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Periodic Potentials and Supersymmetry

We review the current status of one dimensional periodic potentials and also present several new results. It is shown that using the formalism of supersymmetric quantum mechanics, one can considerably enlarge the limited class of analytically solvable one-dimensional periodic potentials. Further, using the Landen transformations as well as cyclic identities for Jacobi elliptic functions discovered by us recently, it is shown that a linear superposition of Lam\'e (as well as associated Lam\'e) potentials are also analytically solvable. Finally, using anti-isospectral transformations, we also obtain a class of analytically solvable, complex, PT-invariant, periodic potentials having real band spectra.


I. INTRODUCTION
One dimensional potential wells have bound states. They are solutions of the Schrödinger equation which satisfy appropriate boundary conditions. When two identical potential wells are very far apart, then each potential has the same energy levels, and each eigenstate is doubly degenerate. As the wells are brought closer together, there is communication between them, and each level is split into two.
Similarly, if one has an array of many identical wells making a periodic potential, then one gets energy bands, which play, for example, a crucial role in determining the electronic properties of crystalline solids. To illustrate this band structure qualitatively, condensed matter physics texts usually treat the problem of a one dimensional periodic array of delta functions, called the Kronig Penney model. Here, one gets a transcendental equation for computing band edges. Another well-studied class of periodic potentials are the Lamé potentials [1,2,3] V (x) = a(a + 1)msn 2 (x, m) , where sn(x, m) is a Jacobi elliptic function [4] of real elliptic modulus parameter m (0 ≤ m ≤ 1). If a is any positive integer, it is well known that these potentials have only a band gaps and hence 2a + 1 band edges which are analytically known (in principle). However, this is an exceptional example. The bottom line is that even in one dimension, there are very few solvable periodic potentials, and it would be desirable to have more, especially some with a richer spatial structure.
We shall describe how to obtain new solvable periodic potentials via several different approaches.
One way, for example is to expand our knowledge of Lamé potentials to the wider class of associated Lamé (AL) potentials [5,6] V (x) = a(a + 1)msn 2 (x, m) + b(b + 1)m cn 2 (x, m) where a, b are constants and without any loss of generality we consider a ≥ b. A second way, for example is to exploit the fact that Jacobi elliptic functions are doubly periodic functions and consider PTinvariant complex periodic potentials [7] obtained from the Lamé and the AL potentials by considering the anti-isospectral transformation x → ix + β (where β is any nonzero real number) [8]. A third way is to consider a linear superposition of several Lamé (AL) potentials which in view of the Landen transformations [9,10] and cyclic identities [11] for Jacobi elliptic functions can be shown to be essentially equivalent to Lamé (AL) potentials with modified parameters. Finally, we can further expand all these classes of solvable potentials by using the techniques of supersymmetric quantum mechanics to generate supersymmetric partner potentials [12].
The outline of this paper is as follows. In Sec. II we review some general properties of periodic potentials as well as the relevant formalism of supersymmetric quantum mechanics. In Sec. III we discuss the band edge energies and eigenfunctions for several Lamé and AL potentials. In particular, for Lamé potentials (1) we derive a remarkable hitherto un-noticed relation connecting band edge energy eigenstates corresponding to modulus parameters m and 1 − m. We also describe how the Lamé potential results can be vastly expanded to get solutions for the AL potentials (2). In particular, we show that for any integral a, b with a > b, there are a bound bands followed by a continuum band, out of which the top b bound bands are unusual in the sense that both band edges are of period 2K(m) (4K(m)) when the integer a − b is even(odd), where K(m) is the complete elliptic integral of the first kind. In Sec. IV we obtain the supersymmetric partners of both the Lamé and the AL potentials, thereby expanding the list of solvable periodic potentials. In Sec. V we show that using Landen transformations [10] and cyclic identities obtained by us recently [11], exact band edge eigenvalues and eigenfunctions can also be obtained in the case of linear superposition of Lamé and AL potentials.
