Resonance spectrum for one-dimensional layered media

# Resonance spectrum for one-dimensional layered media

Alexei Iantchenko Malmö University
School of Technology and Society
SE-205 06 Malmö
Sweden
July 7, 2019
###### Abstract.

We consider the “weighted” operator on the line with a step-like coefficient which appears when propagation of waves thorough a finite slab of a periodic medium is studied. The medium is transparent at certain resonant frequencies which are related to the complex resonance spectrum of

If the coefficient is periodic on a finite interval (locally periodic) with identical cells then the resonance spectrum of has band structure. In the present paper we study a transition to semi-infinite medium by taking the limit The bands of resonances in the complex lower half plane are localized below the band spectrum of the corresponding periodic problem () with or resonances in each band. We prove that as the resonance spectrum converges to the real axis.

###### Key words and phrases:
One-dimensional, layered, truncated periodic, scattering resonances
###### 2000 Mathematics Subject Classification:
47A10, 47A40, 81Q10

## 1. Introduction

In the present paper we consider operator on the line with step-like coefficient which is periodic on a finite interval defined as follows:

 (1) ak(x)=a(x),forx∈[0,k];ak(x)=1b21,forx∉[0,k],

where is periodic function equal to

 (2) a0(x)={b−22forx∈[0,x2)b−21forx∈[x2,1)

for Here and Equation

 Pkψ=−∂xak(x)∂xψ(x)=λ2ψ

appears when the propagation of waves through a finite slab of a periodic medium is studied. Such systems are also called finite or locally periodic media (for revue see [6]).

When is large then the properties of medium is close to an infinite periodic problem in a sense that we are going to discuss in the present paper.

We denote the pure periodic operator, where is periodic function equal to for as in (2). Then the Floquet theory shows the existence of a pair of the quasi-periodic solutions of the equation

 ψ±(λ,x+1)=e±iθψ±(λ,x),

such that for Here is the Bloch phase. We denote

 (3) F(λ)=ρ+12cos{λ(x2b2+(1−x2)b1)}−ρ−12cos{λ(x2b2−(1−x2)b1)}

the Lyaponov function for (see Section 2.1). Here

 ρ=b21+b222b1b2.

The spectrum of the operator has band structure with allowed zones defined as follows:

 (4) λ∈σ(P)⇔|F(λ)|2<1,λ∈R

(see [4] and Section 4). The band edges are given by solutions of

The relation between the Bloch phase and the spectral parameter is called dispersion relation:

 cosθ(λ)=F(λ).

Since the coefficient is constant equal to outside a finite region, we are here concerned with a scattering problem.

We shall denote the reflection and transmission coefficients for the operator by and respectively:

 Pkψ=λ2ψ,ψ=eiλb1x+rke−iλb1x,x<0;ψ=tkeiλb1x,x>k.

Following the ideas in [7] we consider a transition to semi-infinite periodic materials by taking the limit of the reflection coefficient for

The limiting operator

 P∞ψ=−∂xa∞(x)∂xψ(x)

corresponds to the case of such a long slab that it can be considered as half infinite.

In the case of the operator the solution of the scattering problem is defined as the solution of the equation such that

 (5) ψ=eiλb1x+re−iλb1x,x<0;ψ=cψ+(λ,x),x>0,

with some

As in [7] we have that the reflection coefficients and are analytic in the upper half plane and continuous in and when and When is real, converges to in the weak sense (see Theorem 2, Section 2.2).

Numerical calculation shows that in each allowed zone of there are in general frequencies where the transmission probability is one: and the medium is perfectly transparent: There exist an additional frequency when the medium consisting of only one unit cell is transparent and then for all The pics in the transmission probability are related to the complex resonances close to the real axis.

We make the following definition.

The operator defined from to is self-adjoint. For we call the resolvent of For any the operator-valued function

 Rk(λ):L2comp(R)↦L2loc(R)

can be continued to the lower complex half-plane as a meromorphic function of and it has no poles for with positive constant dependent on (see Section 4).

