). Since is bounded, the spectrum of the operator is a discrete sequence of values converging to infinity, denoted by . Then, the spectrum of is given by

 σ(H0)=∞⋃n=1[λn,∞)=[λ1,∞),

and is purely absolutely continuous.

Geometric deformations of such a straight waveguide have been widely studied in recent years, and have numerous applications in quantum transport in nanotubes. The spectrum of the Dirichlet Laplacian in waveguides provides information about the quantum transport of spinless particles with hardwall boundary conditions. In particular, the existence of eigenvalues describes the occurrence of bound states corresponding to trapped trajectories created by the geometric deformations. For a review we refer to [11], where bending against twisting is discussed, and to [8] for a general differential approach. Without being exhaustive we recall some well known situations: a local bending of the waveguide creates eigenvalues below the essential spectrum, as also do a local enlarging of its width ([5, 8]). On the contrary, it has been proved, under general assumptions, that a twisting of the waveguide does not lower the spectrum ([7]), in particular a twisting going to 0 at infinity will not modify the spectrum ([8]). In such a situation it is natural to introduce the notion of resonance and to analyze the effect of the twisting on the resonances near the real axis. There already exist studies of resonances in waveguides: resonances in a thin curved waveguide ([6, 12]), or more recently in a straight waveguide with an electric potential, perturbed by a twisting ([10]). In these both cases, however, the resonances appear as perturbations of embedded eigenvalues of a reference operator, and follow the Fermi golden rule (see [9] for references and for an overview on such resonances). As we will see, in our case the origin of the resonances will be rather due to the presence of thresholds appearing as branch points created by a 1d Laplacian. Our analysis will be close to the studies near 0 of the 1d Laplacian (see for instance [13, 4] where, even if resonances are not discussed, the “threshold” behavior appears). A similar phenomena of threshold resonances was already studied for a magnetic Hamiltonian in [1], where the thresholds are eigenvalues of infinite multiplicity of some transversal problem.

In this article we will consider a small twisting of the waveguide: Let be a non-zero function of class with exponential decay i.e., for some (this hypothesis can be relaxed, see Remark 2), satisfies

 (1) ε(x3)=O(e−α⟨x3⟩),ε′(x3)=O(e−α⟨x3⟩),

where . Then, for we define as the waveguide obtained by twisting with , where , i.e., we define

 Ωδ:={(rθδ(x3)(xt),x3),(xt,x3)∈Ω)},

where is the rotation of angle in . Set

 W(δ):=−δ∂φε∂3−δ∂3ε∂φ−δ2ε2∂2φ=−2δε∂φ∂3−δε′∂φ−δ2ε2∂2φ,

with the notation for . Then, it is standard (see for instance [8, Section 2]) that the Dirichlet Laplacian in is unitarily equivalent to the operator

 H(δ):=H0+W(δ),

defined in with a Dirichlet boundary condition. Since the perturbation is a second order differential operator, is not a relatively compact perturbation of . However the resolvent difference is compact ([3, Section 4.1]), and therefore and have the same essential spectrum. Moreover, the spectrum of coincide with , see [7].

In this article we will show that around there exists, for small enough, a meromorphic extension of the weighted resolvent of with respect to the variable , where is the spectral parameter. In other words, the resolvent , first defined for in , admits a meromorphic extension on a weighted space (space of functions with exponential decay along the tube), for values in a neighborhood of in a 2-sheeted Riemann surface. We will identify the resonances around with the poles of this meromorphic extension in the parameter . We will prove in Theorem 5 that in a neighborhood independent of , there is exactly one pole , whose behavior as is explicit:

 (2) k(δ)=−iμδ2+O(δ3),

where is given by (15) below, and moreover, is on the imaginary axis.

The fact that is on the negative imaginary axis means that in the spectral variable the resonance is on the second sheet of the 2-sheeted Riemann surface, far from the real axis (it is sometimes called an antibound state [14]). In particular such a resonance can not be detected using dilations (a dilation of angle larger than would be needed) and is completely different in nature from those created by perturbations of embedded eigenvalues. For this reason we define resonances as the poles of weighted resolvents, assuming that is exponentially decaying. However, a difficulty comes from the non relatively compactness of the perturbation This problem will be overcome exploiting the smallness of the perturbation and the locality of our problem.

Our analysis provides an analogous result for higher thresholds, in Section 4: Around each there are at most resonances (for all small enough), where is the multiplicity of as eigenvalue of . Moreover, under an additional assumption, each one of these resonances have an asymptotic behavior of the form (2), where the constant is an eigenvalue of a explicit matrix (not necessarily Hermitian). Although Theorem 6 may be viewed as a generalization of Theorem 5, we preferred to push forward the proof for the first threshold for the following reasons: it is easier to follow and contain all the main ingredients needed for the proof in the upper thresholds, the eigenvalues of are generically simple as we know the first eigenvalue is.

