Spectral Theory Of Maxwell Operators With Non-decaying Potentals

This dissertation is an examination of the spectral theory and dynamical behavior of the Hamiltonians (especially the Maxwell operators and the divergence-type operators). The text is divided into three parts.Part One consists of Chap.l and Chap.2.In Chap.1, the focus is on the physical background of the Maxwell operator, its relationship with quantum theory, and its current research situation.In Chap.2, the subject is the fundamental properties of the periodic differential operators (includes periodic Maxwell operators), concerning on the Floquet-Bloch theory, the band-gap structure, and the properties of the band edges. We find that the highly symmetric points on the Brillouin zone that achieve extreme values are stable under small periodic perturbations. Some unsolvedd problems are also presented.Part Two consists of Chap.3,Chap.4,Chap.5 and Chap.6.Chap.3 is devoted to the study of the existence of point spectrum and its bound in the band-gap of locally perturbed 3-D Maxwell operators: We prove that perturbations can creat eigenvalues inside the gap of the background spectrum, provided the perturbation is stronge enough. For perturbations with compact support, the number of the eigenvales created by perturbations can be controlled by the measure of the support of the perturbation. For perturbations occupying the hole space, the number of the eigenvales created by perturbations can be controlled by the volume of the phase space. We specify the results as follows:In the electromagnetic theory of photonic crystals one studies periodic Maxwell operatorswhere p(x) represents the properties of the "background" medium, T is a lattice in R³.We always assume that p(x) is a measurable real-valued function satisfyingfor some constants p±.We suppose that M has at least one spectral gap throughout this paper. However, the periodicity condition on the background medium is not required. We also use q(x) to represent a spatially localized perturbation. It corresponds to an impurity (i.e., a defect) introduced in the background medium. We suppose that there exists a positive constant q+ such thatFormally, we introduce the perturbed Maxwell operatorwhere D:=(?)×q(x)(?)×,p_λ(x):=p(x)+λq(x),andλis a positive coupling constant. Theorem1 Suppose thatτ∈G.Let p(x) and q(x) be given by (1) and (2), respectively. Then there existsι>0 s.t. if q(x)≥q- onβ_ιfor some constant q- > 0,thenτ∈σ(M_λ) for someλ=λ(Ï„) > 0.From Section§3.4, we further suppose thatfor some constant K > 0.In Section§3.4 we proveTheorem2 Let p(x) be given by (3.1.1)and (3.2.1). Let q(x) be given by (3.1.2) and (3.2.1). Suppose that r G G. Then there existsι₀+ι₀(Ï„)>0, such that for suppq(x)(?)β_ι0,there holdsfor allλ>0.Finally, in Section§3.5 we calculate N(λ,q,Ï„) of perturbations decaying at infinity. More precisely, in Subsection§3.5.1 we considerfor some C,s>0.The main result in this subsection isTheorem3 Let p(x) be given by (1) and (3). Let q(x) be given by (2), (3)and (4). Then there exists a positive constant C₁ such thatforλlarge enough, where p- is given by (1).In Subsection§3.5.2 we suppose that(?) for allα=(α₁,α₂),|α|≤1 (5) andThe main result in this subsection isTheorem4 Let p(x) be given by (1) and (3). Let q(x) be given by (2),(3), (5) and (6.2.11). Then there exists a positive constant C₂ such thatforλlarge enough.In Subsection§3.5.3 we suppose that for some constants q_∞and s > 0,We also suppose that the density of states (?) for M exists and is independent of the Dirichlet or Neumann boundary conditions.Theorem5 Let p(x) be given by (1) and (3). Let q(x) be given by (2), (3) and (7). Suppose that the density of states (?)(λ,M) exists and is independent of boundary conditions. Thenwhere XΩdenotes the characteristic function of the setΩ.Chap.4 is devoted to the study of the existence of (probably) continuous spectrum and its bound in the band-gap of the 3-D Maxwell operators with line defects; the properties of the generalized eigenfunctions are also studied. Based on a constructive arguments, we prove that perturbations can creat spectrum inside the gap of the background spectrum, provided the perturbation is stronge enough. Based on the Floquet-Bloch theory and the Combes-Thomas estimate, we prove that the corresponding generalized eigenfunctions are created by line defect are exponentially decaying away from the line defect. We specify the results as follows:We introduce the Maxwell operator M₀on the weighted Hilbert space L²(R³;μ₀(x)dx) which is defined by means of its quadratic formwith the domainNow, we introduce a line defect, denoted as (?)