The poles of the in are called resonances or scattering poles. We denote the set of resonances

Using the explicit construction of the resolvent in [4] the poles are calculated numerically. Some examples are presented in Section A, figures (1), (2) and (3). We summarize the properties of in the following Theorem.

###### Theorem 1.

We consider the finite periodic and periodic operators generated by the same unit cell given in (2). Let denote the resonance spectrum for the finite periodic system with identical cells and denote the band spectrum for given by (4).

The resonance spectrum for the finitely periodic system has band structure related to the bands of the real spectrum for the pure periodic problem as follows:
1) The resonance spectrum of has band structure. Resonances are localized below the bands of the real spectrum of

 λ∈Res(Pk)⇒Re(λ)satisfies(???)⇔Re(λ)∈σ(P∞).

Each resonance band of consists of resonances and eventually an additional resonance with real part such that the one-cell medium is “perfectly transparent” at frequency
2) If the condition

 (6) b2x2=b1(1−x2)⇔b1x2=b21−x2.

is satisfied then is the degenerate band edge (two bands has common edge at ). The resonance spectrum is periodic with the period
3) As then the resonance spectrum of approaches the real axis.

In the present paper we motivate these numerical results.

The band structure of the resonance spectrum for a finitely periodic system and its relation to the band spectrum of the correspondent periodic problem is well-known in physical literature (see [1]).

We say that is periodic if there exists period, such that

 Res(Pk)∩([q+Tn,p+Tn]−iR)=Res(Pk)∩([q+Tm,p+Tm]−iR)

for any and This property follows directly from the equations defining the resonances in Section (4.2) if condition (6) is satisfied.

A special property of the operator with step-like periodic coefficient is that the coefficients in the dispersion relation

 (7) 2cosθ(λ)=(ρ+1)cos{λ(x2b2+(1−x2)b1)}−(ρ−1)cos{λ(x2b2−(1−x2)b1)}

are independent of the spectral parameter Formula (7) implies that the band spectrum is periodic if the profile of verifies (6).

The third part of the Theorem is proved in Section 6.

The convergence of the resonances for a finitely periodic system with cells to the bands of real spectrum for the periodic problem as was discussed by F. Barra and P. Gaspard in [3] in the case of Schrödinger equation.

In our proof we use representations for the reflection and transmission coefficients for a finite slab of periodic medium as in the recent paper of Molchanov and Vainberg [7]. The authors considered transition of truncated medium described by the D Schrödinger operator to semi-infinite periodic materials. By relating the reflection coefficients to the resolvent of we show explicitly that the resonances correspond to the poles of the analytic continuation of to Then we consider the limit of the poles of as

Note that for the reflection coefficient for cells medium is related to for cells medium via where is a linear-fractional automorphism of the unit disk. By considering the fixed point of we get a new proof of the convergence of to when belongs to the spectral gapes and non-degenerate band edges for the operator (see Section 7).

The structure of the paper is the following:
In Section 2 we recall some well-known facts concerning spectral problem for weighted Sturm-Liouville operators (see [2]) and consider scattering by a finite slab of a periodic medium. We follow [7] with minor changes due to the special form of operator We recall exact formulas for the the reflection and transmission coefficients using the iteration of the monodromy matrix. We recall also the result of [7] on a transition to semi-infinite periodic material (limit ). In Section 3 we give explicit expression for the monodromy matrix of In Section 4 we recall the iterative procedure used in [4] for construction of the resolvent and define resonances. In Section 5 the reflection coefficient is expressed using the iteration formulas of [4] and we show that the poles of and the poles of coincide. In Section 6 we prove the convergence of to the real axis. In Section 7 we discuss the convergence of for by considering the limit of a sequence of linear-fractional automorphisms on the unit disk. In Appendix A we present numerical examples.

