Bloch theory for water waves

# Bloch theory and spectral gaps for linearized water waves

Walter Craig Department of Mathematics, McMaster University, Hamilton Ontario, L8S 4K1 Canada,  craig@math.mcmaster.ca Maxime Gazeau Department of Mathematics, University of Toronto, Toronto Ontario, M5S 2E4 Canada,  gazeau@math.toronto.edu Christophe Lacave Institut Fourier, CNRS and Université Grenoble Alpes, F-38000 Grenoble France  Christophe.Lacave@univ-grenoble-alpes.fr  and  Catherine Sulem Department of Mathematics, University of Toronto, Toronto Ontario, M5S 2E4 Canada,  sulem@math.toronto.edu
###### Abstract.

The system of equations for water waves, when linearized about equilibrium of a fluid body with a varying bottom boundary, is described by a spectral problem for the Dirichlet – Neumann operator of the unperturbed free surface. This spectral problem is fundamental in questions of stability, as well as to the perturbation theory of evolution of the free surface in such settings. In addition, the Dirichlet – Neumann operator is self-adjoint when given an appropriate definition and domain, and it is a novel but very natural spectral problem for a nonlocal operator. In the case in which the bottom boundary varies periodically, where , a lattice, this spectral problem admits a Bloch decomposition in terms of spectral band functions and their associated band-parametrized eigenfunctions. In this article we describe this analytic construction in the case of a spatially periodic bottom variation from constant depth in two space dimensional water waves problem, giving a construction of the Bloch eigenfunctions and eigenvalues as a function of the band parameters and a description of the Dirichlet – Neumann operator in terms of the bathymetry . One of the consequences of this description is that the spectrum consists of a series of bands separated by spectral gaps which are zones of forbidden energies. For a given generic periodic bottom profile , every gap opens for a sufficiently small value of the perturbation parameter .

W.C. is partially supported by the Canada Research Chairs Program and NSERC through grant number 238452–16.
C.L. is partially supported by the Agence Nationale de la Recherche through the project DYFICOLTI, grant ANR-13-BS01-0003-01 and the project IFSMACS, grant ANR-15-CE40-0010.
C.S. is partially supported by NSERC through grant number 46179–13.

## 1. Introduction

This paper concerns the motion of a free surface of fluid over a variable bottom, a problem of significance for ocean dynamics in coastal regions where waves are strongly affected by the topography. There is an extensive literature devoted to the effect of variable depth over surface waves and there are many scaling regimes of interest, including in particular long-wave regimes where the typical wavelength of surface waves is assumed to be much longer than the typical lengthscale of the variations of the bathymetry. For purposes of many mathematical studies, the variable bottom topography is assumed either to be periodic, or else to be described by a stationary random ergodic process.

References on the influence of rough bottoms on the free surface include works of Rosales & Papanicolaou , Craig et al  , and Nachbin & Sølna , where techniques of homogenization theory are used to obtain effective long wave model equations. The article  performs a rigorous analysis of the effect of a rapidly varying periodic bottom in the shallow water regime. Using simultaneously the techniques of homogenization theory and long-wave analysis, a new model system of equations is derived, consisting of the classical shallow water equations that give rise to effective (or homogenized) surface wave dynamics, coupled with a system of nonlocal evolution equations for a periodic corrector term. A rigorous justification for this decomposition is given in  in the form of a consistency analysis, in the sense that the constructed approximated solutions satisfy the water wave equations up to a small error term that is controlled analytically. A central issue in this approach is the question of the time of validity of the approximation. It is shown that the result is valid for a time interval of duration in the shallow water scaling only if the free surface is not in resonance with the rapidly varying bottom. However resonances are not exceptional. When resonances occur, secular growth of the corrector terms takes place, and this compromises the validity of the approximation, and in particular, a small amplitude, rapidly oscillating bathymetry will affect the free surface at leading order. The motivation for the present study is to develop analytical tools that will be useful in order to address the dynamics of these resonant situations. As a first step, we consider in this paper the water wave system with a periodic bottom profile, linearized near the stationary state, and we develop a Bloch theory for the linearized water wave evolution. This analysis takes the form of a spectral problem for the Dirichlet – Neumann operator of the fluid domain with periodic bathymetry.