In Sec. VI we show that a class of PT-invariant complex periodic potentials obtained from the Lamé and AL potentials by the anti-isospectral transformation x → ix + β, (β being any nonzero real number) are also exactly solvable problems with real band spectra. Finally, in Sec. VII we discuss the double sine-Gordon equation and show that in some special cases it has unusual band spectra, in that the majority of bound bands have both of their band edges of the same period.
where k denotes the crystal momentum. The spectrum shows energy bands whose edges correspond to kL = 0, π, that is the wave functions at the band edges satisfy ψ(x) = ±ψ(x + L). For periodic potentials, the band edge energies and wave functions are often called eigenvalues and eigenfunctions, and we will also use this terminology.
A general property of eigenstates for a real potential with period L is the oscillation theorem. It states that band edge wave functions, when arranged in order of increasing energy .., have periods L, 2L, 2L, L, L, 2L, 2L, ... [1]. The corresponding number of wave function nodes in the interval L are 0, 1, 1, 2, 2, 3, 3, ... and the energy band gaps are given by where (by convention) the superpotential W (x) is related to the ground state eigenfunction of The corresponding Hamiltonians H ± can be factorized as where so that the spectra of H ± are nonnegative. It is also clear that on the full line, both H ± cannot have zero energy modes, since the two functions ψ (±) 0 given by cannot be simultaneously normalized.
On the other hand, when the superpotential W (x) has period L [W (x + L) = W (x)], and the eigenfunctions of both V ± (x) must satisfy the Bloch condition (3), it is easily shown that in the periodic case, irrespective of whether supersymmetry is broken or unbroken, the spectra of V ± (x) are strictly identical [13,14]. Further, one has unbroken supersymmetry provided and in this case the partner Hamiltonians have identical spectra including zero modes and both ψ (±) 0 belong to the Hilbert space. As a result, unlike on the full line, for periodic potentials, whenever the condition (9) is satisfied, supersymmetry is unbroken and yet the Witten index (which counts the difference between the number of zero modes of ψ (±) 0 ) is always zero. The condition (9) is trivially satisfied when W (x) is an odd function of x and throughout this paper we shall only consider superpotentials W which are odd function of x. Further, using the known eigenfunctions ψ n (x) of V + (x). In particular, from eq. (8) it follows that the ground state of V + (x) is given by [12] ψ (+) while the un-normalized excited states ψ n (x) by using the relation Thus by starting from an exactly solvable periodic potential V − (x), one gets a new isospectral periodic potential V + (x).
A few years ago, the concept of self-isospectral periodic potentials has been defined and developed in detail [13]. A one dimensional potential V − (x) of period L is said to be self-isospectral if its supersymmetric partner potential V + (x) is just the original potential upto a discrete transformationa translation by any constant amount, a reflection, or both. A common example is translation by half a period, in which case the condition for self-isospectrality is V + (x) = V − (x − L/2). In this sense, any self-isospectral potential is rather uninteresting, since the application of supersymmetry just yields a discrete transformation and basically nothing new.
On comparing eqs. (12) and (14) we then have the desired relations relating the eigenstates when the modulus parameter is m and 1 − m: Thus, for any integer a, at m = 1/2, one has remarkable relations On using one can immediately and explicitly verify the relations (15) and (16) for a = 1, 2, 3, 4.
For the a = 2 case, the Lamé potential has 2 bound bands and a continuum band. The energies and wave functions of the five band edges are well known [1,2]. The lowest energy band ranges from 2 + 2m − 2δ to 1 + m, the second energy band ranges from 1 + 4m to 4 + m and the continuum starts For subsequent application of the supersymmetric quantum mechanics formalism, it is convenient to have a potential V − (x) whose ground state energy is zero. This is easily accomplished via a re-defined potential with the ground state energy subtracted out. For the problem under consideration, one has V − (x) = 6msn 2 (x, m) − 2 − 2m + 2δ. This potential Fig. 1. The eigenstates of all the band edges are given in Table 1. Note that in the interval 2K(m) corresponding to the period of the Lamé potential, the number of nodes increases with energy, in agreement with the oscillation theorem. From Table 1, it is easy to check that, as expected, the relations (15) and (16) are indeed true.