Acknowledgements. The author would like to thank Maciej Zworski for suggesting to look at the problem considered in the present paper and for helpful discussions.

## 2. General methods for truncated periodic operators

In this section we following [7] consider the scattering theory for operator combining the Floquet-Bloch theory and scattering theory for D weighted operators.

### 2.1. The monodromy matrix and Bloch quasi-momentum

We recall first some well-known facts concerning the spectral problem of Sturm-Liouville operators on the line (see [2]). We consider equation

 (8) Pψ=−∂xa(x)∂xψ(x)=λ2ψ

on with a strictly positive as in the Introduction, formula (1) or periodic as in (2).

Let be solutions of (8) with initial data

 (9) ψ1(λ,0)=1,(a∂xψ1)(λ,0)=0;ψ2(λ,0)=0,(a∂xψ2)(λ,0)=1.

We define the transfer matrix (propagator) for operator

 (10) Mλ(0,x)=(ψ1(λ,x)λψ2(λ,x)(a∂xψ1)(λ,x)λ(a∂xψ2)(λ,x)).

From (9) it follows that is the identity matrix. For any solution of (8) matrix maps the Cauchy data of at into the Cauchy data of at point

 Mλ(0,x):(ψ(0)(a∂xψ)(λ,0)λ)↦(ψ(x)(a∂xψ)(λ,x)λ).

As the generalized Wronskian associated with

 W[ψ1,ψ2]=ψ1a∂xψ2−ψ2a∂xψ1

is constant, we have

 detMλ(0,x)=W[ψ1,ψ2](1)=W[ψ1,ψ2](0)=1.

Equation (8) with has exactly one solution in normalized by the condition and it has exactly one solution normalized by the same condition. Here are semiaxes Any solution of (8) can be represented as linear combinations of and and from the normalization of it follows, that there exist functions such that

 ψ±=ψ1+m±(λ)ψ2,Imλ>0.

Functions are called Weyl’s functions and we have

 ψ1+m+(λ)ψ2∈L2(R+),ψ1+m−(λ)ψ2∈L2(R−),Imλ>0.

Let be periodic: Consider propagator through one period (monodromy matrix):

 Mλ=Mλ(0,l)=(αβγδ)(λ)=(ψ1(λ,l)λψ2(λ,l)(a∂xψ1)(λ,l)λ(a∂xψ2)(λ,l)).

Denote the Lyapunov function. Both and are entire function of and The eigenvalues of are the roots of the characteristic equation

 (11) μ2−2μF(λ)+1=0.

If then one can select roots of (11) in such a way that where is analytic and

 (12) Imθ(λ)>0whenImλ>0,

i.e.,

 (13) |μ+(λ)|<1,|μ−(λ)|>1,Imλ>0.

The roots for real are defined by continuity in the upper half plane:

 μ±(λ)=μ±(λ+i0),λ∈[0,∞).

Since the trace of is equal to the sum of the eigenvalues ,

 (14) coslθ=F(λ)=12(ψ1+aψ′2)(λ,l)=12(α+δ).

The spectrum of belongs to the positive part of the energy axis and has band structure.

For real the inequality defines the spectral bands (zones)

 bn=[λ2n−1,λ2n],n=1,2,…,

on the frequency axis . The bands are defined by the condition and at any band edge

The function is real valued when belongs to a band. The roots are complex adjoint there, and

The spectrum of (on ) on the frequency axis is

The complimentary open set, given by corresponds to spectral gaps, On gaps, the function is real valued, the roots are real and (13) holds.

A point which belongs to the boundary of a band and the boundary of a gap is called a non-degenerate band edge. If it belongs to the boundary of two different bands, it is called a degenerate band edge.

As in [7] we get that if is a non-degenerate band edge, then If is a degenerate edge, then Both eigenvalues of the monodromy matrix at any band edge are equal to or both are equal to

We normalize the eigenvectors of by choosing the first coordinate of to be equal to one:

 h±(λ)=(1m±(λ)).