The starting point of our analysis is the water wave problem written in its Hamiltonian formulation. Let

 S(b,η)={(x,y):x∈R,−h+b(x)

be the two-dimensional time-dependent fluid domain where the variable bottom is given by , and the free surface elevation by . Following  and , we pose the problem in canonical variables , where is the trace of the velocity potential on the free surface . In these variables, the equations of motion for nonlinear free surface water waves are

 (1.1)

The operator is the Dirichlet-Neumann operator, defined by

 (1.2) G[η,b]ξ=√1+|∂xη|2∂nφ|y=η ,

where is the solution of the elliptic boundary value problem

 (1.3)

and is the acceleration due to gravity. In the present article, we consider the system of water wave equations, linearized near a surface at rest and in the presence of periodic bottom. The bottom defined as where is - periodic in . We assume is in where is the periodized interval . The system (1.1) linearized about the stationary solution is as follows.

 (1.4) {∂tη−G[b]ξ=0∂tξ+gη=0 ,

where now, and for the remainder of this article, we denote by . This is an analog of the wave equation, however with the usual spatial Laplacian replaced by the nonlocal operator whose coefficients are -periodic dependent upon the horizontal spatial variable :

 (1.5) ∂ttη+gG[b]η=0 .

The initial data for the linearized surface displacement are

 (1.6) η(x,0)=η0(x),ηt(x,0)=η1(x),x∈R ,

these being defined on the whole line.

Bloch decomposition, a spectral decomposition for differential operators with periodic coefficients, is a classical tool to study wave propagation in periodic media. For a relatively recent example, Allaire et al.  considered the problem of propagation of waves packets through a periodic medium, where the period is assumed small compared to the size of the envelope of the wave packet. In this work the authors construct solutions built upon Bloch plane waves having a slowly varying amplitude. In a study of Bloch decomposition for the linearized water wave problem over a periodic bed, Yu and Howard  use a conformal map that transforms the original fluid domain to a uniform strip. Using this map, they calculate the formal Fourier series for Bloch eigenfunctions. For various examples of bottom profiles, they compute numerically the Bloch eigenfunctions and eigenvalues, from which they identify the spectral gaps and make several observations of their behavior.

The main goal of our work in the present paper is to develop Bloch spectral theory for the Dirichlet-Neumann operator, in analogy with the classical case of partial differential operators with periodic coefficients. This theory constructs the spectrum as a sequence of bands separated by gaps of instability; it serves as a basis for perturbative calculations that gives rise to explicit formulas and rigorous understanding of spectral gaps, and therefore intervals of unstable modes of the linearized water wave problem over periodic bathymetry.

The principle of the Bloch decomposition is to parametrize the continuous spectrum and the generalized eigenfunctions of the spectral problem for on with a family parametrized by of spectral problems for on the interval , with -periodic boundary conditions. For this purpose, we construct the Bloch eigenvalues and eigenfunctions of the spectral problem

 (1.7) G[b]Φ(x,θ)=Λ(θ)Φ(x,θ) ,

with boundary conditions

 (1.8) Φ(x+2π,θ)=Φ(x,θ)e2πiθ ,

where ; such behavior is termed to be -periodic in .

When the bottom is flat, , the Bloch eigenvalues are given explicitly in terms of the classical dispersion relation for water waves over a constant depth, namely

 (1.9) Λ(0)n(θ)=ω2(n+θ):=g(n+θ)tanh(h(n+θ))

for , and the Bloch parameter , where is the circle . Eigenvalues are simple for and . For half-integer values of , namely , eigenvalues have multiplicity two. If reordered appropriately by their magnitude, the eigenvalues are continuous and periodic in with period . The eigenfunctions satisfy the boundary conditions (1.8). With the ordering of the eigenvalues specified above, the eigenfunctions are periodic in , again of period .

Just as in the case of Bloch theory for many second order partial differential operators, we find that the presence of the bottom generally results in the splitting of double eigenvalues near such points of multiplicity, creating a spectral gap.

For and nonzero, the spectrum of on is composed of a non-decreasing sequence of eigenvalues

 Λ0(θ)≤Λ1(θ)≤⋯≤Λn(θ)≤⋯

which are continuous and periodic in , and continuous in , for in a - neighborhood of the origin. The corresponding eigenfunctions are -periodic in and periodic in . In the case of such that , the eigenvalue is simple, and it and eigenfunction are locally analytic in both and .

The spectrum of the Dirichlet – Neumann operator on the line, namely on , is the union of the ranges of the Bloch eigenvalues , that is

 σL2(R)(G[b])=∪+∞n=0[Λ−n,Λ+n]

where and . It is the analog of the structure of spectral bands and gaps of the Hill’s operator . The ground state satisfies for any bathymetry , and its corresponding eigenfunction is .