We might add here that in case a is not an integer, then for Lamé potentials one has an infinite number of bands and band gaps for which (to the best of our knowledge) no analytic results are available. However, when a is half-integral, then (2a+1)/2 mid-band levels are in principle analytically known and each of them is doubly degenerate [1,2].
(b) Associated Lamé Potentials: We now expand our discussion to the band edges and wave functions of a much richer class of periodic potentials given by eq. (2) called associated Lamé (AL) potentials, since the corresponding Schrödinger equation is called the associated Lamé equation [1].
More precisely, we often refer to the AL potential of eq. (2) as the (p, q) potential where p = a(a + 1) and q = b(b + 1) and note that (p, 0) potentials are just the ordinary Lamé potentials. The AL potentials (2) can also be re-written in the alternative form V (x) = pm sn 2 (x) + qm sn 2 (x + K(m)) [4,5]. Clearly, the potentials (p, q) and (q, p) have the same energy spectra with wave functions shifted by K(m). Therefore, it is sufficient to restrict our attention to p ≥ q, i.e. a ≥ b.
In general, for any values of p and q, the AL potentials have a period 2K(m), but for the special case p = q, the period is K(m). From a physical viewpoint, if one thinks of a Lamé potential (p, 0) as due to a one-dimensional regular array of atoms with spacing 2K(m) and "strength" p, then the AL potential (p, q) results from two alternating types of atoms spaced by K(m) with "strengths" p and q The AL potentials for q = 2, m = 0.5 and several values of p are plotted in Fig. 2. In the critical range of p values 1 ≤ p ≤ 4 , one expects additional extrema, and these are clearly seen.
(c) Parabolas of Solvability: The AL equation is On substituting ψ(x) = [dn(x)] −b y(x) , it is easily shown that y(x), satisfies the Hermite elliptic equation [1]. On further substituting sn(x) = sin t , y(x) ≡ z(t) , one obtains Ince's equation where which is a well known quasi exactly solvable equation [1]. In particular if a + b + 1 = n ( n = 1, 2, 3, ...) then one obtains n solutions, which are given in Table 2. In particular, for any given choice of p = a(a + 1), Table 2 lists the eigenstates of the AL equation for various values of q = b(b + 1).
For q = a(a− 1), there is just one eigenstate with energy ma 2 and wave function ψ = dn a (x). Since the wave function has period 2K(m) and is nodeless, this is clearly the ground state wave function of the (a(a + 1), a(a − 1)) potential for any real choice of the parameter a. The equations p = a(a + 1) and q = a(a − 1) are the parametric forms of the equation of the parabola (p − q) 2 = 2(p + q), which is plotted in Fig. 3 and labeled P 1. For any point on the parabola, one knows the ground state wave function and energy E 0 = ma 2 . The parabola P 1 includes the points (2,0) and (6,2).
For q = (a − 1)(a − 2), we see from Table 2 that two eigenstates with energies 1 + m(a − 1) 2 and 1 + ma 2 are known. Since they have period 4K(m) and just one node in the interval L = 2K(m), they must correspond to the first and second band edge energies E 1 and E 2 of the (a(a + 1), (a − 1)(a − 2)) potential. Eliminating a from the equations p = a(a + 1) and q = (a − 1)(a − 2) gives the "parabola of solvability" (p − q) 2 = 8(p + q) − 12, which is plotted in Fig. 3 and labeled P 2. This parabola includes the points (2,0) and (6,0) which correspond to Lamé potentials. Similarly, the parabolas of solvability P n (n = 0, 1, 2, ...) corresponding to q = (a − n + 1)(a − n) in Table 2 are plotted. Note that n eigenstates are known for any point on the parabola of solvability P n.
