Towards Modelling QFT in Real Metamaterials: Singular Potentials and Self-Adjoint Extensions

1 Content from this work may be used under the terms of theCreativeCommonsAttribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 Towards Modelling QFT in Real Metamaterials: Singular Potentials and Self-Adjoint Extensions L M Nieto1, M Gadella1, J Mateos Guilarte2, J M Mu˜noz-Casta˜neda3, and C Romaniega1,4 1Departamento de F´ısica Te´orica, At´omica y ´ Optica and IMUVA, Universidad de Valladolid, Spain 2Departamento de F´ısica Fundamental, Universidad de Salamanca, Spain 3Departamento de F´ısica ETSIAE, Universidad de Polit´ecnica de Madrid, Spain 4Departamento de F´ısica Te´orica II, Universidad de Complutense de Madrid, Spain E-mail: [email protected], [email protected], [email protected], [email protected], [email protected] Abstract. Solutions of the one-dimensional Schr¨odinger equation are found when point interactions of the type aδ(x−q) + bδ0(x−q) are placed either in a couple of points or in a regular lattice. The results obtained in the present study are a first step toward a rigorous mathematical model of real metamaterials is Solid State Physics. 1. Introduction One dimensional (1D) models with point interactions have received much attention in the recent literature (see Ref. [1] and references therein). They can model small and abrupt defects in materials allowing to perform analytic calculations such as the tunnel effect and the Casimir force. They are also interesting in the analysis of heterostructures in connection to an abrupt effective mass change. The general inquiry of point interactions of the free Hamiltonian H0=−~2 2m d2 dx2is mainly due to Kurasov [2] and it is based on the construction of self-adjoint extensions of symmetric operators with identical deficiency indices. Mu˜noz-Casta˜neda and coworkers [3] reformulated the theory of self-adjoint extensions of symmetric operators over bounded domains in terms of meaningful quantities from a Quantum Field Theory (QFT) point of view, allowing a rigorous study of the theory of quantum fields over bounded domains and the quantum boundary effects. In this context, it is well known that set of self-adjoint extensions of H0defined over the space (−∞, q)∪(q, ∞) is given by the unitary group U(2) and therefore is a 4-parameter family of operators. Some of some of these self-adjoint extensions are associated to the Hamiltonians H0plus point-supported potential aδ(x−q) + bδ0(x−q), see [4]. This perturbation is of great interest in Quantum Mechanics from mathematical and physical points of view, and has been largely discussed in [4,5]. Although it is totally clear which self-adjoint extension corresponds to the 1D-Dirac delta, there is no consensus on which one should be assigned to its derivative. In the framework of QFT the spectra and eigenstates of non-relativistic Schr¨odinger operators provide one-particle states in scalar (1+1)-QFT systems. Point supported potentials can be used in scalar QFT on a line to model impurities and classical singular backgrounds that interact with 2 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 quantum vacuum fluctuations. The most successful results in the frame of QFT are obtained in Casimir-type set-ups where the scattering data allow to compute the distance dependence finite part of the quantum vacuum energy for the quantum vacuum fluctuations that live between two plane parallel plates. In particular, configurations of two pure delta potentials added to the free Schr¨odinger Hamiltonian allowed to compute the quantum vacuum interaction energy between two parallel semitransparent Dirac δ-plates in terms of the scattering data of the pure double delta potential on the real line, see [5]. The same configuration of delta interactions is addressed as