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3 Acknowledgments I wish to express my deep gratitude to my thesis advisor Professor V.Matsaev for his valuable guidance, essential support and permanent attention. It is also a pleasure to thank to Professor N.Kopachevsky who introduced me to the wonderful world of Mathematics and support me during many years. I am also indebted to Professor I.Gohberg for many suggestions and encouragement. I thank the staff of Tel Aviv University for the permanent help in technical problems. Last but not least I want to thank my wife Natalie, my parents and my children for patience and warm moral support. i

4 Abstract The Floquet approach is a main tool of the theory of linear ordinary differential equations (o.d.e.) with periodic coefficients. Such equations arise in many physical and technical applications. At the same time, a lot of theoretical and applied problems that appear in quantum mechanics, hydrodynamics, solid state physics, theory of wave conductors, parametric resonance theory, etc. lead to periodic partial differential equations. The case of ordinary periodical differential equations is in a sharp contrast with a partial one, because a space of solutions of partial differential equation is, generally, an infinite-dimensional space. Methods of the classical Floquet theory (i.e. method of monodromy operator and method of substitution) in general are not applicable and new tools and technique are necessary. The main idea of this thesis is to use the old technique of monodromy operator but from a new point of view. This approach turned out quite effective in the case of elliptic equations periodic with respect to one of variables and allowed to obtain in this case the same results as in the classical o.d.e. one (so, basis property of the set of Floquet solutions and distribution of Floquet multipliers). In Chapter 2 we define the monodromy operator for general elliptic selfadjoint periodic problem as a shift operator on the space of solutions of this problem, similarly to the ordinary case. It is very difficult to deal with this operator since its definition is connected with solving the Cauchy problem for elliptic equations but we overcome this difficulty by considering appropriate elliptic boundary value problem instead of the Cauchy problem. In this way we reduce the spectral problem for the monodromy operator to the spectral problem for the quadratic operator pencil. Then we study this pencil by indefinite inner product techniques. The most of results obtained are related to the symmetric case, that is the case when the domain of the problem has a plane of symmetry and all coefficients of the problem are even w.r.t. this plane. ii

5 The main goal of Chapter 3 is the detailed studying the monodromy operator in the general case (without symmetry). First, we prove that the monodromy operator is closed and has a trivial kernel and a dense domain. Then we establish the quasiisometric property for the monodromy operator and obtain some useful corollaries related to Floquet multipliers and Floquet solutions of the problem. Next, we study properties of Floquet multipliers (that is, spectral properties of the monodromy operator) in a more delicate way. For this purpose we classify the multipliers. Such a classification is standard in the indefinite scalar product theory. This approach has been applied by M.G.Krein for periodic systems of linear differential equations (i.e., in a finite-dimensional case). In the next (and last) Chapter 4 we deal with elliptic selfadjoint periodic problems with a parameter. We are interested in a motion of the multipliers of such problems. Behavior of multipliers under small perturbation is very important in various theoretical and applied problems, generally speaking it is important in all problems that lead to periodic partial differential equations. To describe the motion of the multipliers we use the results obtained in the previous part of our work. iii

6 Contents Acknowledgments Abstract i ii 1 Introduction 2 2 Distribution of Floqet multipliers, property of the set of Floquet solutions The case of Schrödinger type equation The general case Symmetric case and indefinite metric approach The properties of the monodromy operator The basic properties of the monodromy operator Quasi-unitary property of the monodromy operator Classification of the spectrum The problem with a parameter. Motion of multipliers 45 Bibliography 51 1

7 Chapter 1 Introduction 1. In this thesis we consider second order elliptic problems periodic with respect to one of variables and the Floquet theory for such problems. The Floquet approach is a main tool of the theory of linear ordinary differential equations (o.d.e.) with periodic coefficients. Such equations arise in many physical and technical applications (see e.g. [17]). The following result, known as the Floquet- Lyapunov theorem, is central in this theory. Consider the system dy dt = A(t) y, (1.1) where A(t) is a 1-periodic n n matrix-function, i.e. A(t + 1) = A(t) for all t. Then in the space of solutions of system (1.1) there exists a basis that consists of functions of the form r y(t) = e µt u k (t)t k, u k (t + 1) = u k (t) for all t, k. (1.2) k=0 In another words, there is a substitution x = F (t)y with an invertible 1-periodic n n matrix F (t) that converts (1.1) into a system with constant coefficients and Y (t) = e tk F (t) (here Y (t) is a fundamental system of solutions for (1.1) and K is a n n constant matrix). Non-zero complex numbers λ = e µ are said to be Floquet multipliers of equation (1.1). It is not difficult to show that the Floquet multipliers are eigenvalues of monodromy operator acting on the space of solutions of problem (1.1) by U(y(t)) = y(t + 1). (1.3) 2

8 CHAPTER 1. INTRODUCTION 3 Moreover, the set of Floquet solutions (i.e. solutions of the form (1.2)) of problem (1.1) corresponds to the Jordan representation of the monodromy operator. As is well known [3, 17, 24] one can deduce numerous properties of equation (1.1) (stability, solvability of nonhomogeneous equations, exponential dichotomy, spectral theory, etc.) from the distribution of Floquet multipliers. 2. At the same time, a lot of theoretical and applied problems that appear in quantum mechanics, hydrodynamics, solid state physics, theory of wave conductors, parametric resonance theory, etc. lead to periodic partial differential equations. The first result of this type has been proved by Bloch [5]. It pertains to Bloch waves which are plane waves multiplied by periodic functions, and they have formed the basis of the theory of electrons in crystals, i.e. of the theory of solids. Further numerous extensions and generalizations of Bloch s idea gave a good tool for investigation the Schrödinger equation and related topics, see e.g. [15, 26, 29]. However, the case of ordinary periodical differential equations is in a sharp contrast with a partial one, because a space of solutions of partial differential equation is, generally, infinite-dimensional space. Methods of the classical Floquet theory (i.e. method of monodromy operator and method of substitution) in general are not applicable and new tools and technique are necessary. Recently this problem attracts a big deal of attention, see e.g. monograph [21] and references there in. Series of deep results analogous to those from the classical Floquet theory (e.g. completeness of Floquet solutions in special classes of solutions of periodic partial differential equations) are received for elliptic and hypoelliptic equations and systems, parabolic problems, etc. However, the results obtained are not complete. The main idea of our work is to use the old technique of monodromy operator but from a new point of view. This approach turned out quite effective in the case of elliptic equations periodic with respect to one of variables and allowed to obtain in this case numerous results concerning the set of Floquet solutions and the set of Floquet multipliers and to get the complete general picture as in the classical o.d.e. case. 3. Basis property of the set of Floquet solutions and distribution of Floquet multipliers are done in Chapter 2 of the thesis. In Sections 2.1 and 2.2 general el-