All Lamé as well as AL potentials for which a, b are unequal integers, have two parabolas of solvability passing through. This provides a good understanding of why these are completely solvable problems. For instance, the (2, 0) potential is at the intersection of parabolas P1 (1 known state) and P2 (2 known states), thus giving the 3 known band edges. Similarly, the (6, 2) AL potential.
lies on parabola P1 (1 known nodeless state of period 2K(m)) and parabola P4 (4 known states of period 4K(m), two with 1 node and two with 3 nodes). Since we know from the oscillation theorem that 2 states of period 2K(m) are missing, so it would appear that this is an example of a quasi exactly solvable potential. However, using a well known theorem about Ince's equation [1] we discuss below that the two "missing" states are degenerate and have no band gap between them. Hence, AL potentials are also exactly solvable periodic problems with a finite number of band gaps when both a (d) Exact Results: Several exact results are known in the literature about the Ince's equation (19).
In particular, it is well known that the system satisfying Ince's eq. (19) has at most j + 1 band gaps where A, B, D are as given by eq. (20) has either non-negative integral roots, the highest of which is j, or negative integral roots, the lowest of which is −j − 1. The fact that the Lamé potentials (1) have only a band gaps when a is any positive integer is easily understood from here. We now show  Table 3 we have given all five band edge energy eigenstates of the (6, 2) potential.  4) and (5). We first discuss the results for small integer values of a, and then present some eigenstate results for arbitrary integer values of a.
In order to use the supersymmetry formalism, we must shift the Lamé potential by a constant to ensure that the ground state i.e. (the lower edge of the lowest band) has energy E = 0. For a = 1, one = dn(x) and the superpotential is W = msn(x)cn(x)/dn(x). The partner V + (x) turns out to be just V − (x − K(m)), so that this is an example of self-isospectrality. For a = 2, the potential is with a unnormalized ground state wave function ψ (−) 0 (x) = 1+m+δ−3msn 2 (x) [2]. The corresponding superpotential is and hence the partner potential V + (x) for the potential V − (x) given in eq. (22) is Although supersymmetry guarantees that the potentials V ± are isospectral, in this example they are not self-isospectral. Therefore, V + (x) as given by eq. (24) is a new periodic potential which is strictly isospectral to the potential (22) and hence it also has 2 bound bands and a continuum band. In Fig. 1 we have plotted the potentials V ± (x) corresponding to a = 2 for m = 0.8. Using eqs. (10) and (11) and the known eigenstates of V − (x), we can immediately compute all the band-edge Bloch wave functions for V + (x). In Table 1 we have given the energy eigenvalues and wave functions for the isospectral partner potentials V ± (x). In summary, for integral a, Lamé potentials with a ≥ 2 are not self isospectral. They have distinct supersymmetric partner potentials even though both potentials have the same (2a + 1) band edge eigenvalues.
(b) Associated Lamé potentials: It is easily checked from Table 2 that the solution corresponding to q = a(a − 1) as well as one of the q = (a − 2)(a − 3) solutions are nodeless and correspond to the ground state. Hence, for these cases, one can obtain the superpotential and hence the partner potential V + . For example, consider the case of p = a(a + 1), q = a(a − 1). In this case W is given by W = amsn(x)cn(x)/dn(x) , so that the corresponding partner potentials are These partner potentials are self-isospectral and so supersymmetry yields nothing new. As an illustration, the potential (6,2) as discussed in Table 3 is a self-isospectral potential and supersymmetry yields nothing new.
Let us now consider the partner potential computed from the ground state for the p = a(a + The corresponding superpotential W turns out to be Hence the corresponding partner potentials are These potentials are not self-isospectral. Thus one has discovered a whole new class of periodic potentials V + (x) for which three states are analytically known no matter what a is. In particular, the energy eigenfunctions of these three states are easily obtained by taking the corresponding energy eigenstates of V − as given in Table 2 and using eqs. (10) and (11).
We might add here that applying second order Darboux transformations to the Lamé potentials, Fernández et al [15] have obtained an interesting nonlocal effect, by which the transformed potential becomes an exact or approximately displaced copy of the original one.