a perturbation of the Salpeter Hamiltonian. Additionally, generalized Dirichlet boundary conditions were discussed, showing that the use of δ-δ0potentials provides a much larger set of admissible boundary conditions, and the quantum vacuum energies between two plane parallel plates represented by a δ-δ0potential in arbitrary space-time dimension were computed [5]. Since the zero range potentials mimic the plates in a Casimir effect setup, it is interesting to consider a three-plate configuration (Casimir pistons) and investigate the Casimir forces acting of the plate in the middle that is supposed to be mobile in a piston configuration. In particular, it is meaningful to displace the central plate towards one of the other two placed in the boundary; thus, we find the main physical motivation to study the particular situation where two δ−δ0 interactions are superimposed. This is the first problem that will be addressed in Section 2, and the outcome is surprising: a non-abelian addition law emerges and it corresponds to the group law of the Borel subgroup of the SL2(R) group. The other focus of interest is the case where the δ0-couplings are exceptional, i.e., those couplings for which the transmission coefficients are zero and the plates become completely opaque: the left-right decoupling limit. It happens that the distinguished Dirichlet/Robin boundary conditions are implemented in this case by the δ−δ0-potentials but the superposition law just mentioned becomes singular [6]. An ideal model of electric conductivity in solids is provided by a δ-Dirac comb where δ-point interactions sit at the ions sites. This model can be enriched by adding a δ0potentials at every site of the lattice. This is the objective of Sec. 3, which is a work presently in progress [7]. 2. One and two singular perturbations on a line First of all, let us consider first the one-dimensional Schr¨odinger equation with singular potential −~2 2m d2 dx2ψ(x) + aδ(x−d)ψ(x) + bδ0(x−d)ψ(x) = E ψ(x).(1) Introducing the dimensionless quantities x=~ mc y, d =~ mc p, w0=2a ~c, w1=mb ~2, ε =2E mc2,and ϕ(y) = ψ(x), the Hamiltonian is simply −d2 dy2ϕ(y) + w0δ(y−p)ϕ(y) + 2w1δ0(y−p)ϕ(y) = ε ϕ(y).(2) The point potential we are interested in is usually defined via the theory of self-adjoint extensions of symmetric operators of equal deficiency indices, so that the total Hamiltonian in (2) is selfadjoint. The crucial point is to find the domain of wave functions ϕ(y) that makes H0selfadjoint over the domain R/{p}and characterizes the potential V(y). As these functions and their derivatives should have a discontinuity at y=p, we have to define the products of the form δ(y−p)ϕ(y) and δ0(y−p)ϕ(y). These can be done in several ways, and we choose the following: δ(y−p)ϕ(y) = ϕ(p+) + ϕ(p−) 2δ(y−p),(3) δ0(y−p)ϕ(y) = ϕ(p+) + ϕ(p−) 2δ0(y−p)−ϕ0(p+) + ϕ0(p−) 2δ(y−p).(4) 3 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 To obtain a self-adjoint determination of H=H0+V(y) = H0+w0δ(y−p) + 2w1δ0(y−p), we have to find a self-adjoint extension of H0. To do it, we have to find a domain on which this extension acts, which is given by the Sobolev space W2 2(R\{p}) of absolutely continuous functions f(y) with absolutely continuous derivative f0(y), both having arbitrary discontinuities at p, such that the Lebesgue integral given by Z∞ −∞{|f(y)|2+|f00(y)|2}dy converges, and such that at psatisfy the following matching conditions: ϕ(p+) ϕ0(p+)!=   1 + w1 1−w1 0 w0 1−w2 1 1−w1 1 + w1    ϕ(p−) ϕ0(p−)!.(5) These results could be extended to interactions of the type P"aiδ(y−pi)+2biδ0(y−pi), where the sum could be either finite or infinite. 2.1. Two singular perturbations on a line We assume now that the total Hamiltonian is H=H0+V+W, with H0=−d2 dy2, V =v0δ(y) + 2v1δ0(y), W =w0δ(y−p) + 2w1δ(y−p).