9 CHAPTER 1. INTRODUCTION 4 liptic selfadjoint problem of second order periodic with respect to one of variables is studying by monodromy operator technique. Section 2.1 is devoted to a particular case, namely to the case of Dirichlet problem for Schrödinger type equation u + qu = 0. It is the simplest case but transition to general one doesn t demand any additional idea. However to generalize our considerations some technical complication is necessary. It is done in Section 2.2. The main points of these Sections are follows. An analogue of the monodromy operator is constructed on the space of solution of elliptic problem, namely, there is an operator U that acts similarly to (1.3). One can get the answer to the questions raised above (so, basis and distribution properties) by means of studying the spectrum and the set of eigenfunctions and associated functions of the operator U. However, it is very difficult to deal with this operator since its definition is connected with solving the Cauchy problem for elliptic equation. To overcome this difficulty we consider appropriate elliptic boundary value problem instead of the Cauchy problem. Solving this problem we come to quadratic operator pencil with a spectral problem equivalent to one for the monodromy operator in certain sense. The pencil is L(λ) = λ 2 A λb + A, (1.4) where A is a compact Hilbert-Schmidt operator with trivial kernel, B is a selfadjoint bounded from below operator with compact resolvent. First conclusion on distribution of the Floquet multipliers follows from this equivalence immediately: Corollary (and Theorem 2.2.2) The set of Floquet multipliers of second order elliptic selfadjoint problem periodic with respect to one of the variables (see problem (2.13)) is or C (the whole complex plane) either a countable subset of C with accumulation points 0 and. This set is symmetric with respect to the unit circle, that is if λ is the Floquet multiplier of problem (2.13) then λ 1 is too, and the multiplicity of λ is the same as the multiplicity of λ 1. In Section 2.3 operator pencil (1.4) is studied by indefinite inner product techniques. Under one essential restriction, namely, some kind of symmetry for domain and coefficients of our problem, the main results of Chapter 1 are obtained. These results are formulated in the following two theorems:

10 CHAPTER 1. INTRODUCTION 5 Theorem ) Under the symmetry assumption (see assumption (A) in Section 2.3) the set of Floquet multipliers of second order elliptic selfadjoint problem periodic with respect to one of the variables (see problem (2.13)) is a discrete set with two accumulation points 0 and, this set has double symmetry with respect to the real axis and with respect to the unit circle; 2) the set of non-real multipliers is finite, if κ is the number (with multiplicity) of non-positive eigenvalues of operator F 22 (see Section 2.1), then exist at most 4κ multipliers belonging to neither the real axis nor the unit circle; 3) there are at most 2κ multipliers different from ±1 with Floquet solution of positive order and the order is not grater than 2κ (if λ = ±1 is a multiplier then its order can reach 4κ + 1). Theorem Under assumption (A) an arbitrary solution of problem (2.13) can be expanded in a series over Floquet solutions, the series converges in the sense of W s 2,loc(Ω) topology for all s 0 and in the sense of C loc(ω) topology. 4. In Chapter 3 we study the properties of the monodromy operator and distribution of the multipliers in more detail. First, we establish that the monodromy operator is closed and its domain and range are dense sets. It is done in Section 3.1 (see Theorem 3.1.1). In Section 3.2 we obtain a quasi-unitary property of the monodromy operator (see Theorem 3.2.2). It is the well-known property, in a finite-dimensional case it means that a transformation preserve symplectic structure defined by a skew-scalar product. In Section 3.2 we mention some useful results on spectral properties of the monodromy operator. They are direct consequences of its quasi-unitarity. Section 3.3 is devoted to classification of the multipliers. Such a classification is standard in the indefinite scalar product theory. Namely, we introduce the following definition (see Definitions and 3.3.2): A function f is called positive (negative or neutral) if i[f, f] > 0 ( i[f, f] < 0 or [f, f] = 0, where the skew-scalar product [f, f] is defined in Section 3.2 (see (3.9)). A Floquet multiplier is called multiplier of the first kind (second kind or neu-

11 CHAPTER 1. INTRODUCTION 6 tral) if all corresponding eigenfunctions are positive (negative or neutral). This classification was successfully applied for periodic differential equations by M.G.Krein. In work [20] he studied by this approach periodic systems of linear differential equations. We use here the idea but not the methods of Krein s paper because a finite-dimensional case of system of linear o.d.e is in a sharp contrast with an infinite-dimensional case of partial differential problems that we are interested on. For example, we can not represent the monodromy operator in the form of matriciant and such representation is essential for Krein s work. Therefore, we use the results obtained in the previous sections of this thesis to establish some important properties of the Floquet multipliers and the eigenfunctions of the monodromy operator and these properties are done in the terms of the classification defined above, see Propositions 3.3.3, and 3.3.6, Theorems and , Corollaries The main goal of the last Chapter 4 is to study the motion of multipliers of a periodic elliptic problem with a parameter. Here we refer, once again, to Krein s work [20]. Periodic differential problems with a parameter were studied thoroughly in this paper for systems of linear differential equations, so in a finite-dimensional case. This chapter is a generalization of [20] on an infinite-dimensional case of partial differential problems. Behavior of multipliers under small perturbation is very important in various theoretical and applied problems, generally speaking it is important in all problems that lead to periodic partial differential equations. The results obtained in Section 3.3 and standard general reasons say us that the general properties of nonunimodular multipliers do not depend on small perturbation, therefore the only case of great interests is the case of multipliers belonging to the unit circle. We assume that all coefficients of the problem (3.1) depend on a complex parameter ɛ in such a way that the monodromy operator U is an analytic function of ɛ in a neighborhood of ɛ = 0, the domain of definition of the operator U(ɛ) does not depend on ɛ and the conditions (i) - (iv) of (3.1) hold true for real values of ɛ. We assume, as well, that the energy of the problem (see (4.5)) is an increasing function of the parameter. Under these assumption we obtain the complete local description of motion of all multipliers which have not neutral eigenfunctions, namely:

12 CHAPTER 1. INTRODUCTION 7 All unimodular multipliers which have not neutral eigenfunctions remain on the unit circle under small perturbation, multipliers of the first kind move clockwise on the unit circle as ɛ increases and multipliers of the second kind move counterclockwise. Thus a multiplier can jump off the unit circle only if it has a neutral eigenfunction. The indefinite metric approach that we used in Chapter 3 and Chapter 4 was applied successfully to elliptic periodic problems by V.Derguzov, see [9]-[11]. But our case differs from Derguzov s one by three very essential points. First, Derguzov studied the problems periodic with respect to all variables and the problems periodic with respect to one of the variables, but such that the domains of the operator coefficients do not depend on this variable. It is impossible to reduce our problems to the problems of Derguzov. Second, quasi-unitary property does not imply automatically any property of the multipliers because the monodromy operator and its inverse are not bounded. To obtain the properties of the multipliers we need additional accurate reasonings, in Derguzov s works these reasonings and these properties are missing (actually, Derguzov does not prove even quasi-unitarity). Third, Derguzov studied the motion of multipliers for the case of linear dependence on the parameter only. Thus the results of Chapter 3 and Chapter 4 are new and can not be obtained from the papers of Derguzov. The part of the results of this thesis were obtained jointly with V.Matsaev.

13 Chapter 2 Distribution of Floqet multipliers, property of the set of Floquet solutions 2.1 The case of Schrödinger type equation 1. We consider the following problem: u + qu = 0 u = 0 (Ω) ( Ω), (2.1) where Ω R 3 is an infinite and 1-periodic with respect to x 1 (R 3 x = (x 1, x 2, x 3 )) domain with infinitely smooth boundary, such that any section of Ω by plane orthogonal to x 1 is bounded, q(x) is a real infinitely smooth 1-periodic with respect to x 1 function, i.e. q(x 1 + 1, x 2, x 3 ) = q(x 1, x 2, x 3 ) for all x. Similarly to (1.2) the Floquet solution of problem (2.1) is said to be a solution of the form r u(x 1, x 2, x 3 ) = e µx 1 x k 1v k (x 1, x 2, x 3 ), v k (x 1 + 1, x 2, x 3 ) = v k (x 1, x 2, x 3 ) for all x, k. As in the ordinary case the numbers e µ are called the Floquet multipliers of the problem. k=0 8

14 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 9 Properties of the Floquet solutions and the Floquet multipliers are strongly connected with spectral properties of the monodromy operator which we define in the following way. Let us take the section of Ω by a smooth enough simple surface Γ 1 non-tangential to Ω. Let Γ 2 be the shift of Γ 1 by 1 along x 1. We denote the part of Ω between these sections by Ω. Thus Ω is the elementary cell of the domain Ω or the fundamental domain of the group of shifts by z Z on Ω. Let function u(x) satisfy the equation and the boundary condition from (2.1) on the closure of the domain Ω. We define the monodromy operator as U u 1 u n1 = u 2 u n2, (2.2) where u j is the trace of u(x) on Γ j and u nj is the normal derivative of u(x) on Γ j, j = 1, 2. We take here normals in the same directions, namely n 1 (on Γ 1 ) is an internal normal for the domain Ω and n 2 (on Γ 2 ) is an external one. This definition is correct due to uniqueness theorem for Cauchy problem for second order elliptic equations (this theorem states that if u vanishes of infinite order at some point then u vanishes identically, see e.g. [16, theorem ]), thus U is the linear operator on an infinite-dimensional space. But U is an unbounded operator connected with Cauchy problem so, as mentioned in the Introduction, it is very difficult to study U by direct methods. However it is possible to overcome this difficulty. Namely, instead of Cauchy problem we consider the elliptic boundary value problem in the elementary cell Ω: u + qu = 0 u = 0 ( Ω) ( Ω) u = u 1 (Γ 1 ) u = u 2 (Γ 2 ). (2.3) Remark. The domain Ω has non-smooth boundary. The nature of the solution u of the Dirichlet problem changes as the domain becomes less smooth and the existence of the solution requires additional study. The solution is best described in terms of a notion called harmonic measure (see [8] and [18]). The domain Ω is Lipschitz domain and it is proved in [8] that on Lipschitz domain harmonic measure and surface measure are mutually absolutely continuous. Using this fact one can

15 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 10 solve the Dirichlet problem on Lipschitz domain for all u L 2 ( Ω). The details can be found in [8] and [18]. 2. In order to not complicate our reasonings by minor technical details, let us assume that homogeneous problem (2.3) (i.e. u 1 = u 2 = 0) has only the trivial solution. To overcome this unessential restriction and pass to the general case one can use generalized Green s function, for details see Remark 1 at the end of this section. Under this assumption problem (2.3) has unique (in the sense of L 2 ( Ω)) solution. Let us consider the operator F u 1 u 2 = u n 1 u n2, or in the matrix form F 11 F 12 F 21 F 22 u 1 u 2 = u n 1 u n2 Thus F acts as follows: we solve problem (2.3) and take the normal derivatives of the solution on Γ 1 and Γ 2, this couple is the image of the couple of boundary data (u 1, u 2 ) under the action of the operator F. Obviously, the operators U and F are strongly connected and we can study the spectral problem for the operator U with help of the operator F. Our nearest goal is to study properties of the operators F jk. We can identify Γ 1 and Γ 2, so let us consider the functions u k and F jk (u k ) as functions on the surface Γ 1 = Γ 2 with the standard two-dimensional Lebesgue measure. Let the operators F jk act on the space of such functions L 2 (Γ 1 ) = L 2 (Γ 2 ). Note that F 11 acts as follows: we solve (2.3) with u = 0 on Γ 2 and then take the normal derivative of the solution on Γ 1, this function is the image of the boundary data on Γ 1 under the action of F 11. Similarly, to find the image of the boundary data on Γ 1 under the action of F 21, we solve the same boundary problem as for F 11 but take the normal derivative of the solution on Γ 2. To find the image of the boundary data on Γ 2 under the action of F 22, we solve (2.3) with u = 0 on Γ 1 and then take the normal derivative of the solution on Γ 2, to find the image of the boundary data on Γ 2 under the action of F 12, we solve the same problem as for F 22 but take the normal derivative of the solution on Γ 1. The operators F jk, j = 1, 2, k = 1, 2 are closed operators. Proposition The operators F 12 and F 21 are compact Hilbert-Schmidt operators (i.e. operators of the class S 2 ) on the space L 2 (Γ 1 ) (or, the same, L 2 (Γ 2 ))..