V. SUPERPOSITION OF LAMÉ AND ASSOCIATED LAMÉ POTENTIALS
Using the recently discovered cyclic identities [11] and Landen transformation formulas [9,10] for Jacobi elliptic functions, we shall now further expand the list of analytically solvable periodic potential problems. In particular, we obtain the exact band edge eigenstates for potentials obtained via a certain specific type of linear superposition of Lamé as well as AL potentials.
Superposition of Lamé potentials: Consider the following linear superposition of p Lamé potentials with translated arguments: where x j ≡ x + 2(j − 1)K(m)/p. Note that for the special case of p = 2, this corresponds to the AL potential with a = b. On expressing sn 2 (x, m) in terms of dn 2 (x, m), the corresponding Schrödinger equation is given by On making use of the Landen transformation formula [9,10] one can rewrite the Schrödinger eq. (29) in the form Here A d ≡ p j<k=1 dn(x j , m)dn(x k , m). Using recently discovered cyclic identities, A d can be shown to be [11] where cs(x, m) ≡ cn(x,m) sn(x,m) , while Z(x, m) is the Jacobi zeta function [4]. On making the transformation x = αy, the Schrödinger eq. (33) can be rewritten as a Lamé equation but for the modulus parameter m and with energy E (L) (m): − ψ ′′ (y) + [a(a + 1)msn 2 (y,m)]ψ(y) = E (L) (m)ψ(y) .
Here the true energy eigenvalues E(m) of the superposed potentials (28) are related to the eigenvalues where α and A d are as given by eqs. (31) and (34) respectively whilem and m are related by eq. (32).
Similarly, the true eigenfunctions of the superposed potentials (28) are related to those of eq. (35) by As an illustration, for p = 2 (i.e. a = b AL case), it is well known [9,10] that and hence the energy eigenvalues of the a = b AL potentials (28) (with p = 2) are given in terms of those of Lamé potential (1) (but with the modulus parameterm) by For a = 1, 2 we can immediately verify that the energy eigenvalues and eigenfunctions of the (2,2) and (6,6) potentials as calculated from here are identical (as they should be) to those calculated by us by an entirely different method where no use of either Landen transformations or the cyclic identities was made [5].
In fact we can relate the energy eigenstates of the a = b AL potentials at different modulus parameters by using the relationship (15) between the corresponding Lamé energy eigenstates. For this purpose, it is best to reexpress the relation (39) entirely in terms ofm by noting that on using eq. (38) we have Using eqs. (38) to (40) in eq. (15) we find that the AL eigenvalues are related by (where for simplicity we have replacedm everywhere by m) where For a = 1, 2 we immediately verify that these relations are indeed true. Further, the corresponding eigenfunctions are related by where y = 2ix Similarly, for p = 3, it is well known that and hence the energy eigenvalues of (28) (with p = 3) are given in terms of those of Lamé's with parameterm by Thus by using Landen transformations and recently discovered cyclic identities one has discovered new solvable periodic potentials. On following the last section, it is easy to see that for any integer a ≥ 2, corresponding to every superposed potential, supersymmetry will give us another new exactly solvable periodic potential.

Superposition of Associated
where x j ≡ x+2(j−1)K(m)/p. Now proceeding exactly as above and using the Landen transformation (30) and the cyclic identity (34), one can show that the energy eigenvalues and eigenfunctions of the superposed AL potentials (46) are related to those of AL potential (2) (but with modulus parameter m) by where α,m and A d are given by eqs. (31), (32) and (35) respectively. Thus we have further enlarged the list of new exactly solvable periodic problems. Needless to say that using supersymmetry, one can further enlarge this list by obtaining partner potentials of the superposed potentials which will have the same band spectra.
As an illustration, for p = 2, 3, the energy eigenvalues of the superposed potentials (46) are given below in terms of the corresponding AL potentials (with modulus parameter m changed tom): where use has been made of eqs. (38) and (44).