(6) Vis supported at the origin y= 0 and Wis supported at p > 0. The potential V+Wis related to the Casimir effect. One of the motivations of this work is the study of the effect of the superposition of the two point potentials Vand Wat the same point (taking the limit as p→0). We shall see that we obtain a point potential of the type u0δ(y) + 2u1δ0(y), where u0 and u1are not the sums v0+w0and v1+w1. 2.1.1. Matching conditions and scattering coefficients. In order to analyze ϕ(y), the solution of (6), let us split the real line into the regions (1) y < 0, (2) 0 < y < p, and (3) p < y, where ϕi(y) = Aie−iky +Bieiky , ϕ0 i(y) = −ik(Aie−iky −Bieiky), i = 1,2,3, k2=ε > 0. Since the point potential Vis defined by matching conditions like those in (5), we have A2 B2=K−1MvKA1 B1, Mv=   1 + v1 1−v1 0 v0 1−v2 1 1−v1 1 + v1   , K =1 1 −ik ik .(7) This expression gives the matching conditions at y= 0. At the point y=pwe get A3 B3=Q−1K−1MwK Q K−1MvKA1 B1=TpA1 B1,(8) where the definition of Tpis obvious, while Qand Mware respectively: Q=e−ipk 0 0eipk , Mw=   1 + w1 1−w1 0 w0 1−w2 1 1−w1 1 + w1   .(9) 4 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 The Tp-matrix in (8) relates the asymptotic behavior of the two linearly independent Jost scattering solutions at y << 0 with their counterparts at y >> 0. Remark that detTp=T11 pT22 p−T12 pT12 p= 1 , T11 p=¯ T22 p, T12 p=¯ T21 p⇒Tp∈SL2(C). From the Tp-matrix one obtains the 2 ×2 unitary scattering matrix Sp: Sp=1 T11 p1−T12 p T21 pdetTp, S† p=1 ¯ T11 p1¯ T21 p −¯ T12 pdetTp, S† pSp=1 0 0 1 .(10) The transmission and reflection coefficients provide the usual form of writing the Spmatrix: Sp=t(k)rR(k) rL(k)t(k),(11) where t(k) is the amplitude of the transmitted waves coming from the far left or from the far right. Time-reversal invariance implies tR(k;p) = tL(k;p) = t(k;p). However, as the potential is not parity invariant, the reflection amplitudes are different for incoming waves either from the far right or the far left: rR(k;p)6=rL(k;p). We explicitly find: rL(k;p) = −[e−2ipk "2k"v2 1+ 1+iv0(4kw1−iw0) + (4kv1−iv0)"2k"w2 1+ 1−iw0]/∆(k), rR(k;p) = [e2ipk "2k"v2 1+ 1−iv0(4kw1+iw0) + (4kv1+iv0)"2k"w2 1+ 1+iw0]/∆(k), t(k;p) = 4k2(1 −v2 1)(1 −w2 1)/∆(k), where ∆(k) = e2ikp(v0+ 4ikv1)(w0−4ikw1) + (2kv2 1+ 2k+iv0)(2kw2 1+ 2k+iw0) is a function of kand the other parameters of the problem. 2.2. Bound/antibound states and resonances Complex zeroes of T11 p(k) = ∆(k)/[4k2(1−v2 1)(1−w2 1)] (poles of the Sp-matrix) give rise to bound or antibound states if are located on the purely imaginary, respectively positive or negative halfaxis. Complex zeroes coming in pairs having opposite real part and identical imaginary part correspond to resonances. Then the relevant physical information is encoded in the analysis of complex zeroes of ∆(k). If we make the changes 2kp =z, σ =pv0 1 + v2 1 , τ =pw0 1 + w2 1 , v =v1 1 + v2 1 , w =w1 1 + w2 1 ,(12) then, ∆(k) = 0 is equivalent to eiz =−z+iσ σ+ 2ivz  z+iτ τ−2iwz , z =zr+izi, zr, zi∈R.(13) Equation (13) is a generalization of the so-called generalized Lambert equation, from where we can obtain two real equations, corresponding to its real and imaginary parts 4vwz2 r−(2vzi−σ)(2wzi+τ)cos zr−2zr(τv + 4vwzi−σw) sin zr=(zi+σ)(zi+τ)−z2 rezi. 2zr(τv + 4vwzi−σw) cos zr+4vwz2 r−(2vzi−σ)(2wzi+τ)sin zr=−zr(2zi+σ+τ)ezi. A simpler compatibility condition is this Lambert equation in two real variables e2zi=(4v2z2 r+"2vzi−σ)2"4w2z2 r+ (2wzi+τ)2 (z2 r+ (zi+σ)2) (z2 r+ (zi+τ)2).(14) We can easily check that a simple solution of the system is zr= 0, zi= 0, but this corresponds to k= 0, which is a pole of T11 p(k). The typical behavior of the solutions of these equations is shown in Fig. 1, where the blue curves are the real part, the yellow ones the imaginary part and the green curve is the compatibility condition (14). 