16 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 11 Proof. Due to Green s function of problem (2.3) we have for any smooth enough solution of (2.3): G(x, y) u(y) = u 1 dx + Γ 1 n x Γ 2 u 2 G(x, y) n x Here G(x, y) is the classical Green s function of the problem u + qu = f u = 0 ( Ω) ( Ω) This Poisson s formula is the simple consequence of Green s formula for the operator of problem (2.3) (pay attention that n 1 is an internal normal for domain Ω and n 2 is an external one). This formula implies explicit formulas for the operators F jk. In particular, we have after closure for any u 1, u 2 L 2 (Γ 1 ) = L 2 (Γ 2 ). dx, F 12 (u 2 )(x) = Γ 2 2 G(y,x) n 1,x n 2,y u 2 (y) dy; F 21 (u 1 )(x) = Γ 1 2 G(y,x) n 2,x n 1,y u 1 (y) dy. (2.4) Here x y (they belong to different parts of the boundary), so the kernels of this operators are continuous. It proves our proposition. Proposition F 12 = F 21, F 11 and F 22 are symmetric operators, F 22 is bounded from below, F 11 is bounded from above. Proof. Let u(x) and v(x) be solutions of problem (2.1) and they are smooth enough (e.g. u, v C 2 (R 3 )). We apply second Green s formula for the operator u qu and the domain Ω. From (2.1) u = v = 0 on Ω, n 1 is the internal normal for the domain Ω, so we have Γ 1 u 1 v n 1 + u n 1 v 1 = Γ 2 u 2 v n 2 + u n 2 v 2. (2.5) Here u 1, u 2, v 1, v 2 are the traces of u(x) and v(x) on Γ 1 and Γ 2. But (2.3) has the unique solution, hence u(x) and v(x) are the solutions of problem (2.3) with boundary values u = u j on Γ j, v = v j on Γ j respectively. By taking u 2 = v 1 = 0 we have from (2.5) Γ 1 u 1 F 12 (v 2 ) = Γ 2 F 21 (u 1 )v 2,

17 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 12 and after closure with respect to the norm of L 2 u 1, F 12 (v 2 ) = F 21 (u 1 ), v 2. (2.6) for all u 1 L 2 (Γ 1 ), v 2 L 2 (Γ 2 ) (here, is the usual inner product in the space L 2 (Γ 1 ) (or, the same, L 2 (Γ 2 )). Similarly, by the suitable choice of u and v, we have for all functions from the domains of operators F 11 and F 22 u 1, F 11 (v 1 ) = F 11 (u 1 ), v 1 and u 2, F 22 (v 2 ) = F 22 (u 2 ), v 2. (2.7) Equalities (2.6) and (2.7) prove the symmetry properties of the operators F 11 and F 22 and the formula F 12 = F 21. It still remains to prove the semi-boundedness for the operators F 22 and F 11 First Green s formula for the Laplace operator and the domain Ω implies Γ 1 u 1 u n 1 + Γ 2 u 2 u n 2 = u u + Ω u 2 = Ω (q u 2 + u 2 ). Ω By taking u 1 = 0 we have after closure for all u 2 from the domain of the operator F 22 F 22(u 2 ), u 2 = (q u 2 + u 2 ). (2.8) Ω On the other hand for all solutions of (2.3) we have the standard estimate u L2 ( Ω) c u 2 L2 (Γ 2 ), (2.9) where c doesn t depend on u. In smooth domain this estimate follows from the classical regularity result for elliptic boundary value problem, namely: let u solve the Dirichlet problem for elliptic operator Lu = 0 on Ω and u = f on Ω then the operator A : L 2 ( Ω) L 2 ( Ω), A f = u is bounded. Our domain Ω has non-smooth boundary, however the regularity result for problem (2.3) holds true (see Remark 2 at the end of this section) and implies inequality (2.9). Let q 0 = inf q(x), x Ω. Substitution (2.9) into (2.8) gives us F 22 (u 2 ), u 2 q 0 and F 22 is bounded from below. u 2 q 0 c 2 u 2 Ω Γ 2 = c 1 u 2 2 L 2 (Γ 2 ) 2

