3 hoods of q = 0 and z = E$_{0}$ , R m 0 ,n $_{0}$( z, q ) behaves like ∼ 1 / [( z − E$_{0}$ − αq )( z − E$_{0}$ + αq )], where α is a constant and E$_{0}$ ± αq are the two eigenvalues that cross and be- come a defective eigenvalue at q = 0. Let us now choose the spectral functions χ$_{m}$ ( q ) to vanish for m = m$_{0}$ , and ϕ$_{n}$ ( q ) to vanish for n = n$_{0}$ ; conversely, χ$_{m}$$_{0}$ ( q ) and ϕ$_{n}$$_{0}$ ( q ) are assumed to be nonvanishing and bounded functions of q in the interval ( − π/a, π/a ). The functions | χ 〉 and | ϕ 〉 in direct space that yield such spectral functions are determined by the corresponding inversion relations given above. For instance, if χ$_{m}$$_{0}$ ( q ) and ϕ$_{n}$$_{0}$ ( q ) assume a constant value in the interval ( − π/a, π/a ), then | χ ( x ) 〉 = N [sin( πx/a ) / ( x )] exp( ik$_{B}$m$_{0}$x ) and | ϕ ( x ) 〉 = N [sin( πx/a ) / ( x )] exp( ik$_{B}$n$_{0}$x ), where N is a normalization constant. Under such a choice, the sum in Eq.(4) reduces to the only term m = m$_{0}$ and n = n$_{0}$ . Owing to the behavior of R ( z, q ) near z = E$_{0}$ and q = 0, the corresponding integral on the right-hand-side of Eq.(4) diverges as z → E$_{0}$ ± i 0 $^{+}$, i.e. G$_{χ,ϕ}$ ( z ) is unbounded in the neighborood of z = E$_{0}$ , which proves the theorem. quence of spectral singularities is to prevent the construc- tion of a biorthogonal eigensystem for H , i.e. to resolve the identity in terms of the biorthogonal basis associated to H . By definition, H is diagonalizable if φ$_{α}$ ( x, q ), to- gether with a set of (generalized) eigenfunctions φ † $_{α}$( x, q ) of the adjoint H $^{†}$, form a complete biorthogonal sys- tem, i.e. they satisfy 〈 φ$_{α}$ ( x, q ) | φ † $_{β}$( x, q $^{′}$) 〉 = δ$_{α,β}$δ ( q − q $^{′}$) and ∑ α ∫ π/a − π/a dq | φ$_{α}$ ( x, q ) 〉〈 φ † $_{α}$( x, q ) | = I . As H † = − ∂ 2 x + V $^{∗}$( x ), it can be easily shown that one has φ † $_{α}$( x, q ) = N $_{α}$( q ) φ ∗ $_{α}$( x, − q ), where N $_{α}$( q ) is a multiplying term that needs to be determined. A direct computation of the scalar product 〈 φ$_{α}$ ( x, q ) | φ † $_{β}$( x, q $^{′}$) 〉 using expansion (2) yields 〈 φ$_{α}$ ( x, q ) | φ † $_{β}$( x, q $^{′}$) 〉 = δ ( q − q $^{′}$) δ$_{α,β}$ N $_{α}$( q ) D $_{α}$( q ), where we have set D $_{α}$( q ) = 〈 $_{w}$( α $^{)}$( q ) | $_{w}$( α ) $^{†}$( q ) 〉 and $_{w}$( α ) $^{†}$( q ) = $_{w}$( α ) $^{∗}$( − q ). Therefore, completeness of the biorthogonal system is ensured by letting N $_{α}$( q ) = 1 / D $_{α}$( q ) provided that that D $_{α}$( q ) = 0. Because D $_{α}$( q ) vanishes if and only if the energy E$_{α}$ ( q ) is a defective eigenvalue of the matrix H ( q ), from theorem 1 one con- cludes that a spectral singularity prevents the construc- tion of a biorthogonal eigensystem for H . Finally, it is also worth mentioning the following the- orem, which shows that the defective eigenvalues of H correspond to the spectral singularities of H , defined as the divergence points of the resolvent G ( z ) = ( z − H ) − 1 on the continuous spectrum. Theorem 2 . Any defective eigenvalue E of H is a diver- gence point for the resolvent G ( z ) = ( z − ) − $^{1}$. Proof. Let | χ 〉 and | ϕ 〉 be two square integrable functions of the Hilbert space, and let us introduce the complex function