###### Remark 1.

Independent of the size of the perturbation , a more global definition of resonances would be possible by showing that a generalized determinant (as in [2] or in [15, Definition 4.3]) is well defined on and admits an analytic extension. Then the resonances would be defined as the zeros of this determinant on a infinite-sheeted Riemann surface (as in [1, Definitions 1-2]).

## 2. Preliminary decomposition of the free resolvent

Let us describe the singularities of the free resolvent. Setting , we have that

 (3) H0−λ1=(−Δω−λ1)⊗Ix3+Ixt⊗D23.

For , define

 R0(k):=(H0−λ1−k2)−1,

and similarly for . If for , is the orthogonal projection onto ker, using (3) for , we have that

 (4) R0(k)=(H0−λ1−k2)−1=∑q=1πq⊗(D23+(λq−λ1)−k2)−1.

The integral kernel of is explicitly given by

 (5) i2keik|x3−x′3|.

Let be an exponential weight of the form , for . Also, for and set . Then, as in [1, Lemma 1] it can be seen that the operator valued-function , initially defined on , has a meromorphic extension in for any , with a unique pole, of multiplicity one, at . More precisely,

 (6) ηR0(k)η=1kπ1⊗α0+A0(k),

where is the rank one operator and : is the analytic operator-valued function

 (7) A0(k):=π1⊗r1(k)+∑q=2πq⊗η(D23+(λq−λ1)−k2)−1η,

with being the operator in with integral kernel given by

 iη(x3)(eik|x3−x′3|−1)2kη(x′3).

Clearly, for , the family of operators is uniformly bounded on .

###### Remark 2.

Note that the condition on the function , enters here in order to have analytic properties in the ball , . This assumption can be relaxed to , but the results will be restricted to with .

In order to define and study the resonances, we will consider a suitable meromorphic extension of , using the identity

 (8) ηR(k)η=ηR0(k)η(Id+η−1W(δ)R0(k)η)−1.

Since has no eigenvalue below (see [7]), the above relation is initially well defined and analytic for . It is necessary then to understand under which conditions this formula can be used to define such an extension. Since we can not apply directly the meromorphic Fredholm theory ( is not -compact), we will need to show explicitly that is meromorphic in some region around zero.

Let be such that , (then ), and define

 (9) Φδ:=−i2((∂φψ1⊗η−1ε′)+δ(∂2φψ1⊗η−1ε2)).
###### Lemma 3.

Let . There exists such that for any and

 η−1W(δ)R0(k)η=δkK0+δT(δ,k),

where is the rank one operator

 (10) K0:=|Φδ⟩⟨ψ1⊗η|,

and : is an analytic operator-valued function. Moreover,

 (11) sup0<δ≤δ0,k∈B(0,r)||T(δ,k)||<∞.
###### Proof.

Thanks to (6),

 (12) η−1W(δ)R0(k)η=1kη−1W(δ)η−1(π1⊗α0)+η−1W(δ)η−1A0(k).

Since the range of the operator is spanned by constant functions, we have , and therefore

 η−1W(δ)η−1(π1⊗α0)=i2|η−1(−δε′∂φ−δ2ε2∂2φ)η−1(ψ1⊗η)⟩⟨ψ1⊗η|=δ|Φδ⟩⟨ψ1⊗η|=δK0.

We now treat the last term of (12): Setting we immediately get

 T(δ,k)= −2(∂φπ1⊗η−1ε∂3η−1r1(k)+∑q≥2∂φπq⊗η−1ε∂3(D23+(λq−λ1)−k2)−1η) −δ(∂2φπ1⊗η−1ε2η−1r1(k)+∑q≥2∂2φπq⊗η−1ε2(D23+(λq−λ1)−k2)−1η).

It is clear that the two last terms are analytic and uniformly bounded in . For the first one, we note that the kernel of is , and therefore admits an analytic expansion which is uniformly bounded. The same arguments run for .

## 3. Meromorphic extension of the resolvent and study of the resonance

###### Proposition 4.

Let be a compact neighborhood of zero. With the notation of Lemma 3, for sufficiently small, let us introduce the functions and

 (13) wδ(k)=δ⟨~Φδ|ψ1⊗η⟩.