VI. PT INVARIANT POTENTIALS WITH REAL BAND SPECTRA
We shall now show that the complex PT-invariant periodic potentials obtained by applying the anti-isospectral transformation [16] x → ix + β (where β is a nonzero real number) to the Lamé and AL potentials with a finite number of band gaps are also exactly solvable and have a finite number of band gaps. We give explicit expressions for the band edges of some of these potentials [8]. For the special case of the potential V (x) = −2msn 2 (ix + β, m), we analytically obtain the corresponding dispersion relation.

Let us begin with the simple observation that if ψ(x) is a solution of the Schrödinger equation for
the real potential V (x) with energy E, then ψ(ix + β) is a solution of the Schrödinger equation for the complex potential −V (ix + β) with energy −E. The new potential −V (ix + β), generated by the anti-isospectral transformation x → ix + β, is clearly PT-symmetric and will be denoted by V (P T ) (x).
It is important to understand that the key point in obtaining these results is that unlike trigonometric and most other periodic functions, the Jacobi elliptic functions are doubly periodic functions.
This allows both V (x) and V (P T ) (x) to be simultenously periodic, even though the periods are different. Note that the arbitrary nonzero constant β in the transformation x → ix + β helps us to avoid the singularities of Jacobi elliptic functions.
Let us first apply our approach to the Lamé potentials (1). On applying the anti-isospectral transformation x → ix + β [16], where β is any non-zero real number, it is easily shown that the band-edge eigenvalues and eigenfunctions of the PT-invariant potentials V (P T ) (x) = −a(a + 1)msn 2 (ix + β, m) , a = 1, 2, 3, ... , are related to those of the Lamé potentials (1) by [8] It may be noted here that the PT-invariant complex potential (51) is a periodic potential with period 2K ′ (m)(≡ 2K(1 − m)). Thus we have shown that the PT-invariant periodic potential (51) too has precisely a band gaps and hence 2a + 1 band edges at energies given by relation (52). Special mention may be made of the remarkable fact that for any integer a, all bands and band gaps exchange their role as one goes from a Lamé potential to its PT-transformed version.
For any band structure problem, one important quantity is the discriminant ∆ [1] which gives information about the number of band gaps as well as their widths. The question is if we can relate ∆ (P T ) with the discriminant ∆ for the corresponding Lamé potential. We now show that using eqs.
(15) and (52) this is indeed possible. In particular, on using these two equations we deduce that and hence the corresponding discriminants are related by [8] As an illustration, in Fig. 4 we plot the real and imaginary parts of the PT-invariant complex potential −6msn 2 (ix + β, m). Using the well known results for the Lamé potential with a = 2 (see Table 2) and eq. (52), the ground state (lowest band edge) eigenvalue and eigenfunction is easily shown to be where δ = √ 1 − m + m 2 . In Table 4 we have given all the five band edge eigenvalues and eigenfunctions of this PT-invariant potential, where we have subtracted off the ground state energy from the potential so that the lowest band edge by construction is at zero energy. Observe from the table that the band edges are both periodic as well as anti-periodic with periods of 2K ′ (m) and 4K ′ (m) respectively thereby showing that contrary to the suggestion of Bender et al. [17], the absence of anti-periodic band edges is not a general property of PT-invariant periodic potentials.
Additional analytically solvable finite band gap potentials can be obtained from here by using supersymmetry. In particular, consider the potential V state eigenfunction is given in eq. (55), and we find that the corresponding superpotential is Thus the supersymmetric partner potential is and it has the same energy spectrum as V Table 4]. In this way, one has discovered yet another PT-invariant complex potential with a finite number of band gaps. The corresponding band edge energy eigenfunctions are immediately obtained from those of V and then obtaining the corresponding PT-invariant potential by the anti-isospectral transformation x → ix + β. Thus one has found three distinct complex PT-invariant periodic potentials having the same band spectra. One can show that this is true for any integral a ≥ 2. On the other hand, for a = 1 one can show [8] that the three potentials are not distinct but are self-isospectral (i.e. differ from each other by a translation by a constant) [13,15].