_ι,along the x₁-direction:whereΩ_ι=ιΩis the support of the line defect in the transverse plane (x₂,x₃).We suppose thatΩis a measurable bounded subset of R².Without loss of generality, we also suppose that the origin is an inner point of the setΩ.In this paper, we consider that the perturbed medium is homogenous inside the line defect (?)_ι.More precisely,whereε₁,μ₁ are two positive constants. We adapt M as the perturbed operator according to the perturbed medium which is defined analogously to M₀. Theorem6 Suppose G is a band gap in the spectrum of the operator M₀, and the interval I_Ï„,d≡[Ï„-d,Ï„+d](?)G.Then interval I_Ï„,d contains at least one point in the spectrum of the perturbed operator M if the following inequality is satisfiedι²Îµ₁μ₁≥Then we show that the generalized eigenfunction created by a linedefect decays exponentially away from the defect strip. The goal can be realized by using the so-called Combes-Thomas estimates.Theorem7 Suppose that G is a band gap in the spectrum of the operator M,and z∈σ(M)∩G,u is a generalized eigenfunction of the M according to z, then we havewhere X_y is the characteristic function of the cube {y | |y_j-x_j|≤1,j=1,2,3}centered at x.Π₁(z) andΠ₂(z) are two positive constantsdepending on z.In Chap.5, the stability of the essential spectrum of the 2-D Maxwell operator under line defect is presented, using the argument of interior estimates and the Combes-Thomas estimates, the exponentially decaying property are presented. We specify the results as follows:Let A_ε0(β) and A_ε(β)be the unperturbed and perturbed Maxwell operators,respectively.Theorem8 (Stability of essential spectrum) Let X_x,h be the characteristic function of a square of side 2h centeredat x,i.e.,withandTheorem9 Let z∈p(A₀(β)),n∈N,h>0 and 0 < v < 1,then we havewhereandμ₀^- is defined in (5.1.18). The norm in the left hand side is the operator norm in H_ε0,where H_ε0 is analogous to H_εdefined in (5.1.9).We useε₀ andμ₀ to describe the background medium. It is reasonable physically that there exist constantsε₀,±andμ₀,±such thatwe describe the perturbed medium by (?)(x) and (?)(x).We adapt A_?(β) and A_?(β) as the perturbed operator according to A_ε0(β) and A_ε0(β),respectively.In this section we consider the case that the perturbed medium is homogeneous inside the line defect (?)_ι,more precisely whereε₁ andμ₁ are two positive constants which are assumed to satisfyLet B=(a, b) be a band gap in the spectrum of the operator A_ε0(β) associated with the periodic background medium, where 0 0 such thatTheorem10 Let B = (a, b) be a band gap in the spectrum of the operator A_ε0(β).For any interval I_Ï„,d satisfying (12), ifthen the interval I_Ï„,d contains at least one eigenvalue of the perturbed operator A_?(β).Particular, forΩ={x∈R²| |x|<1},we can further estimate byusingφ=(?) as an approximate function. We also chooseζ= (1,0)^Τ.Thus (13) can be expressed as the following simple formWe write R(z) = (A_ε0(β)-zI)^-1 for z∈p(A_ε0(β)).Let X_x,h be the characteristic function of a square of side 2h centered at x,i.e.,where Now we give the estimate on the decay of the operator (?)_β×R(z)Theorem11 For any z∈p(A_ε0(β)),the operator (?)_β×R(z):V_ε0→V_ε0 has the bound:where d=dist(z,σ(A_ε0(β))).Furthermore, for any h>0 and 02 with |x-y|≥2h,其中and the definition of ###₀ and d are the same as in Theorem 5.2.2. The norm in the left hand side of (14)is the operator norm in L²(R²;C³).Finally we show that any eigenfunctions created by a line defect strip decay exponentially away from the defect strip, we describe the background medium byε₀ andμ₀,and the perturbed medium by (?)(x) and (?)(x), we also adapt A_?(β) as the perturbed operator according to A_ε0(β).Theorem12 For any eigenvalue z∈σ(A_?