The second coordinates of the vectors coincide with the Weyl’s functions defined above. In fact, if are solutions of the equation with the initial Cauchy data given by the eigenvector then

 (15) ψ±(λ,x+l)=e±ilθ(λ)ψ±(λ,x)

and (13) implies that when From here it follows that coincide with Weyl’s solution introduced for general Hamiltonians and that the second coordinates of the vectors are Weyl’s functions.

Since

 (α−e±ilθ(λ)βγδ−e±ilθ(λ))(1m±)=0,

the following two representations are valid for Weyl’s functions:

 (16) m±(λ)=e±ilθ(λ)−α(λ)β(λ)=γ(λ)e±ilθ(λ)−δ(λ).

### 2.2. Reflection coefficient for the truncated periodic operator

We consider operator with the truncated periodic coefficient

 Pkψ=−∂xak(x)∂xψ(x),ak(x)=a(x),%forx∈[0,kl];ak(x)=1b21,forx∉[0,kl],

which appears when the propagation of waves through a finite slab of a periodic medium is studied. We shall also consider the limiting case

 P∞ψ=−∂xa∞(x)∂xψ(x),

which corresponds to the case of such a long slab that it can be considered as half infinite.

We shall denote the reflection and transmission coefficients for the operator (with compactly supported coefficient ) by and respectively:

 Pkψ=λ2ψ,ψ=eiλb1x+rke−iλb1x,x<0;ψ=tkeiλb1x,x>kl.

In the case of the operator the solution of the scattering problem is defined as the solution of the equation such that

 (17) ψ=eiλb1x+re−iλb1x,x<0;ψ=cψ+(λ,x),x>0,

with some We have the following version of Theorem 3 of S. Molchanov, B. Vainberg in [7]:

###### Theorem 2.

1) The transfer matrix over periods has the form

 (18) Mkλ=(αkβk\parγkδk)=sinklθ(λ)sinlθ(λ)Mλ−sin(k−1)lθ(λ)sinlθ(λ)I,

where is the Bloch function. The elements of satisfy the relations

 αk−δk=sinklθ(λ)sinlθ(λ)(α−δ),βN=sinklθ(λ)sinlθ(λ)β, (19) γk=sinklθ(λ)sinlθ(λ)γ,αk+δk=2cosklθ(λ).

2) The reflection coefficients have the forms

 (20) rk(λ)=−(α−δ)+i(b1γ+βb1)2sinlθ(λ)cosklθ(λ)sinklθ(λ)+i(b1γ−βb1),
 (21) r(λ)=βb1+b1γ−i(α−δ)2sinlθ(λ)−(b1γ−βb1).

3) The transmission probability have the form

 (22) |tk(λ)|2=4sin2lkθsin2lθ((α−δ)2+(b1γ+βb1)2)+4=1sin2lkθsin2lθ|r1|2|t1|2+1.

4)The reflection coefficients and are analytic in the upper half plane and continuous in For any we have When converges to in the weak sense:

for any test function

Proof: We reproduce here the proof of [7] for the sake of completeness with only minor changes due to the “weight” in the definitions of and Formula (18) follows by induction from relation (11):

 M2λ−2coslθ(λ)Mλ+I=0.

The first three relations of (19) are immediate consequences of (18). In order to get the fourth one we note that the eigenvalues of are Thus,

 (23) αk+δk=trMkλ=eiklθ(λ)+e−iklθ(λ)=2cosklθ(λ).

Next we prove (20). The relation between Cauchy data for the left-to-right scattering solution at and are given by

 (αkβkγkδk)(1+rkib1(1−rk))=(tkitkb1)⇔⎧⎪⎨⎪⎩αk(1+rk)+βk(ib1(1−rk))=tkγk(1+rk)+δk(ib1(1−rk))=itkb1

By dividing the second equation by the first one we arrive at

 γk(1+rk)+δkib1(1−rk)αk(1+rk)+βkib1(1−rk)=ib1.