18 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 13 By taking u 2 = 0 we have from first Green s formula F 11 (u 1 ), u 1 = (q u 2 + u 2 ) Ω instead of (2.8) and the same arguments prove now boundedness of F 11 from above. The proposition is proved. Proposition F 11 and F 22 are selfadjoint operators with compact resolvent. Proof. Let us prove this statement for the operator F 22. We prove it under assumption Ker F 22 = {0} (see arguments after (2.3) in behalf of this unessential restriction and see Remark 1 at the end of this section for it overcoming). Hence we have to prove F 1 22 is compact. Consider the following problem: u + qu = 0 ( Ω) u = 0 ( Ω Γ 1 ) u = ϕ (Γ n 2). (2.10) The operator F 1 22 solves problem (2.10) and takes the trace of the solution on Γ 2, so F 1 22 (ϕ) = u 2. Problem (2.10) is a mixed problem with non-smooth boundary, there is a lot of papers devoted to regularity properties of such problems, we refer to the results of [28]. The main theorem of this paper implies boundedness of the operator T of problem (2.10) (so, T ϕ = u) as operator from W 1/2 2 (Γ 2 ) to W 1 2 ( Ω) (here W s 2 is a notation for the Sobolev space). We can present F 1 22 as F 1 22 = V 2 γ T V 1, where V 1 is the embedding operator from L 2 (Γ 2 ) to W 1/2 2 (Γ 2 ), V 2 is the embedding operator from W 1/2 2 (Γ 2 ) to L 2 (Γ 2 ), γ is a trace operator, γu = u 2, γ acts from W2 1 ( Ω) to W 1/2 2 (Γ 2 ). It follows from the classical embedding theorems that the operators V 1 and V 2 are compact and the operator γ is bounded. Hence F22 1 is compact operator on L 2 (Γ 2 ). The first statement of this proposition (selfadjointness) follows from the second one proved above and from symmetry of F 22 (symmetric operators with compact resolvent is selfadjoint, obviously). The proposition is proved for F 22. The proof for F 11 is the same. Next proposition, the last one, follows immediately from the uniqueness theorem for Cauchy problem.

19 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 14 Proposition The operators F 12 and F 21 have trivial kernel. Indeed, if F 21 (u 1 ) = 0, we get for the operator u + qu Cauchy problem with vanishing initial data on Γ 2, due to uniqueness its solution is trivial, so u 1 (which is the trace of this solution on Γ 1 ) is zero function. The same arguments are valid for the operator F Let us return to the spectral problem for monodromy operator (2.2). As we mentioned in the Introduction, to describe properties of the set of Floquet solutions and the set of Floquet multipliers one can study properties of eigenvalues and eigenfunctions of this operator, these two problems are equivalent (see Section 2.3 for details). So, we want to study the problem U u 1 u n1 = λ u 1 u n1 or, the same u 2 u n2 = λ If we put u 1 = u, u n1 = v, the last equation is equivalent to F u = v λu λv After multiplying the first equation by λ we have or u 1 u n1 F 11 u + λf 12 u = v F 21 u + λf 22 u = λv. λf 11 u + λ 2 F 12 u = F 21 u + λf 22 u.. (2.11) Denote F 12 = A, (F 22 F 11 ) = B. The last consideration and Propositions prove the following result: Theorem The spectral problem for monodromy operator (2.2) is equivalent to the spectral problem for the quadratic operator pencil L(λ) = λ 2 A λb + A, (2.12) where A is a compact Hilbert-Schmidt operator with trivial kernel, B is a selfadjoint bounded from below operator with compact resolvent. The equivalence of these problems means the following: operator (2.2) and pencil (2.12) have the same set of eigenvalues and u is an eigenfunction (associated function) corresponding to the eigenvalue λ of (2.12) if and only if it is a first component of the eigenfunction (associated function) corresponding to the same eigenvalue of (2.2).

20 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 15 For eigenvalues and eigenfunctions the statement is done above, for associated functions one can verify this statement by simple direct calculation, we omit it here. The first conclusion on distribution of the Floquet multipliers follows from this theorem immediately: Corollary The set of Floquet multipliers of problem (2.1) is or C (the whole complex plane) either a countable subset of C with accumulation points 0 and. This set is symmetric with respect to the unit circle, that is if λ is the Floquet multiplier of problem (2.1) then λ 1 is too, and the multiplicity of λ is the same as the multiplicity of λ 1. Indeed, due to Theorem the set of Floquet multipliers (or, the same, the set of eigenvalues of monodromy operator (2.2)) coincides with the set of eigenvalues of pencil (2.12). Proposition implies existence of λ 0 such that (B λ 0 I) 1 is compact. Multiplying (2.12) by λ 1 (B λ 0 I) 1 (this operation does not change the set of nonzero eigenvalues) we get the analytic operator valued function on the domain C \ {0}, L 1 (λ) = 1 λ L(λ)(B λ 0I) 1 = λa(b λ 0 I) 1 I λ 0 (B λ 0 I) λ A (B λ 0 I) 1, and for every λ C \ {0} L 1 (λ) is Fredholm operator of index zero. The first conclusion of the Corollary follows now from the classical result on spectrum of analytic Fredholm operator valued functions (see e.g. [12, theorem XI.8.2]) and Proposition (to show that λ = 0 is not eigenvalue and so the set of different eigenvalues is infinite). To check the second conclusion of the Corollary we assume that λ is the Floquet multiplier of multiplicity r of problem (2.1). Then due to Theorem λ is an eigenvalue of multiplicity r of the pencil L(λ) = λ 2 A λb + A. But (L(λ)) λ 2 = A λ B + λ A = L(λ ). Therefore, λ 1 is an eigenvalue of multiplicity r of (2.12) and then it is Floquet multiplier of the same multiplicity of problem (2.1). The Corollary is proved. We note that the alternative mentioned above (the set of Floquet multipliers is or C either a countable subset of C with accumulation points 0 and ) is known

21 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 16 for various elliptic periodic problems, see for example [21, theorems 5.4.4, 5.4.9], [9], [10]. It is possible to deduce the first conclusion of the Corollary from these results, but instead, we prefer to give the direct proof, much more simple. 4. Remark 1. There were two restrictions done in the previous considerations. As was mentioned, one can overcome these restrictions easily. The first one is the uniqueness of solution of boundary value problem (2.3). If it is not so, problem (2.3) has a finite dimensional (due to Fredholmness) kernel and all results obtained above are valid with two minor changes: firstly, the operator F takes normal derivatives (on Γ 1 and Γ 2 ) of the unique solution of (2.3), which has the boundary values u 1, u 2 and belongs to orthogonal complement to the kernel of problem (2.3), and secondly, to get Poisson s formula (see the proof of Proposition 2.1.1) we have to use so-called generalized Green s function instead of the classical one, see e.g. [7, section V.14]. The second restriction was done in the proof of Proposition We assumed that F 22 has the trivial kernel. As above, if it is not the case, the operator F 22 has a finite dimensional (due to Fredholmness) kernel and F 22 = diag[p F 22 P, (I P )F 22 (I P )], where P is an orthogonal projection on this kernel. The operator F 22 = (I P )F 22 (I P ) is invertible on the complement to the kernel and we can 1 apply all arguments of the proof of Proposition to prove F 22 is compact, hence (the kernel of F 22 has finite dimension!) F 22 has a compact resolvent and Proposition is valid. Remark 2. The following regularity result has been used in this section. Let u solve the Dirichlet problem for elliptic operator Lu = 0 on Ω and u = f on Ω (see Remark after (2.3)) then the operator A : L 2 ( Ω) L 2 ( Ω), A f = u is bounded. The domain Ω is Lipschitz domain and regularity properties for boundary value problems on Lipschitz domains generalize the classical results for smooth domains. The statement formulated above follows immediately from [18, pp.62-63]. 2.2 The general case 1. The main goal of this section is to extend the results obtained in the previous one to a more general selfadjoint case. Namely, we consider a second order boundary