In Section 4 we give a perturbation analysis of spectral behavior and we compute the gap opening for , asymptotically as a function of . As an example we consider , in analogy with the case of Matthieu’s equation, and we calculate the asymptotic behavior of the first several spectral gaps. We find that, as in Floquet theory for Hill equation, the first spectral gap obeys . However, in contrast to the case of the Matthieu equation, the second spectral gap only opens at order . In addition, we show that the centre of the gap is strictly decreasing in .

A generic bottom profile will open all spectral gaps. Clearly, the band endpoints satisfy unless which certainly does not occur in a perturbative regime. For sufficiently small generic bathymetric variations , we also know that , which is the case for Hill’s operator, and although we conjecture this to be the case for the Dirichlet – Neumann operator for large general , we do not have a proof of this fact. Furthermore, for Hill’s operator, the band edges of the gap correspond to the periodic spectrum, while we do not have a proof of the analogous result for the Dirichlet – Neumann operator. The reality condition implies that , and therefore for even, when the gap opens. The same holds for and odd. The existence of a spectral gap implies that the spectrum is locally simple. Hence the general theory  of self-adjoint operators implies analyticity of both and . In fact, for , the unperturbed spectrum is simple and the same statement of local analyticity holds for .

Gaps are not guaranteed to remain open as the size of the bottom variations increases, as shown in the numerical simulations performed in , Fig.4 (second gap).

## 2. The Dirichlet – Neumann operator

The goal is to study the spectral problem

 (2.1) G[b]Φ(x,θ)=Λ(θ)Φ(x,θ) ,

where is the Dirichlet – Neumann operator for the fluid domain . We impose -periodic boundary conditions

 (2.2) Φ(x+2π,θ)=Φ(x,θ)e2πiθ

for the Bloch parameter. It is convenient in Bloch theory to define

 (2.3) ψ(x,θ)=e−iθxΦ(x,θ)

to transform the original problem to an eigenvalue problem with periodic boundary conditions. Indeed, condition (2.2) implies that is periodic in of period . The spectral problem is now rewritten in conjugated form

 (2.4) e−iθxG[b]eiθxψ(x,θ)=Λ(θ)ψ(x,θ) , (2.5) ψ(x+2π,θ)=ψ(x,θ) .

### 2.1. Analysis of the Dirichlet – Neumann operator

The following proposition states the basic properties of the Dirichlet – Neumann operator with -periodic boundary conditions.

###### Proposition 2.1.

For each , the operator is self-adjoint from to with periodic boundary conditions. It has an infinite sequence of eigenvalues , which tend to as tends to in such a way that .

Writing , the Dirichlet-Neumann operator is written as

 G[b]=G0+DL[b]

where is the Dirichlet-Neumann operator with a flat bottom, and is the correction due to the presence of the topography. In , it was shown that

 (2.6) DL[b]=−DB[b]A[b] ,

where

 A[b]f(x)=∫Reikxsinh(b(x)k)sech(hk)^f(k)dk (2.7) B[b]f(x)=1π∫∂x′b(x′)(x′−x)+h−b(x′)(x−x′)2+(b(x′)−h)2f(x′)dx′ −12π∫ln((x−x′)2+(h−b(x′))2)~G[−h+b]f(x′)dx′ ,

where is the usual Dirichlet-Neumann operator in the domain that associates Dirichlet data on the boundary with Neumann boundary condition at , to the normal derivative of the solution to the Laplace equation on .

In the following, we use the notation

 e−iθxG[b]eiθx=Gθ+M

where

 Gθ=e−iθxG0eiθx (2.8) M=−e−iθxDL[b]eiθx=−(e−iθxDB[b]eiθx)(e−iθxA[b]eiθx) .

The operator is diagonal in Fourier space variables, with

 Gθeijx=(j+θ)tanh(h(j+θ))eijx .

In the next proposition, we prove that preserves the class of - periodic functions.

###### Proposition 2.2.

Given -periodic bottom topography, , suppose that is a -periodic function defined on , namely that

 ξ(x+2π)=e2πiθξ(x) .

Then the result of application of the Dirichlet – Neumann operator is also -periodic. That is

 (G[b]ξ)(x+2π)=e2πiθG[b]ξ(x) .
###### Proof.