Solving for we obtain

 (24) rk=δk−αk−i(b1γk+βkb1)δk+αk+i(b1γk−βkb1).

Using (19) we get

 rk=−sinklθsinlθ(α−δ)+i(b1sinklθsinlθγ+1b1sinklθsinlθβ)2cosklθ+i(b1sinklθsinlθγ−1b1sinklθsinlθβ).

This justifies (20).

In order to get (21) we note that (17) implies that

 (1+rib1(1−r))=c(1m+)⇔1+rib1(1−r)=1m+

and therefore,

 r=ib1−m+ib1+m+.

From here and (16) it follows that

 r=ib1−eilθ−αβib1+eilθ−αβ=α+ib1β−eilθeilθ−(α−ib1β).

and

 r=ib1−γeilθ−δib1+γeilθ−δ=ib1(eilθ−δ)−γib1(eilθ−δ)+γ=eilθ−(δ−ib1γ)eilθ−(δ+ib1γ).

Hence,

and

 r =(α+ib1β)−(δ−ib1γ)2eilθ−(α−ib1β)−(δ+ib1γ)=(α−δ)+i(βb1+b1γ)2eilθ−(α+δ)+i(βb1−b1γ)= =(α−δ)+i(βb1+b1γ)2isinlθ+i(βb1−b1γ),

where the last equality is a consequence of (14) and it implies (21).

We prove the third statement of the theorem. From (24) it follows

 (25) |rk|2=(δk−αk)2+(b1γk+βkb1)2(δk+αk)2+(b1γk−βkb1)2.

Using that we get

 (26) |tk|2=1−|rk|2=4δkαk−4γkβk(δk+αk)2+(b1γk−βkb1)2.

We use and get

From formulas (25) and (26) we get

 |rk|2|tk|2=14((δk−αk)2+(b1γk+βkb1)2)

and hence, putting

 |tk|2=1sin2lkθsin2lθ|r1|2|t1|2+1.

The analyticity of and in and their continuity in follow from the explicit formulas (20), (21). For we have and is pure imaginary on the gaps Furthermore, if then

and this justifies the convergence of to when

The weak convergence for is a consequence the convergence in the complex half plane.

The proof of Theorem 2 is complete.

Note also the following relations:

 |rk|2|tk|2=sin2lkθsin2lθ⋅|r1|2|t1|2,
 |tk|2 =44cos2klθ+(b1sinklθsinlθγ−1b1sinklθsinlθβ)2=4sin2lθsin2klθ4sin2lθcos2klθsin2klθ+(b1γ−βb1)2.

The last formula follows from (26) by using (23).

Formula (22) implies that the perfect transmission () occurs whenever () or if

 (27) sin2lkθsin2lθ=0.

For equation (27) is satisfied when for

Therefore, in the general case (), the transmission probability has peaks with in each allowed energy band as increases by Since the peaks in the transmission probability (or in general in the cross section) are associated with resonances, we expect to find resonances near each allowed energy band.

On the gaps, the function is real valued. Then the transmission probability is given by

 (28) |tk(λ)|2=1sinh2lkiθsinh2liθ|r1|2|t1|2+1.

As for then in the forbidden zone unless Hence there are no resonances below the gaps.

On the gaps, the reflection coefficient for the half-periodic system satisfy

 |r(λ)|2=(βb1+b1γ)2+(α−δ)2e2iθ+e−2iθ−2+(b1γ−βb1)2=(βb1+b1γ)2+(α−δ)2(α+δ)2−4+(b1γ−βb1)2=1.

In [7], Theorem 5, was shown that

###### Lemma 1.

If is a degenerate band edge, i.e. then the reflection coefficient is zero,

The proof uses the fact that at any degenerate band edge the monodromy matrix and This allows to pass to the limit in (20) as