Resonances near thresholds in slightly Twisted Waveguides
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,
Keywords: Twisted waveguide, Dirichlet Laplacian, Resonances near thresholds.
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
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 , where bending against twisting is discussed, and to  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 (), in particular a twisting going to 0 at infinity will not modify the spectrum (). 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 (). In these both cases, however, the resonances appear as perturbations of embedded eigenvalues of a reference operator, and follow the Fermi golden rule (see  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 , 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
where . Then, for we define as the waveguide obtained by twisting with , where , i.e., we define
where is the rotation of angle in . Set
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
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 .
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:
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 ). 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.
Independent of the size of the perturbation , a more global definition of resonances would be possible by showing that a generalized determinant (as in  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
For , define
and similarly for . If for , is the orthogonal projection onto ker, using (3) for , we have that
The integral kernel of is explicitly given by
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,
where is the rank one operator and : is the analytic operator-valued function
with being the operator in with integral kernel given by
Clearly, for , the family of operators is uniformly bounded on .
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
Since has no eigenvalue below (see ), 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
Let . There exists such that for any and
where is the rank one operator
and : is an analytic operator-valued function. Moreover,
Thanks to (6),
Since the range of the operator is spanned by constant functions, we have , and therefore
We now treat the last term of (12): Setting we immediately get
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
Let be a compact neighborhood of zero. With the notation of Lemma 3, for sufficiently small, let us introduce the functions and
There exists such that for any , ,
is a positive constant, and is an analytic function in satisfying
When , there holds .
We use the Taylor expansion and Lemma 3 to see that
where is holomorphic operator-valued function that is uniformly bounded for and small.
By definition of , we have:
The first term is zero because tends to zero at infinity. Using integration by part, since satisfies a Dirichlet boundary condition, we deduce
Noticing that , from (16) we get
where is holomorphic and uniformly bounded for and small.
We now compute . First recall that . Next, note that since ,
and therefore, using the definition of in (9), we get
which in turn implies that
In consequence, having in mind (9) again, we deduce
We compute the main term of the last expression using integration by parts, both in the and the variables:
Now, we notice that
In addition, since and , we have that
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. ∎
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
with given by (15) and .
For let us set
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
Therefore, is invertible if and only if , and
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:
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
does not vanish for sufficiently small.
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
and let be the matrix .
Denote by and .
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
where are the eigenvalues of the matrix .
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
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 ):
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
where we have set Assume that is invertible, then by (25)