Define . The harmonic extension of which satisfies the bottom boundary conditions is unique. Comparing with , we have that

 0=Nb(x)⋅∇φ1(x,y) ,0=Nb(x+2π)⋅∇e2πiθφ(x,y)=Nb(x)⋅∇e2πiθφ(x,y) ,

by the periodic nature of the bottom profile . Thus

 N⋅∇φ1(x,0)=∂yφ1(x,0) ,N⋅∇e2πiθφ(x,0)=e2πiθ∂yφ(x,0) ,

and therefore the two coincide. Namely

 e2πiθ(G[b]ξ)(x)=(G[b]ξ1)(x)=G[b](e2πiθξ(x)) .

The next statement shows that the operator is bounded on , and in fact is strongly smoothing.

###### Proposition 2.3.

There exists such that for , the ball centered at the origin and of radius of and , is also periodic of period , and satisfies the estimate

 (2.9) ∥Mf∥L2 ≤ C0(|b|C1)∥f∥L2 ,

where the constant depends on the -norm of , .

In addition, the operator is strongly smoothing

 (2.10) ∥Mf∥Hs ≤ Crs(|b|C1)∥f∥H−r .

for all .

The proof of this proposition is given in Section 5.

### 2.2. Floquet theory

The spectrum of the Dirichlet – Neumann operator acting on the domain is real, non-negative, and is composed of bands and gaps. It is the union over of the Bloch eigenvalues , the analog to Bloch theory for the Schrödinger operator.

When , the spectrum of on consists of the Bloch eigenvalues which are labeled in order of increasing magnitude. The eigenvalues are periodic in of period one, and are simple when . For , the spectrum is double (see Fig.1). Denoting , the eigenvalues and eigenfunctions associated to are given as follows:

 For −12≤θ<0, Λ(0)2n(θ)=g−n(θ); ψ2n(x,θ)=e−inx , for 0≤θ≤12, Λ(0)2n(θ)=gn(θ); ψ2n(x,θ)=einx ,

and

 for −12≤θ<0, Λ(0)2n−1(θ)=gn(θ); ψ2n−1(x,θ)=einx , for 0≤θ≤12, Λ(0)2n−1(θ)=g−n(θ); ψ2n−1(x,θ)=e−inx .

With this definition, both and are periodic in , and is continuous in while has discontinuities at .

The goal of our analysis is to show that in the presence of a variable periodic topography, spectral curves which meet when typically separate, creating spectrum gaps corresponding to zones of forbidden energies. For this purpose, assume that the bottom topography is given by where is a -periodic function in the ball , with . For our analysis, the circle of Floquet exponents is divided into regions in which unperturbed spectra are simple (outer regions), and regions that include the unperturbed multiple spectra (inner regions). To apply the method of continuity, these regions are defined so that they overlap.

###### Theorem 2.4.

For all , the -spectrum of on the domain is composed of an increasing sequence of eigenvalues that are simple, and analytic in and . The corresponding eigenfunctions are -periodic in , and analytic in and .

The result in Theorem 2.4 is a direct consequence of the general theory of perturbation of self-adjoint operators (Rellich ). However in Section 3.2, we provide a straightforward alternate proof by means of the implicit function theorem, which also serves to motivate the proof of the following result.

###### Theorem 2.5.

In the neighbourhood of the crossing points , i.e for , the spectrum of on the domain is composed of an increasing sequence of eigenvalues which are continuous in . For , the lowest eigenvalue is simple, and it and the eigenfunction are analytic in and .

Both Theorems 2.4 and 2.5 are local in . Their domains of definition overlap on the intervals .

By uniqueness, in these intervals the eigenvalues and eigenfunctions agree. Hence they are globally defined periodic functions of in the interval . We will focus on results concerning the opening of spectral gaps at for eigenvalues and ; this is the topic of Section 3.3. The analysis near the double points is similar.

An illustration of eigenvalues as functions of is given in Figure 1. The left hand side shows the unperturbed first five eigenvalues in the case of a flat bottom labeled in order of magnitude. The right hand side shows these eigenvalues in the presence of a small generic bottom perturbation and the gap openings. Figure 1. First five eigenvalues in order of magnitude: (left) flat bottom; (right) in the presence of a small generic bottom perturbation. The vertical axis is the Λ axis, and the spectra of the operators G and G[b] are represented by the vertical red line and the red intervals, respectively.