5 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 Figure 1. 2.3. The addition law at the p= 0 limit We can summarize what happens in the limit p→0 as follows: the Tpmatrix for two-pairs of δ−δ0interactions displaced from each other a distance pbecomes: A3 B3=K−1Mw·MvKA1 B1=K−1MuKA1 B1.(15) Then, the superposition of these two δ−δ0is equivalent to a single δ−δ0where the matching conditions are characterized by the Kurasov matrix Mu=   1 + u1 1−u1 0 u0 1−u2 1 1−u1 1 + u1   , u1=v1+w1 1 + v1w1 , u0=v0(1 −w1)2+w0(1 + v1)2 (1 + v1w1)2.(16) This addition law is quite interesting. It can be proved [6] that the Kurasov matrices is the Borel subgroup of SL2(R), with the product law Mv·Mw=Mugiven by (16). It can be also shown that the p→0 limit of the scattering coefficients of two a priori separated pairs of δ−δ0 interactions is lim p→0Sp(Mv, Mw) = S(Mv·Mw) = S(Mu) (17) Hence, the scattering matrices produced by two superimposed pairs of δ−δ0interactions give rise to a representation of the non-abelian composition law of Kurasov matrices. In other words, the non-abelian addition law (16) also works for the scattering coefficients. 2.4. The left-right decoupling values of the δ0couplings It is interesting to stress that there are singularities of the Kurasov matrix Mv, Mwin (7) and (9) at the critical points v1=±1 and w1=±1. One may expect that these critical cases, where the transmission coefficients are zero, do not contribute to the group structure that we have previously explained. The completely opaque potentials have one of the following forms: V=v0δ(y)±2δ0(y) and/or W=w0δ(y−p)±2δ0(y−p).(18) In order to define these potentials, we cannot use the same matching conditions as before. Instead, following the work of Kurasov, we shall impose: (i) For v1= 1: ϕ(0−) = 0 , ϕ0(0+) = v0ϕ(0+)/4. (ii) For w1= 1: ϕ(p−) = 0 , ϕ0(p+) = w0ϕ(p+)/4. (iii) For v1=−1: ϕ(0+) = 0 , ϕ0(0−) = −v0ϕ(0−)/4. (iv) For w1=−1: ϕ(p+) = 0 , ϕ0(p−) = −w0ϕ(p−)/4. 6 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 Thus, we set Dirichlet boundary conditions on one side of the two points y= 0 and y=p, whereas Robin boundary conditions are chosen on the other. There are eight possible configurations involving at least one decoupling configuration of the couplings: either the two δ−δ0interactions build opaque walls both at y= 0 and y=p, or there is no transmission only at one point. We discuss first the four cases when the two δ0couplings take the decoupling limit. 2.4.1. Two δ0couplings in the decoupling limit. There are four cases in which we have decoupling situations both at y= 0 and y=p. Let us consider them separately. Case 1:v1= 1, w1= 1. The boundary conditions are given by ϕ(0−) = 0, ϕ0(0+) = v0ϕ(0+)/4, ϕ(p−) = 0, ϕ0(p+) = w0ϕ(p+)/4. For y= 0+, the condition is written as: −ik(A2−B2) = v0 4(A2+B2) =⇒A2 B2 =−v0−4ik v0+ 4ik =−exp −2iarctan 4k v0.(19) For y=p−, we obtain A2e−ikp +B2eikp = 0. Then, we have a transcendental equation, which can be written either in the form of a generalized Lambert equation e2ikp = (v0−4ik)/(v0+4ik), or tan(kp) = −4k/v0. This equation has a countably infinite number of solutions, kn, that give the energy levels corresponding to this case. In the limit p= 0, we have only one solution, k= 0. Outside the interval [0, p], we have the equations ϕ(0−) = 0 =⇒A1+B1= 0 ϕ0(p+) = w0 4ϕ(p+) =⇒A3 B3 =−e2ikp w0−4ik w0+ 4ik .(20) These are relations between coefficients, which do not provide of any further information, so that we shall not refer for similar situations which will appear for all other cases. Case 2:v1= 1, w1=−1. The boundary conditions are ϕ(0−) = 0, ϕ0(0+) = v0ϕ(0+)/4, ϕ(p+) = 0, ϕ0(p−) = −w0ϕ(p−)/4. The condition at y= 0 has already been studied. From the new boundary condition at y=p: −kp = arctan 4k w0 + arctan 4k v0 =⇒tan(kp) = −4k(w0+v0) w0v0−16k2.