Then:

1. There exists such that for any , ,

 (14) wδ(k)=iμδ2+O(δ3)+δ2kgδ(k)

where

 (15) μ:=12∑q≥2(λq−λ1)⟨∂φψ1|πq∂φψ1⟩⟨ε|(D23+λq−λ1)−1ε⟩

is a positive constant, and is an analytic function in satisfying

 supδ∈(0,δ0)supk∈D|gδ(k)|<+∞.
2. When , there holds .

###### Proof.

We use the Taylor expansion and Lemma 3 to see that

 (16) (Id+δT(δ,k))−1=Id−δT(δ,0)+δkGδ(k)+O(δ2),

where is holomorphic operator-valued function that is uniformly bounded for and small.

By definition of , we have:

 ⟨Φδ|ψ1⊗η⟩=−i2(⟨∂φψ1|ψ1⟩L2(ω)⟨η−1ε′|η⟩L2(R)+δ⟨∂2φψ1|ψ1⟩L2(ω)⟨η−1ε2|η⟩L2(R)).

The first term is zero because tends to zero at infinity. Using integration by part, since satisfies a Dirichlet boundary condition, we deduce

 ⟨Φδ|ψ1⊗η⟩=δi2∥∂φψ1∥2∥ε∥2.

Noticing that , from (16) we get

 (17) wδ(k)=δ2i2∥∂φψ1∥2∥ε∥2−δ2⟨T(δ,0)Φδ|ψ1⊗η⟩+δ2kgδ(k)+O(δ3),

where is holomorphic and uniformly bounded for and small.

We now compute . First recall that . Next, note that since ,

 π1∂φψ1=0,

and therefore, using the definition of in (9), we get

 (π1⊗r1(0))Φδ=−δi2π1∂2φψ1⊗r1(0)η−1ε2,

which in turn implies that

 ⟨(δ−1η−1W(δ)η−1)(π1⊗r1(0))Φδ|ψ1⊗η⟩=O(δ).

In consequence, having in mind (9) again, we deduce

 (18) ⟨T(δ,0)Φδ,ψ1⊗η⟩
 = ⟨η−1(−2ε∂φ∂3−ε′∂φ−δε2∂2φ)η−1(∑q≥2πq⊗η(D23+(λq−λ1))−1η)Φδ|ψ1⊗η⟩+O(δ) = i2⟨η−1(2ε∂φ∂3+ε′∂φ)(∑q≥2πq⊗(D23+(λq−λ1))−1)∂φψ1⊗ε′|ψ1⊗η⟩+O(δ).

We compute the main term of the last expression using integration by parts, both in the and the variables:

 (19) ⟨η−1(2ε∂φ∂3+ε′∂φ)(∑q≥2πq⊗(D23+(λq−λ1))−1)∂φψ1⊗ε′|ψ1⊗η⟩=∑q≥2⟨∂φψ1|πq∂φψ1⟩×⟨ε′|(D23+λq−λ1)−1ε′⟩.

Now, we notice that

 ⟨ε′|(D23+λq−λ1)−1ε′⟩ =⟨ε|(D23+λq−λ1)−1D23ε⟩ =∥ε∥2−(λq−λ1)⟨ε|(D23+λq−λ1)−1ε⟩.

In addition, since and , we have that

 (20) ∑q≥2⟨∂φψ1|πq∂φψ1⟩=∥∂φψ1∥2.

Then, from (18) and (19) we get

 (21) ⟨T(δ,0)Φδ,ψ1⊗η⟩=i2∥ε∥2∥∂φψ1∥2−i2∑q≥2(λq−λ1)⟨∂φψ1|πq∂φψ1⟩⟨ε|(D23+λq−λ1)−1ε⟩+O(δ).

Putting together (17) and (21), we deduce (14). Moreover, is clearly non-negative, and from (20), there exists such that . Since is a positive operator, we get .

Let us prove ii. For all , has a real integral kernel, see (7). Therefore if is real valued, so is . In consequence, since has values in , so is , and we deduce that has values in as well.

###### Theorem 5.

Let be a non-zero -function satisfying (1) and be a compact neighborhood of zero. Then, for sufficiently small, , initially defined in , admits a meromorphic operator-valued extension on , whose operator-values act from into . This function has exactly one pole in , called a resonance of , and it is of multiplicity one. Moreover, we have the asymptotic expansion

 k(δ)=−iμδ2+O(δ3),

with given by (15) and .

###### Proof.

Consider the identity (8), and note that from Lemma 3 for and   sufficiently small we can write

 (22) (Id+η−1W(δ)R0(k)η)=(Id+δT(δ,k))(Id+δk(Id+δT(δ,k))−1K0).

For let us set

 K:=δk(Id+δT(δ,k))−1K0=δk|~Φδ><ψ1⊗η|,

which is a rank one operator. Then, we need to study the inverse of .