## 3. Gap opening

### 3.1. A finite-dimensional model of gap opening

We describe now, on a simplified model, the mechanism through which there is the opening of a gap between the two eigenvalues and near in the presence of a periodic bathymetric variation . Denote by the orthogonal projection in onto the subspace spanned by and decompose the operator as

 (3.1) Gθ+M =Pn(Gθ+M)Pn+(I−Pn)(Gθ+M)Pn +Pn(Gθ+M)(I−Pn)+(I−Pn)(Gθ+M)(I−Pn) .

We consider a matrix model of the component of (3.1), showing that the presence of a periodic perturbation involving nonzero Fourier coefficients of the bathymetry leads to a gap between eigenvalues and at . This model also exhibits how the corresponding eigenfunctions and are modified. The precise model consists in dropping the three last terms in the rhs of (3.1), reducing to its first term . In addition, we simplify the correction term by replacing by its one term Taylor series approximation in , i.e. namely

 DL1[b]=−Dsech(hD)b(x)Dsech(hD) .

Recall that has a convergent Taylor expansion in powers of , for in of and the successive terms can be calculated explicitly. Acting on Fourier coefficients of a periodic function , the operator is represented by the matrix

 (3.2) A=(gn(θ)b2nsn(θ)sn(−θ)¯¯b2nsn(θ)sn(−θ)gn(−θ)) ,

where we have defined . We conjugate this matrix to diagonal form as where

 (3.3) Λ=(Λ2n(θ)00Λ2n−1(θ)) ,

and where so that is a rotation. The eigenvalues are explicitly given as

 12(gn(θ)+gn(−θ)±√(gn(θ)−gn(−θ))2+4|b2n|2sn(θ)2sn(−θ)2) .

Assuming , the eigenvalues split near , and in particular . The corresponding eigenfunctions are and , where for and for . The eigenvalues , are Lipschitz continuous in . For nonzero , both the matrix and the eigenvalues , are continuous and indeed even analytic.

### 3.2. Perturbation of single eigenvalues

We now take up the spectral problem for the full Dirichlet – Neumann operator . Consider values of the parameter in the interval , for which is a simple eigenvalue for all . With no loss of generality, suppose that even (for odd and the only change has to do with the indexing, as is the case of ). By analogy with finite dimensional problem, we seek a conjugacy that when described in terms of the Fourier transform, will reduce the operator to a matrix operator whose entries are zero in the row and column. Specifically, we seek a transformation parametrized by operators satisfying , such that in acting on Fourier series the matrix will be block diagonal. For this purpose, use the orthogonal projection onto the span of the Fourier mode in , and decompose the operator as

 (3.4) e−T(Gθ+M)eT=Pne−T(Gθ+M)eTPn+(I−Pn)e−T(Gθ+M)eTPn +Pne−T(Gθ+M)eT(I−Pn)+(I−Pn)e−T(Gθ+M)eT(I−Pn) .

We are seeking that is an anti-Hermitian operator such that the block off-diagonal components of (3.4) satisfy

 (3.5) (I−Pn)e−T(Gθ+M)eTPn+Pne−T(Gθ+M)eT(I−Pn)=0 .

The existence of such will follow from the implicit function theorem in a space of operators . Define , where

 (3.6) F1(T,M)=Pne−T(Gθ+M)eT(I−Pn) ,F2(T,M)=F∗1(T,M) , (3.7) F(T,M)=F1(T,M)+F2(T,M) .

The goal is to solve , describing as a function of in appropriate functional spaces. We look for a solution of (3.5) restricted to the space of anti-Hermitian operators with the additional mapping property that

 (3.8) T(PnL2)⊆(I−Pn)L2,andT((I−Pn)L2)⊆(PnL2) .

The following analysis is performed in Fourier space coordinates, which is to say in a basis given in terms of Fourier series. Denote the space of -periodic functions in , represented in Fourier series coordinates. Alternately using Plancherel, this characterizes by its sequence of Fourier coefficients;

 hr={(ψl)l∈Z : ∑l∈Z⟨l⟩2r|ψl|2<+∞} ,

where as usual we write .

In Fourier space variables, operators have a matrix representation in the basis , defined by

 Aeilx=∑j∈ZAjleijx .

For the most part, we will be concerned with the Hermitian and anti-Hermitian operators. A scale of operator norms for Hermitian (and anti-Hermitian) operators represented in Fourier coordinates is given by

 ∥A∥r=supj(∑l∈Z|Ajl|⟨j−l⟩r) .

This norm quantifies the off-diagonal decay of the matrix elements of . The identical expression for a norm is used for anti-hermitian operators , while for a general operator we need to use

 ∥Y∥r:=(supj∑k|Yjk|⟨j−k⟩r)1/2(supk∑j|Yjk|⟨j−k⟩r)1/2.