22 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 17 problem of the following form: 3 Lu = L(x, D)u = T u = 0 ( ajk x j (x) u j,k=1 x k ) + a0 (x) u = 0 (Ω) ( Ω), (2.13) where (i) Ω R 3 is an infinite and 1-periodic with respect to x 1 (R 3 x = (x 1, x 2, x 3 )) domain with infinitely smooth boundary, such that any section of Ω by plane orthogonal to x 1 is bounded, (ii) all coefficients of L and T are infinitely smooth 1-periodic with respect to x 1 functions, (iii) L is an elliptic operator and the differential expression of L is formally selfadjoint, that is a jk (x) = a kj (x) for all x and j, k = 1, 2, 3 and a 0 (x) is a real valued function, (iv) T is a boundary differential operator of the first order or of the order zero, T satisfies so-called normality and covering conditions (see e.g. [22, Chapter 2, Section 1]), T is formally selfadjoint with respect to Green s formula. The last condition means that for any sections of Ω by smooth enough non intersecting surfaces α 1 and α 2 there exists boundary operator S, ordert +orders = 1, such that for any smooth enough functions u(x) and v(x) vanishing on α 1 and α 2 we have after integration by parts ( ) 3 u v (Lu)v = a jk a 0 uv + SuT v. (2.14) Ω Ω x j x k Ω j,k=1 Here Ω is the part of Ω between the sections α 1 and α 2 and Ω is the part of Ω between these sections. Thus (2.13) is a regular elliptic problem, periodic with respect to x 1 and selfadjoint. The definitions of the Floquet solutions and the Floquet multipliers for problem (2.13) are the same that in the model problem, see the beginning of Section We define the monodromy operator in the same way as in Section 2.1. So, we take the sections of Ω by smooth enough simple surfaces Γ 1 and Γ 2 non-tangential to Ω, where Γ 2 is the shift of Γ 1 by 1 along x 1, and define the elementary cell Ω as the part of Ω between these sections. For functions u(x) that satisfy the equation

23 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 18 and the boundary condition from (2.13) on the closure of the domain Ω we define the monodromy operator as U u 1 u ν1 = u 2 u ν2. (2.15) Here u j is the trace of u(x) on Γ j, u νj = u on Γ ν j, j = 1, 2, and ν is so-called conormal for the operator L, that is by definition ν = 3 j,k=1 a jk N j, x k where N = (N 1, N 2, N 3 ) is a unit vector of an internal (with respect to Ω) normal on Γ 1 and of an external normal on Γ 2. Thus ν 1 (on Γ 1 ) is an internal conormal for the domain Ω and ν 2 (on Γ 2 ) is an external one, and definition (2.15) coincides with definition (2.2) for the model case up to change the normals n 1, n 2 to conormals ν 1, ν 2. By standard technique integration by parts one can obtain (using (2.14)) the following generalized Green s formulas for the operator L on the domain Ω (pay attention that ν 1 is an internal conormal for the domain Ω and ν 2 is an external one): here (Lu)v = a(u, v) + Ω a(u, v) = Γ 2 u ν2 v 2 3 Ω j,k=1 ( Γ 1 u ν1 v 1 + Ω SuT v, ) u v a jk a 0 uv. x j x k This formula implies immediately the second one Ω (Lu)v u(lv) = Γ 2 (u ν2 v 2 u 2 v ν2 ) Γ 1 (u ν1 v 1 u 1 v ν1 ) Here Ω is a part of Ω between Γ 1 and Γ 2. (2.16a) + Ω (SuT v T usv). (2.16b) Repeating the arguments of Section 2.1 we replace Cauchy problem connected with definition (2.15) by the elliptic boundary value problem in the elementary cell Ω (see Remark after (2.3): Lu = 0 ( Ω) T u = 0 ( Ω). (2.17) u = u 1 (Γ 1 ) u = u 2 (Γ 2 )

24 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS Reasoning as in Section 2.1, we assume that the homogeneous problem has only the trivial solution 1 and consider the operator F u 1 u 2 = u ν 1 u ν2 here u is a solution of problem (2.17), F is 2 2 operator-matrix F = (F jk ), j, k = 1, 2. Thus we just repeat the arguments of Section 2.1 with some technical complication. Producing in this manner let us prove the following Proposition Propositions are valid for the operators F jk defined above. Proof. Properly speaking, we have to repeat all the arguments from the proofs of Section 2.1 with the following modifications and remarks. In the same way as was done in the proof of Proposition we get Poisson s formula for solutions of (2.17) u(y) = Γ 1 u 1 G ν1 dx Here G(x, y) is Green s function of the problem Lu = f T u = 0 ( Ω) ( Ω), u = 0 (Γ 1 Γ 2 ) Γ 2 u 2 G ν2 dx. (2.18) and we get this Poisson s formula by the simple substitution of G instead of v in (2.16b) as u is the solution of (2.17). This formula implies explicit formulas for F jk, just the same as (2.4) up to replacement of operator by operator n ν Next, if u and v are solutions of (2.17), we have from (2.16b) u ν1 v 1 u 1 v ν1 Γ 1 = u ν2 v 2 u 2 v ν2. Γ 2 This formula plays the same role as (2.5) in the proof of Proposition Instead of (2.8) we have from (2.16a) for u = v (u is a solution of (2.17)) a(u, u) = 3 Ω j,k=1 ( ) u u a jk a 0 u 2 x j x k = Γ 2 u ν2 u 2 Γ 1 u ν1 u 1 1 one can overcome this restriction in the same manner as in the model case, see Remark 1 at the end of Section 2.1