(β))∩B,and (?) be the corresponding eigenfunction, we havefor all x,y∈R² with |x-y|≥2h,and 0x,h is the characteristic function of the set {y||x-y|≤h} with h>0,the constant C₀ depends on v,z,ε₀,μ₀,ε,μand the distance from z to the edge ofσ(A_ε0(β)) In Chap.6,we continue the model studied in Chap.5, we first prove the completeness, then wo alse show asymptotic distributiosn of the eigenvalues under various perturbations. Particularly, we find that weak perturbation can not create any eigenvalue in the gap of the background spectrum. We specify the results as follows:We adopt the following notation,We suppose thatε(?) is invariant under any translation in the third normal direction x₃,i.e.,ε(?) is independent of x₃,In particular, ifε(x) is a periodic function satisfying(whereΓis a lattice in R²) this structure is called a (two dimensional) photonic crystal, (or, band gap material) [79].Furthermore, a photonic crystal fiber can be created if a line defect along x₃-direction is introduced. We describe the defect strip bywhereΩι:=ιΩ,0<ι≤∞is the support of the defect in the cross-section. We suppose that 0 is a measurable subset of R².Without loss of generality, we also suppose that 0 is an inner point ofΩ.We use p(?) to describe the properties of the background medium. Since the medium is assumed to be invariant under any translation in the third normal direction x₃,we have p(?) = p(x). We always assume that p(x) is a measurable function satisfyingfor some constants p±.Typically, one may think of M as a periodic Maxwell operator according to a two-dimensional photonic crystal, namely,whereΓis a lattice in R².However, the above periodicity condition is unnecessary in this paper. As we mentioned above, we shall suppose in this paper that M has at least one spectral gap. We shall also use q(?)=q(x) to describe the properties of a line defect (which serves as the core for light guiding) along the x₃-direction. Note that the line defect is also invariant under any translation in the x₃-direction.We suppose that q(x) is a measurable function satisfyingandfor some constant q+ .Then, we formally introduce the perturbed Maxwell operatorFor some given fixed levelÏ„in a gap ofσ(M),the main questions in this paper concern the set ofλ's s.t.Ï„is an eigenvalue of M+λD and the growth of the number of eigenvalues asλ→∞.We can also think this problem in term of the generalized eigenvalue problem: given D = (?)_β×q(x)(?)_β×andÏ„in a gap ofσ(M),we solve (λ,(?))∈R⁺×L²(R²;C³) such thatIn the sequel, we suppose G=(a, b),0λ)∩G is a (i.e., the lower edge of G). Moreover, the eigenvaluesÏ„₁<Ï„₂<…of the operator M_λin G coincide with the set of the solutions of the equationswhere {ζ_j(Ï„)}_j=1^∞the negative eigenvalues of the operator (?)(Ï„).Furthermore,if (?)_i is an eigenfunction of the operator (?)(Ï„_i) according to the eigenvalueζ_ji(Ï„_i)=-1,thenis an exponentially localized eigenfunction of M_λaccording to the eigenvalueÏ„_i.Note that Theorem 6.1.1 is just a very elementary result for the study of perturbed spectra.To state our further results precisely, we need some notation. We introduce the following eigenvalue distribution counting functionwhereτ∈G with G be a gap inσ(M),and M_S=M+sD.Theorem14 Suppose that a levelÏ„lies in G. Then there exists a numberι>0 such that if q(x)>0 on the disk {x∈R²:|x|<ι} thenτ∈σ(M_λ) for someλ=λ(Ï„)>0.After Section§6.4, we further suppose thatfor some constant C>0.Theorem15 Suppose that a levelÏ„lies in G. Let q(x) satisfy (26). For any given fixed coupling constantλ>0, there existsι₀=ι₀(Ï„)>0 small enough such that if suppq(x)(?){x∈R²:|x|<ι₀},thenwhere the counting function N(λ,q,Ï„) is defined in (25). Theorem16 Let q(x) satisfy (26). Suppose that supp q(x)(?){x∈R²:|x|<ι} for someι>0.Then we havewhere p- is defined in (18)and the constant C is independent of A and q(x).Finally, in Section§6.7, we study various asymptotic behaviors of the perturbations. In Subsection§6.7.1 we suppose thatandThe main result in this subsection isTheorem17 Let q(x) satisfy (26), (27)and (28). Then there exists a positive constant C₁ such thatforλlarge enough.In Subsection§6.7.2 we suppose thatThe main result in this subsection isTheorem18 Let q(x) satisfy (26) and (29). Then there exists a positiveconstant C₂ such thatforλlarge enough,where p- is defined in (18). In Subsection§6.7.3 we suppose that for some constants q_∞and s>0,We also suppose that the density of states (?) for M exists and is independent of the Dirichlet and Neumann boundary conditions.Our main result in this subsection is given by the following Theorem19 Let q(x) satisfy (31) and (26). Suppose that the density of states (?)