Let us consider , the projection onto into the direction and , the projection onto into the direction normal to . We can easily see that

 (Id+K)Π⊥δ=Π⊥δand(Id+K)Πδ=(1+δk⟨~Φδ|ψ1⊗η⟩)Πδ=k+wδ(k)kΠδ.

Therefore, is invertible if and only if , and

 (23) (Id+K)−1=Π⊥δ+kk+wδ(k)Πδ.

Let us consider the equation . Using (14), for all , for small enough, the equation has no solution for and . We then apply Rouché Theorem inside the ball : consider the analytic functions and . The function has exactly one root, and on the circle , using again (14), there holds for small enough. Thus, we deduce that the equation has exactly one solution in , for each fixed small enough. In consequence, putting together (8), (6), (22) and (23) we have that for all

By the definition of , we have that and then:

 ηR(k)η=1k+wδ(k)(π1⊗α0)Πδ(Id+δT(δ,k))−1
 +kk+wδ(k)A0(k)Πδ(Id+δT(δ,k))−1+A0(k)Π⊥δ(Id+δT(δ,k))−1.

Therefore, for sufficiently small, admits a meromorphic extension to , where the pole is giving by the solution of .

Using (14), the asymptotic expansion of follows immediately. Further, the multiplicity of this resonance is the rank of the residue of , which coincides with the rank of . It is one because is of rank one with its range in span and

 ((π1⊗α0)+k(δ)A0(k(δ)))~Φδ=i2⟨~Φδ|ψ1⊗η⟩(ψ1⊗η)+O(δ2)=−δμ2(ψ1⊗η)+O(δ2)

does not vanish for sufficiently small.

Finally let us prove that . As a consequence of Proposition 4.ii, we have that the function , defined on by is real valued. Moreover, using (14) for small, and . In consequence, this function admits a root which is real. By uniqueness, . ∎

## 4. Upper Thresholds

We now extend our analysis to the upper thresholds. We will show that if is an eigenvalue of multiplicity of , then is a bound for the number of resonances around .

Let be a normalized basis of . In analogy with (15), for define

 (24) μj,l=⟨∂φψq0,j|πq0∂φψq0,l⟩||ε||2+12∑q,λq≠λq0(λq−λq0)⟨∂φψq0,j|πq∂φψq0,l⟩⟨(D23+λq−λq0)−1ε|ε⟩,

and let be the matrix .

Denote by and .

###### Theorem 6.

Suppose that is an eigenvalue of multiplicity of , that is a non-zero -function satisfying (1) with , and that is a compact neighborhood of zero. Then, for all sufficiently small, the operator-valued function , initially defined in , admits a meromorphic extension on . This extension has at most poles, counted with multiplicity. These poles are among the zeros of some determinant, which satisfy

 kl(δ)=−iνq0,lδ2+o(δ2),δ↓0,

where are the eigenvalues of the matrix .

###### Proof.

Some points in this proof are close to what has been done for the first threshold. We will keep the same notations and explain how to modify the arguments of the previous sections. In analogy with section 2 set

 Φj,δ:=−i2((∂φψq0,j⊗η−1ε′)+δ(∂2φψq0,j⊗η−1ε2))  and  K0:=∑j|Φj,δ⟩⟨ψq0,j⊗η|.

Then, the analog of Lemma 3 still holds. Here, since is in the interior of the essential spectrum, the resolvent is initially defined for near and the extension of the weighted resolvent is done with respect to from to a neighborhood of .

Also, as in the proof of Theorem 5, we have for , with (and similar notation for ):

 (25) ηR(k)η=ηR0(k)η(Id+K)−1(Id+δT(δ,k))−1,

where

 K:=δk(Id+δT(δ,k))−1K0=δkm0∑j=1|~Φj,δ⟩⟨ψq0,j⊗η|

is now of rank , with obvious notation for .

Next, let be the projection over in the direction of and . Then, the matrix of in the basis is given, for , by

 (26) 1k⎡⎢ ⎢ ⎢⎣k+ω1,1,δ(k)…ωm0,1,δ(k)⋮⋱⋮ω1,m0,δ(k)…k+ωm0,m0,δ(k)⎤⎥ ⎥ ⎥⎦:=1kMδ(k)

where we have set Assume that is invertible, then by (25)

 ηR(k)η= (i2k∑j|ψq0,j⊗η⟩⟨ψq0,j⊗η|+A0(k))(Π⊥δ+kMδ(k)−1Πδ)(Id+δT(δ,k))−1 = (i2∑j|ψq0,j⊗η⟩⟨ψq0,j⊗η|Mδ