25 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 20 and (by taking u 1 = 0) F 22 (u 2 ), u 2 = a(u, u) Here we use the fact that ellipticity of L implies a 0 u 2 c 0 u 2, c 0 = inf x Ω ( a 0 (x)). Ω Ω 3 j,k=1 a jk u x j u x k 0 for all u smooth enough, for all x Ω. Now we can proceed in the same way as in the proof of Proposition Next, we replace problem (2.10) by the problem Lu = 0 ( Ω) T u = 0 ( Ω). u = 0 (Γ 1 ) u = ϕ (Γ ν 2) It is the unique change that we have to do in the proofs of Propositions and The proposition is proved. 4. To study properties of the Floquet solutions and the Floquet multipliers we reduce the spectral problem for monodromy operator (2.15) to the spectral problem for a quadratic operator pencil. We do it in the same manner as in Section 2.1. Instead of spectral problem (2.11) we have U u 1 u ν1 = λ u 1 u ν1 or, the same The last equation we can write in the form or u 2 u ν2 = λ F u = v where u 1 = u, u ν1 = v, λu λv F 11 u + λf 12 u = v F 21 u + λf 22 u = λv. u 1 u ν1 After multiplying the first equation by λ we get once again quadratic pencil (2.12) with the properties described in Theorem We summarize the consideration of this section by following.

26 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 21 Theorem Theorem is valid for monodromy operator (2.15), so the set of Floquet multipliers of problem (2.13) with properties (i)-(iv) coincides with the set of eigenvalues of pencil (2.12), the set of Floquet solutions corresponds in a very natural way to the set of eigenfunctions and associated functions of this pencil, namely u(x 1, x 2, x 3 ) is a Floquet solution of (2.13) of order r that corresponds to a Floquet multiplier λ = e µ if and only if u 1 = u Γ1 is the first component of r- th associated function (eigenfunction, if r = 0) of monodromy operator (2.15) that corresponds to its eigenvalue λ or, the same, if and only if u 1 is an r-th associated function (eigenfunction, if r = 0) of operator pencil (2.12), that corresponds to its eigenvalue λ. Corollary is also valid for problem (2.13). 2.3 Symmetric case and indefinite metric approach 1. In this section we study problem (2.13) with properties (i)-(iv) under one additional restriction, namely we assume that the domain Ω has a plane of symmetry orthogonal to x 1 and all coefficients of L and T are even functions with respect to this plane. (A) Let us choose the elementary cell Ω to be symmetric with respect to this plane (then Γ 1 and Γ 2 are planes orthogonal to the boundary Ω). Under assumption (A) the functions G ν1 and G ν2 from (2.18) are equal so that F 12 = F 21 and F 11 = F 22. This fact together with Proposition gives F 12 = F12 or, the same, the operator A from (2.12) is selfadjoint. Thus we have the pencil L(λ) = λ 2 A λb + A, (2.19) where A is a compact selfadjoint operator with trivial kernel on the space L 2 (Γ 1 ) or, the same, L 2 (Γ 2 ), B = 2F 22 is a selfadjoint bounded from below operator with compact resolvent on the same space. The following theorem on spectral properties of pencil (2.19) is proved by indefinite metric technique. Theorem 2.3.1

27 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 22 1) The spectrum of operator pencil (2.19) consists of two branches of its eigenvalues {λ 1j } and {λ 2j }, λ 1j 1, λ 2j 1, λ 1j λ 2j = 1, {λ 1j } tends to infinity, {λ 2j } tends to zero; 2) if κ is the number (with multiplicity) of non-positive eigenvalues of operator B, there are at most 2κ (with multiplicity) non-real eigenvalues of (2.19) inside the unit circle (and, so, at most 2κ outside the unit circle) and non-real spectrum is symmetric with respect to the real axis; 3) there are at most 2κ eigenvalues different from ±1 and such that corresponding eigenfunctions have associated functions and the length of a chain of eigenfunction and associated functions cannot be grater than 2κ + 1 (if λ = ±1 is an eigenvalue of (2.19) than corresponding eigenfunctions always have associated functions and the length of a chain can reach 4κ + 2); 4) the same system of eigenfunctions and associated functions corresponds to each branch of eigenvalues, this system forms a Riesz basis in the energetic space of the operator B (see the proof). Proof. We shall first prove the theorem in the case KerB = {0}. From (2.19) the spectral problem L(λ)u = 0 is equivalent to the problem (λ 2 + 1)Au = λbu. By the substitution µ = λ/(λ 2 + 1) we obtain the linear pencil Au = µ Bu. (2.20) If B is positive we get immediately the spectral problem for the compact selfadjoint operator B 1/2 AB 1/2 ϕ = µ ϕ, ϕ = B 1/2 u. In the general case B has a finite number of negative eigenvalues and we proceed in the following way. Denote by H + ( H ) the linear span of all eigenfunctions of B corresponding to the positive (negative) eigenvalues. Since B is a selfadjoint bounded from below operator with compact inverse, we obtain the following decomposition: According to this decomposition B = B B L 2 (Γ 1 ) = H + H, dimh = κ <., B ± > 0, B is an operator of finite rank.