(λ,M) exists and is independent of boundary conditions. Thenwhere (?)_Ωdenotes the characteristic function of the setΩ.Part Three consists of Chap. 7: The spectral theory and dynamical behavior of the ergodic Hamiltonians (especially the Maxwell operators and the divergence-type operators) on the lattice are analyzed. We first give the Wegner estimate, then we obtain the correlation estimate (namely, the second moment estimate) via the spectral averaging trick. Finally, by using the multiscale analysis and fractional moment analysis prove the simplicity of eigenvalues and its Poisson distribution; the distribution of localization centers is also studied. We prove that: in the localization region, localization centers of the eigenfuncations become far apart, as corresponding energies are close together. We specify the results as follows:First we will give the rigorous definition of the operator that we considered here. The nonnegative self-adjoint operator M onι²(Z^d) can be uniquely defined by the following quadratic form,where b={x,y},x,y∈Z^d,andγ(b)>0 are real i.i.d.r.v.'s on the probability space of probability measure P with bounded density p:||p||_∞≤Const..We assume that ess Ranγ(b)=[γ_s,γ_b],where 0<γ_s<γ_b<∞.We requireγto satisfyBy [115] we know that M is essentially self-adjoint a.s. (i.e., with probability 1).We also suppose that p is exponentially decaying at infinity.We first show the Wegner estimate . For any cubeΩin Z^d centered at 0, we suppose that |Ω|=k.Let (?) denote all of the (unoriented) edges inΩ.Clearly,|(?)|< kd (since the number of the edges connected with the points onδΩare smaller than the inner points).Formally, define the nonnegative self-adjoint operator M_Ωonι²(Ω),It can be uniquely defined by the quadratic formwhere u∈ι²(Ω). Theorem20 Suppose 7 satisfy (33) and (34). Then for any 0<ε<λ,we haveNow we give the correlation estimate.Theorem21 For the operator M_Ω,defined above and any bounded interval I=[a,b],there holdswhere p is the density of the probability distrubation of the random variblesγ.Suppose {E_j}_{j=1,...,|Ω|} are the eigenvalues of the operator M_Ω(counting muliplicity).The IDS defined in (7.2.4) of the operator M_Ωdefined above exists(cf.[115]).The corresponding density is n(E)=(?)In 2007,Hislop and Müller [70] proved that under certain conditions (e.g., about the random potential), the density of states of Schr(?)dinger operator is strictly positive a.e..Here we also suppose the density of states n(E) of M is strictly positive a.e..Definewhereδis the Dirac function in the lattice case.Based on the results above and the correlation estimate (Theorem7.3.1), we haveTheorem22 Let E satisfy where the constantsC,κare all independent ofΩ(?)Z^d.We also assume that n(E) exists and is positive. IfΩ↑Z^d,then the point process e(Ω,E)converge to the point process e of density n(E). Via the MSA, we haveTheorem23 For the operator M defined above, all the eigenvalues that located in the localization region are simple a.s..The last result is about the distribution of the localization centers of the eigenfunctions. We will use the following notation and definitions.·Define the boundary of the cubeΩbyIn particular,we useΩ_ι(n) to denote the cube of side lengthιcentered atn∈Z^d.·We callΩ_ι(p) are (θ,z)-good,if z(?)σ(M_Ωι(p)) and for any m∈δΩ_ι(p)we haveotherwise they are called to be (θ,z)-bad.·For any bound state u≠0,define its localization center by(?)(36)·In the following we always suppose that R is the localization region of the operator M.By the Combes-Thomas estimate (similar as in Subsect.5.1.5) and the resolution of identity, we can proveProposition24 For any interval∑(?)R,there existsθ>0,η>2d andι₀>>1 s.t. In fact, this proposition combining with the Wegner estimate, guarantee to use MSA [37]:defineΩ_j(p)=Ω_ιj(p),p∈Z^d,then there exists s∈(1,2) s.t. for the two-scale relationι_j=ι_j-1^S there holdsLemma25 Suppose the upper bound ||p||_∞of the density is small enough (namely, high disorder), thenTheorem26 With probability 1,for any given m∈Z^d,letα_j=(?),j∈N.For any I,|I|≤α_j in R and an eigenvalue z₀ in I,there exists N sufficiently large s.t.if j≥N,then there is no eigenvalue of M,except z₀,such that its localization center locates inΩ_j.In the other words, the distance from its localization center to the localization center according to z₀ is at leastι_j/2.