(21) This new transcendental equation gives another set of energy eigenvalues, seen in the left hand side of next Figure. In the limit p→0, we have only one solution: k= 0. Case 3:v1=−1, w1= 1. The boundary conditions are such that A2+B2= 0 and ϕ(p−) = 0, which is A2/B2=−e2ikp, so that A2 B2 =−1 = −e−2ikp =⇒k=πn p.(22) Again, in the limit p= 0, the only solution is k= 0. Case 4:v1=−1, w1=−1. This correspond to ϕ(0+) = 0, ϕ0(0−) = −v0ϕ(0−)/4, ϕ(p+) = 0, ϕ0(p−) = −w0ϕ(p−)/4. Then, the transcendental equation giving the energy levels is given by e2ikp =w0−4ik w0+ 4ik ⇐⇒ tan(kp) = −(4k)/(w0).(23) This transcendental equation also has a countably infinite number of solutions, kn, that give the energy levels corresponding to this situation, seen in the right hand side of Fig. 2. Once more, in the limit p→0, only the solution k= 0 remains. 7 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 Figure 2. 2.4.2. Only one δ0coupling in the decoupling limit. Let us consider the following four cases: Case 1:v1= 1, w16=±1. Skipping the technical details, the system behaves as if there is an impenetrable barrier at y= 0. The coefficients A3, B3are obtained from A2, B2: A3 B3!=    2k(1 + w2 1) + iw0 2k(1 −w2 1)e2ikp 4kw1+iw0 2k(1 −w2 1) e−2ikp 4kw1−iw0 2k(1 −w2 1) 2k(1 + w2 1)−iw0 2k(1 −w2 1)     A2 B2!.(24) All the relevant information about bound states, anti-bound states and resonances is obtained imposing the purely outgoing boundary condition, which in our case is A3= 0. Then, we get e2ikp =v0−4ik v0+ 4ik 2k(1 + w2 1) + iw0 4kw1+iw0 .(25) This equation is the equivalent of ∆(k) = 0, which provides the relevant information in the non-decoupling case. Indeed, (25) is obtained making v1= 1 in ∆(k) = 0. The analysis in this case is similar to the one carried out previously. We are very interested in the limit p→0. Then, from (25) we get (v0+ 4ik)(4kw1+iw0) = (v0−4ik)(2k(1 + w2 1) + iw0). The complex solutions of this equation are k0= 0, k1=−i(4w0+v0(1 −w1)2)/(4(1 + w1)2). The solution k= 0 is not relevant, but the other produces interesting results: •If 4w0+v0(1 −w1)2<0, k1is on the positive imaginary axis and it is a bound state. •If 4w0+v0(1 −w1)2>0, k1is on the negative imaginary axis and it is an anti-bound state. The curves solving the equations Re(∆) = 0 (blue) and Im(∆) = 0 (yellow) are represented on the left hand side of the next Figure (the green line is the real exis). Case 2:v1=−1, w16=±1. The system behaves also as if there were an impenetrable barrier at y= 0. Now, the purely outgoing boundary condition A3= 0 implies that e2ikp = (2k(1 + w2 1) + iw0)/(4kw1+iw0). See the right hand side of Fig. 3. In the limit p→0 we get the unique solution k= 0. Case 3:v16=±1, w1= 1. It is similar to the previous one. Case 4:v16=±1, w1=−1. This is similar to the first case studied before. Imposing the purely outgoing boundary condition B1= 0, we get e2ikp =−w0−4ik w0+ 4ik 2k(1 + v2 1) + iv0 4kv1−iv0 .(26) In the limit p→0, we have the complex solutions k0= 0, k1=−i(4v0+w0(1+v1)2)/(4(1−v1)2). The solution k= 0 is not relevant, but the other one, k1, corresponds either to a bound state if 4v0+w0(1 + v1)2<0, or to an anti-bound state if 4v0+w0(1 + v1)2>0. 8 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 Figure 3. 3. A hybrid δ−δ0Dirac comb A very well known example of exactly solvable periodic potential used in Solid State Physics is the Kronig-Penney model, which describes electron motion in a period array of rectangular barriers. The Dirac-Kronig-Penney model can be considered as a special case of the KronigPenney model obtained by taking the appropriate limit in such a way that the rectangular barriers become Dirac delta distributions. Hence, very naturally comes the idea of considering a modified Dirac comb adding a δ0in every singular point. The dimensionless Schr¨odinger equation for such an hybrid Dirac comb becomes: −d2ψ(y) dy2+ X n∈Zw0δ(y−na) + 2w1δ0(y−na)!