28 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 23 Let us denote J = P + P = I 0 0 I and B = B B ( here P ± are projections on H ± ), Thus B = B 1/2 J B 1/2 and after the substitution u = B 1/2 ϕ and multiplication by B 1/2 from the left we obtain from (2.20) or, multiplying by J, ˆL(µ)ϕ = (K µj)ϕ = 0, K = B 1/2 A B 1/2, (JK)ϕ = µϕ, K is compact and selfadjoint. (2.21) Problem (2.21) is a spectral problem for a compact selfadjoint operator on Pontrjagin space Π κ, it is the well known problem in the theory of indefinite scalar product spaces and from the classical results (see e.g. [6, cor.vi.6.3, th.ix.4.6,4.8,4.9]) we obtain the following properties of the spectrum of (2.21): the spectrum is discrete with one accumulation point zero, it is symmetric with respect to the real axis, total algebraic multiplicity of eigenvalues from the upper (lower) half-plane is at most κ, there are no more than κ eigenvalues such that corresponding eigenfunctions have associated functions and maximal length of a Jordan chain of a real eigenvalue is 2κ + 1. In addition, from [4, theorem IV.2.12, remark IV.2.13] follows that the system of eigenfunctions and associated functions of (2.21) forms an unconditional (Riesz) basis in the Hilbert space of all functions {ϕ = B 1/2 u u L2 (Γ 1 ) = L 2 (Γ 2 )}. Now return to problem (2.19) by the inverse substitutions λ 1,2 (µ) = 1 ± 1 4µ 2 ; ϕ = B 1/2 u. 2µ It is obvious that λ 1 λ 2 = 1 and λ 1,2 (µ) = λ 1,2 (µ). Thus for each real eigenvalue µ of (2.21) such that µ < 1/2 we have a pair of conjugate with respect to the unit circle real eigenvalues of (2.19), for each real eigenvalue µ of (2.21) such that µ > 1/2 (there exists a finite number of such µ!) we have a pair of complex conjugate eigenvalues of (2.19) on the unit circle and

29 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 24 for each pair of complex conjugate eigenvalues of (2.21) (there exist at most κ such pairs) we have 4 eigenvalues of (2.19) and they are symmetric with respect to the real axis and with respect to the unit circle. Moreover, it is easy to see (e.g., from [25, lemma 11.3]) that the space of eigenfunctions and associated functions corresponding to eigenvalues of (2.21) different from ±1/2 is preserved under the substitution µ = µ(λ), so the system of eigenfunctions and associated functions that corresponds to each of two branches of eigenvalues of (2.19) different from ±1 (the same system for each branch) coincides with the system of eigenfunctions and associated functions of (2.21) (for µ ±1/2) up to the substitution ϕ = B 1/2 u. The case µ = ±1/2 is exceptional. The direct calculation shows that if µ = ±1/2 is an eigenvalue of (2.21) of multiplicity m then λ = ±1 is an eigenvalue of (2.19) of multiplicity 2m and this eigenvalue has a double system of eigenfunctions and associated functions. It means that if u belongs to this system than u appears in this system twice up to multiplication by scalar and ϕ = B 1/2 u belongs to the root subspace of µ = ±1/2. Thus we can split the eigenvalue λ = ±1 in such a way that it will belong to both branches of eigenvalues of (2.19) and corresponding subspace of eigenfunctions and associated functions, the same for each branch, coincides with the root subspace of the eigenvalue µ = ±1/2 of (2.21) up to the substitution ϕ = B 1/2 u. Thus in any case the same system of eigenfunctions and associated functions corresponds to each of two branches of eigenvalues of (2.19) and it coincides with the system of eigenfunctions and associated functions of (2.21) up to the substitution ϕ = B 1/2 u and for this reason it forms a Riesz basis in the space with the norm = B 1/2, B 1/2, that is in the so-called energetic space of the operator B. This completes the proof in the case KerB = {0}. If KerB {0}, denote by P 0 the projection on KerB, then P 1 = I P 0 is the projection on KerB = ImB. For λ ±i problem (2.19) is equivalent to the problem (λ 2 + 1)A 1 u = λb 1 u, (2.22) where A 1 = P 1 AP 1 A 0, A 0 is a selfadjoint operator of finite rank and B 1 = P 1 BP 1. The operators A 1 and B 1 have the same properties as A and B and KerB 1 = {0}, therefore all the arguments given above are valid for (2.22) on the space ImB. To complete the proof we add to the set of eigenvalues of (2.22) λ = ±i, its eigenspace from (2.19) is exactly KerB.

30 CHAPTER 2. DISTRIBUTION OF FLOQUET MULTIPLIERS 25 The theorem is proved. Remark. If the spectrum of the pencil (2.12) does not coincide with the whole complex plane then the property 1) of this Theorem is valid without symmetry condition (A), so it is valid in the general case (see Corollary and Theorem 2.2.2). 2. We are ready to prove now the main results of this paper. We recall that a solution of (2.13) of the form r u(x 1, x 2, x 3 ) = e µx 1 x k 1v k (x 1, x 2, x 3 ), k=0 v k (x 1 + 1, x 2, x 3 ) = v k (x 1, x 2, x 3 ) for all x, k is called the Floquet solution of order r that corresponds to the Floquet multiplier e µ. The direct calculation and Theorem show that u(x 1, x 2, x 3 ) is a Floquet solution of (2.13) of order r that corresponds to a Floquet multiplier λ = e µ if and only if u 1 = u Γ1 is the first component of r-th associated function (eigenfunction, if r = 0) of monodromy operator (2.15) that corresponds to its eigenvalue λ or, the same, if and only if u 1 is an r-th associated function (eigenfunction, if r = 0) of operator pencil (2.12) (operator pencil (2.19), if assumption (A) is fulfilled), that corresponds to its eigenvalue λ. Thus from Theorem we obtain the following result on distribution of the Floquet multipliers of our problem: Theorem ) Under assumption (A) the set of Floquet multipliers of problem (2.13) is a discrete set with two accumulation points 0 and, this set has double symmetry with respect to the real axis and with respect to the unit circle; 2) the set of non-real multipliers is finite, if κ is the number (with multiplicity) of non-positive eigenvalues of operator F 22, then exist at most 4κ multipliers belonging to neither the real axis nor the unit circle; 3) there are at most 2κ multipliers different from ±1 with Floquet solution of positive order and the order is not grater than 2κ (if λ = ±1 is a multiplier then its order can reach 4κ + 1). Our last result deals with basis properties of the set of Floquet solutions. Let us denote by W2,loc(Ω) s the topological space of all functions that locally belong