For the special case of the a = 1 Lamé potential, the dispersion relation is analytically known [3].
We now show [8] that it is possible to derive the dispersion relation for the PT-invariant potential (51) with a = 1. To that end we start from the Schrödinger equation: where we have subtracted the ground state energy from the potential so that the new potential has zero ground state energy. On substituting y = ix + β, eq. (58) takes the form Now it is well known that two independent solutions of this equation are given by [3] ψ where H, θ, Z are Jacobi eta, theta and zeta functions while α 1 is related to E of eq. (58) by On using the fact that while θ(ix + β) is periodic function with period 2K ′ (m), H(ix + β) is only quasi-periodic [3], i.e.
and using the Bloch condition, it is easily shown that for the PT-invariant complex potential (1) with a = 1, the dispersion relation is given by where α 1 is given by eq. (61).
We now turn to the associated Lamé potentials given by eq. (2) where without any loss of generality we consider a > b with both being positive integers. On using the anti-isospectral transformation, it is easy to see that the band edges of the potential (2) and its PT-transformed one as given by are again related by the relation (52).
For the special case a = b, using relations (40), (41) and (52) we obtain an interesting relation betwen the eigenvalues of the AL potential and the corresponding complex PT-invariant potential given by where m 1,2 are as defined by eq. (42). Hence we can immediately relate the discriminant ∆ for the a = b AL potential and the corresponding complex PT-invariant potential, i.e. we have VII. UNUSUAL BAND STRUCTURE FOR THE DOUBLE SINE-GORDON EQUA-TION Finally, we would like to consider the band spectra of the double sine-Gordon equation characterized by the periodic potential where a, b are real, and without loss of generality a is taken to be nonnegative. This equation arises in several areas of condensed matter physics. It is well known that this system has an infinite number of bands and band gaps. Further, it is also well known that if a is an integer then a band edges of period π(2π) are analytically known depending on whether a is an odd(even) integer [18,19]. However, what is not so well known [1] is that in case a is an integer, then the potential has unusual band spectra.
We start from the Schrödinger equation for the potential (67). On using the ansatz in the Schrödinger equation, it is easily shown that φ satisfies Ince's eq. (19) As discussed before, this is a quasi-exactly solvable equation, i.e. a band edges of period π(2π) are analytically known when a is an odd (even) integer [18,19]. We can now compute the polynomials Q(µ) and Q * (µ) defined in eq. (21), and from them we conclude that if a is an odd (even) integer then there are at most a+1 2 ( a+2 2 ) band gaps of period π(2π). It is worth pointing out here that similar conclusions can also be drawn about the complex PTinvariant periodic potential Note that this potential is invariant under the parity transformation x → x + π/2 followed by T . As has been shown in ref. [19], when a is an even integer then PT symmetry is spontaneously broken and there are no quasi-exactly solvable states of period 2π since in this case all the eigenvalues are now complex conjugate pairs. However, if a is an odd integer then there are a quasi-exactly solvable states of period π and in this case PT symmetry is not spontaneously broken. On completing the above analysis, it follows that if a is an odd integer then there are at most (a + 1)/2 band gaps of period π. Thus in this case the band spectrum is rather unusual in that the majority of the infinite number of band edges are anti-periodic and have period 2π, thereby again showing that the absence of anti-periodic band edges is not a characteristic feature of PT-invariant potentials.    The five eigenvalues and eigenfunctions for the self-isospectral associated Lamé potential corresponding to a = 2, b = 1 which gives (p, q) = (6, 2). The potential is V − (x) = 6msn 2 (x) + 2m cn 2 (x) dn 2 (x) − 4m, and has period 2K(m). The number of nodes in the interval 2K(m) is tabulated.      Table 2. Parabola P n is for q = (a − n + 1)(a − n) for n = 1, 2, 3, .... One knows n eigenstates for any point on parabola P n from  The potential has a period 2K ′ (0.8) = 3.3192.