The space of linear operators from the Sobolev space to itself that have finite -norm is denoted by . The space of Hermitian symmetric operators with finite -norm is denoted by while the space of anti-Hermitian symmetric operators with finite -norm is denoted by . When , this norm dominates the usual operator norm on , while for the expression gives a norm which is a bound for . One notes that this is indeed a proper operator norm, such that . In fact if this same inequality holds when considering for any .

###### Proposition 3.1.

Taking , then for all , the Fourier representation of the operator defined in (2.1) satisfies

 ∥M∥r=supj(∑l∈Z|Mjl|⟨j−l⟩r)≤Cr(|b|C1)=O(|b|C1) .

This bound follows directly from Proposition 2.3.

We also must consider unbounded operators on sequence spaces, for instance , for which the operator norm that we use is given by

 ∥A∥hr→hr−1:=(supj∑k|Ajk|1⟨j⟩⟨j−k⟩r)1/2(supk∑j|Ajk|1⟨j⟩⟨j−k⟩r)1/2.

It is worth the remark that our diagonal operator satisfies .

###### Proposition 3.2.

For and the operator composition satisfies

 ∥AB∥hr→hr−1≤∥A∥hr→hr−1∥B∥r

Proof: We note that in our proof, we principally encounter Hermitian and/or anti-Hermitian operators and . Their product is not of this form, however, and we must use the general expression for the norm for operators . Take and as in the proposition, where we suppose that and are finite, and calculate

 ∥AB∥hr→hr−1 =(supj∑k∣∣∑lAjlBlk∣∣1⟨j⟩⟨j−k⟩r)1/2(supk∑j∣∣∑lAjlBlk∣∣1⟨j⟩⟨j−k⟩r)1/2 ≤(supj∑k∑l|Ajl|1⟨j⟩⟨j−l⟩r|Blk|⟨l−k⟩r(⟨j−k⟩r⟨j−l⟩r⟨l−k⟩r))1/2 ×(supk∑j∑l|Ajl|1⟨j⟩⟨j−l⟩r|Blk|⟨l−k⟩r(⟨j−k⟩r⟨j−l⟩r⟨l−k⟩r))1/2 .

Since

 ⟨j−k⟩r⟨j−l⟩r⟨l−k⟩r≤1 ,

then the rhs is bounded by . ∎

In addition, the family of operators defined in (3.7) satisfies the mapping properties (3.8); these properties define a linear subspace . The operator defined in (2.1) is a bounded Hermitian symmetric operator, as expressed in Proposition 2.1 and Proposition 3.1. We seek a solution of (3.5) in Fourier coordinates, where , which also satisfies the mapping property (3.8). The set of anti-Hermitian operators satisfying (3.8) is a linear subspace (it is not however closed under operator composition).

Describing in terms of its matrix elements in Fourier variables, because of the property (3.8), , while for then . Thus the operator , identified by its matrix elements , is nonzero only in the row and column, thus its operator norm is given by

 ∥T∥r=∑l≠n|Tnl|⟨n−l⟩r .

Similarly, the operator is defined in terms of its matrix elements by its action on Fourier modes, .

###### Theorem 3.3.

Let , , and defined as in (2.1). There exists and a continuous map

 M∈Bρ(0)⊆Hr→T=T(M)∈Pr⊆Ar

such that satisfies (3.8) and for all . Since is analytic in , for , then is also analytic in .

The result is that is a unitary transformation that conjugates the operator to diagonal on the eigenspace associated with the eigenvalues and .

The proof of Theorem 3.3 is an application of the implicit function theorem, for which we need to verify the hypotheses. As a starting point, clearly since is diagonal in a Fourier basis. We proceed to verify the relevant properties of the mapping and its Jacobian derivative . This is the object of Lemma 3.4.

###### Lemma 3.4.

Let . (i) For an open subset of containing , , which is continuous in , and in .

(ii) The Frechet derivative at is invertible, namely for , there is a unique such that

 ∂TF(0,0)V=N .
###### Proof.

We will show that has the required smoothness and invertibility properties as an element of , in order to invoke the implicit function theorem. Firstly, the function is built of operators with the following properties: Firstly the operator maps to , and it is diagonal in the Fourier basis. The operator , in fact it is unitary. Also, . Because is unbounded, We also have

 e−T(Gθ+M)eT∈Hhr→hr−1 ,

therefore

 F:=Pn[e−T(