of z , G$_{χ,ϕ}$ ( z ), defined by G$_{χ,ϕ}$ ( z ) ≡ 〈 χ | G ( z ) ϕ 〉 . To prove the theorem, it is enough to show that there exists at least a couple of functions | χ 〉 and | ϕ 〉 such that G$_{χ,ϕ}$ ( z ) is unbounded as z → E$_{0}$ , E$_{0}$ being a defective eigenvalue of H . To this aim, after expanding | χ 〉 and | ϕ 〉 on the plane-wave basis | q, n 〉 , one can readily show that An important physical implication of spectral sin- gularities in complex crystals is the appearance of a secular growth of the amplitudes of waves scattered off the lattice when it is excited by a plane wave at special incident angles. Such an anomalous behavior has been previously noticed by Berry [20, 21] for certain absorptive potentials; here we prove that this is a very general feature of complex crystals related to the existence of spectral singularities. Namely: Theorem 3 . Let ψ ( x, t ) = exp( − iHt ) ψ$_{0}$ ( x ) be the Bragg diffraction pattern corresponding to crystal excitation, at t = 0, with a plane wave ψ ( x, 0) = exp( ikx ) of wave number k . Then H has spectral singularities if and only if for some integer n and k = nk$_{B}$/ 2 the solution ψ ( x, t ) contains secular (linearly growing) terms in t . Proof. Let us set k = q + l$_{0}$k$_{B}$ , where l$_{0}$ is an integer and − π/a ≤ q < π/a . Then the solution ψ ( x, t ) of Eq.(1) with the initial condition ψ ( x, 0) = exp( ikx ) is given by ψ ( x, t ) = ∑ ∞ l = $_{−∞}$c$_{l}$ ( t ) exp[ i ( q + lk$_{B}$ ) x ], where c$_{l}$ ( t ) sat- isfy the coupled equations i ( dc$_{l}$/dt ) = ∑ m H $_{l,m}$( q ) c$_{m}$ ( t ) with the initial conditions c$_{l}$ (0) = δ$_{l,l}$$_{0}$ . Physically, the coefficients c$_{l}$ are the amplitudes of diffracted waves at various orders at plane t [see Fig.1(a)]. Hence c$_{l}$ ( t ) = M l,l $_{0}$( q, t ), where M ( q, t ) = exp[ − it H ( q )]. The exponential matrix M can be calculated from the Jordan decomposition of H as M = T exp( − it J ) T − $^{1}$. Having in mind the form of the exponential of a Jordan matrix [28], it follows that secular growing terms in some of the coefficients c$_{l}$ ( t ) [and hence in ψ ( x, t )] appear if and only if H ( q ) has at least one defective eigenvalue, the largest B. Spectral singularities and secular Bragg diffraction G$_{χ,ϕ}$ ( z ) = ∑ $_{n,m}$∫ π/a − π/a dqχ ∗ $_{m}$( q ) ϕ$_{n}$ ( q ) R $_{m,n}$( z, q ) , (4) where χ$_{m}$ ( q ) ≡ 〈 q, m | χ 〉 , ϕ$_{n}$ ( q ) ≡ 〈 q, n | ϕ 〉 and R ( z, q ) = [ z − H ( q )] − 1 = T ( q )[ z − J ( q )] − $^{1}$T − $^{1}$( q ) is the resolvent of the matrix H ( q ). Note that the spectral functions χ$_{m}$ ( q ) and ϕ$_{n}$ ( q ) are simply re- lated to the Fourier transforms of | χ 〉 and | ϕ 〉 as described in the proof of Theorem 1, and hence the inversion relations | χ ( x ) 〉 = ∑ n ∫ π/a − π/a dqχ$_{n}$ ( q ) | q, n 〉 and | ϕ ( x ) 〉 = ∑ n ∫ π/a − π/a dqϕ$_{n}$ ( q ) | q, n 〉 hold. If E$_{0}$ is a defective eigenvalue of H ( q ), say at q = 0, with algebraic multiplicity 2 and geometric multiplicity 1, then there exists one element of the matrix R ( z, q ), say R m 0 ,n $_{0}$( z, q ), which has a second-order pole at z = E$_{0}$ when q = 0. This follows from the well-known form of the resolvent [ z − J ( q )] − 1 of a Jordan matrix possessing a Jordan block with dimension 2. In the neighbor-