ψ(y) = ε ψ(y).(27) To solve it, first, we will solve the equation in (−a, 0) (region I) and (0, a) (region II): ψI(y) = AIeiky +BIe−iky, ψII (y) = AII eiky +BII e−iky , ε =k2.(28) Hence, ψ0 I(y) = ikAIeiky −ikBIe−iky,ψ0 II (y) = ikAII eiky −ikBII e−iky. In compact matrix form, these solutions can be written as follows (J=I, II): ψJ(y) ψ0 J(y)=K QyAJ BJ, K =1 1 ik −ik , Qy=eiky 0 0e−iky .(29) Now, we impose Kurasov’s matching conditions at y= 0, as in the previous Section: AII BII =K−1TwKAI BI.(30) Finally, as the potential is periodic, Floquet-Bloch theory establishes that for a periodic potential of period a, the wave functions are quasi-periodic, a plane wave times a periodic function: ψ(y) = eiqyφ(y), q ∈R, q ≃q+2π an , n ∈Z, φ(y+a) = φ(y), which implies that ψ(y+a) = eiqaψ(y), henceforth ψ0(y+a) = eiqaψ0(y). The Floquet index q is conventionally referred to as quasi-momentum. Thus, we write for y∈(−a, 0), ψII (y+a) ψ0 II (y+a)=eiqa ψI(y) ψ0 I(y).(31) Using the explicit form of the solutions in that equation we get KQyQaAII BII =eiqaKQyAI BI.(32) 9 1234567890 International Conference on Quantum Phenomena, Quantum Control and Quantum Optics IOP Publishing IOP Conf. Series: Journal of Physics: Conf. Series 839 (2017) 012007 doi :10.1088/1742-6596/839/1/012007 In order to find non-trivial solution, we must have det eiqa I−QaK−1TwK= 0. After some algebra, we obtain the following dispersion relation guaranteeing that the determinant is null: cos qa =f(w1)cos ka +a 2w0g(w1)sin ka ka , f(w1) = 1 + w2 1 1−w2 1 , g(w1) = 1 1 + w2 1 .(33) If the δ0interactions are switched off, w1= 0, the band spectral equation for the standard Dirac comb reappears. On the other hand, if we define the 2 ×2-matrix Ω(1) =MaK−1TwK, the band spectrum condition or dispersion relation can re-written as cos(qa) = tr(Ω(1))/2. Since cos(qa)∈(−1,1) only those momenta/energies that satisfy tr(Ω(1))≤2 comply with the spectral condition and therefore characterize the allowed band spectrum of the ideal 1D crystal described by the δ−δ0comb. 3.1. Analysis of the structure of the band spectrum The analysis of the band structure of the energy spectrum of a periodic but not even potential with period ademands the solution of the transcendent equation: cos qa =t2(k)−rR(k)rL(k) + 1cos ka +it2(k)−rR(k)rL(k)−1sin ka 2t(k)(34) =cos [ka +δ(k)] |t(k)|, which depends only on the scattering data for a single point-potential. In the case of a single δ−δ0interaction in the y∈[0, a) interval these magnitudes are: t(k) = (1 −w2 1)k (1 + w2 1)k+iw0/2,|t(k)|=2|k||1−w2 1| p4k2(1 + w2 1)2+w2 0 , rR(k) = −2kw1+iw0/2 (1 + w2 1)k+iw0/2, rL(k) = 2kw1−iw0/2 (1 + w2 1)k+iw0/2, δ(k) = 1 2ilog t2(k)−rR(k)rL(k)=1 2ilog 2ik(1 + w2 1) + w0 2ik(1 + w2 1)−w0 +θ(w2 1−1)π, where θ(x) is the Heaviside step function. Notice that the reflection amplitudes for incoming particles from either the left or the right are different since the δ0coupling breaks parity invariance. Time reversal invariance, however, is preserved and the transition amplitude is the same whatever the incoming direction of the particle is. It follows that a given value of kbelongs to an allowed band if 1 |t(k)|cos (ka +δ(k)) ≤1. The right hand side of (34) is an oscillating function of kthat tends asymptotically to a periodic function of amplitude 1/|t(k→ ±∞)| ≥ 1. Thus, there are pairs of points kjand kj+1 where cos (kja+δ(kj)) /|t(kj)|=±1 and cos (kj+1a+δ(kj+1)) /|t(kj+1)|=∓1, which are the edges of allowed bands: in the intervals [kj, kj+1] the spectral function varies between ±1 and ∓1, and there are intersections with the k-independent functions g= cos qa ∈ [−1,1], giving solutions of the spectral equation. In the interval (kj+1, kj+2), where kj+2 is the