On the Spectra of Real and Complex Lam\'e Operators

We study Lam\'e operators of the form $$L = -\frac{d^2}{dx^2} + m(m+1)\omega^2\wp(\omega x+z_0),$$ with $m\in\mathbb{N}$ and $\omega$ a half-period of $\wp(z)$. For rectangular period lattices, we can choose $\omega$ and $z_0$ such that the potential is real, periodic and regular. It is known after Ince that the spectrum of the corresponding Lam\'e operator has a band structure with not more than $m$ gaps. In the first part of the paper, we prove that the opened gaps are precisely the first $m$ ones. In the second part, we study the Lam\'e spectrum for a generic period lattice when the potential is complex-valued. We concentrate on the $m=1$ case, when the spectrum consists of two regular analytic arcs, one of which extends to infinity, and briefly discuss the $m=2$ case, paying particular attention to the rhombic lattices.


Introduction
The Lamé equation (1) − d 2 ψ dz 2 + m(m + 1)℘(z)ψ = λψ, where m ∈ N and ℘(x) is Weierstrass' elliptic function, satisfying (2) (℘ ) 2 = 4(℘ − e 1 )(℘ − e 2 )(℘ − e 3 ), is a classical object of 19 th century mathematics. Viewed as an equation in the complex domain, its solutions, which have a number of remarkable properties, were described explicitly by Hermite and Halphen, see e.g. [22]. Note that for x on the real line the potential ℘(x) has singularities. If, however, e 1 , e 2 , e 3 are all real, one can make a pure imaginary half-period shift by z 0 = ω 3 and consider the Lamé operator (3) L = − d 2 dx 2 + m(m + 1)℘(x + z 0 ), whose potential is real, 2ω 1 -periodic and regular on the whole of R. Consequently, one can apply Bloch-Floquet theory, which implies that the spectrum has a band structure, as described e.g. in [17].
Generically, the spectrum of a periodic Schrödinger operator (or Hill operator) on the real line consists of a countably infinite number of bands. It was Ince [12] who in 1940 first pointed out the remarkable fact that the spectrum of L has a band structure with not more than m gaps. Erdélyi [8] later showed that in fact all m gaps are open. Nowadays, this is just the simplest example in the large class of finite-gap operators discovered in the 1970s, see [7,13,16]. It turned out that all such operators can be described explicitly in terms of hyperelliptic Riemann theta functions, with the Lamé operator (3) corresponding to the elliptic case. However, it seems that the question of exactly which gaps in the spectrum are open has not been explicitly discussed in the literature even in this case.
The first result of this paper, obtained in Section 2, demonstrates that in the Lamé case it is precisely the first m gaps that are open. Although this fact may not be surprising for experts, we could not find a rigorous proof in the literature. For m = 1, the result is illustrated in Figure 1, showing that all the closed gaps are indeed located in the infinite spectral band.
In the second part of the paper, we consider the Lamé operator with a complexvalued periodic potential (4) V (x) = m(m + 1)ω 2 ℘(ωx + z 0 ), where ω is any half-period of ℘(x), and the only assumption on z 0 ∈ C is that the corresponding potential is non-singular. The spectral theory of Schrödinger operators with a complex periodic potential has been studied in [1,9,19,21]. In particular, Rofe-Beketov [19] showed that the spectrum of a Schrödinger operator with periodic, regular, but complex-valued potential u(x) can be defined as the set of λ ∈ C such that at least one solution of the equation Lψ = λψ is bounded on the whole real line. Equivalently, the corresponding Floquet multiplier ρ(λ), given by where T is a period of u(x), should lie on the unit circle: |ρ(λ)| = 1.
In the case of the Lamé operator, it follows from Gesztesy and Weikard [9,21], who studied the general elliptic finite-gap case, that the spectrum consists of finitely many regular analytic arcs, with at most one arc extending to infinity.
Note that if the shift z 0 = ω 3 in the Lamé operator (3), we have in general a periodic, regular, but complex-valued potential. However, it is easy to see that the Floquet multipliers do not depend on z 0 , so that the spectrum is the same as in the self-adjoint case. Moreover, setting ω = ω 1 , ω 3 in (4), we have essentially Ince's result, since the corresponding operator is equivalent to the previous case.
In Section 3, we consider genuinely complex instances of the Lamé operator. More specifically, to begin with we allow any half-period ω in the potential (4) and impose no reality conditions on the elliptic curve (2).
Using Hermite's solutions of (1), we demonstrate that the spectrum of such a complex Lamé operator is independent of the value of z 0 . We prove this result for all m ∈ N.
Then we focus on the m = 1 case. We recall that Hermite's solutions of the corresponding Lamé equation where k ∈ C and σ(z) and ζ(z) are the Weierstrass σ-and ζ-function, see e.g. [22]. Since the solutions have the Floquet property with η = ζ(ω), they remain bounded on the line z = ωx + z 0 , x ∈ R, if and only if By the result of Rofe-Beketov, the corresponding values of λ = −ω 2 ℘(k) constitute the spectrum of the Lamé operator . Using Morse theory to determine the topological nature of the zero set (6), we prove that, under certain non-degeneracy and non-singularity assumptions, the spectrum of (7) consists of precisely two regular analytic arcs, with one arc extending to infinity and the remaining endpoints being −ω 2 e j , j = 1, 2, 3. We also show that the arc extending to infinity is asymptotic to the positive half of the real line.
We have used the software R to numerically compute the spectrum of the complex Lamé operator (7) in particular cases. One example is displayed in Figure 2, along with a plot of the corresponding zero set (6). Further examples are provided in Section 3, see Figures 4 to 7.
We study in further detail Lamé operators with either rectangular or rhombic period lattices. In particular, for ω = ω 2 or ω = ω 1 , respectively, we show that all closed spectral gaps are contained in the spectral arc extending to infinity.

Spectral gaps for the real Lamé operator
In this section, we consider the Lamé operator in the Jacobian form , with 0 < k < 1, m ∈ R and sn x one of the Jacobi elliptic functions, see e.g. [22]. The spectrum of L is the set of λ ∈ R such the corresponding Lamé equation (9) − d 2 ψ dx 2 + m(m + 1)k 2 sn 2 (x, k)ψ = λψ has at least one solution ψ = ψ(x) that is bounded for all x ∈ R. (For λ / ∈ R there are no bounded solutions.) This equation is of Hill type: the potential m(m + 1)k 2 sn 2 (x, k) is a real periodic function with (primitive) real period 2K given by Hence, by the oscillation theorem (see e.g. [14]), there exists a sequence of real numbers where λ j , λ j → ∞ as j → ∞, such that (9) has a 2K-periodic solution if and only if λ = λ j , j ∈ Z ≥0 , and a 2K-antiperiodic solution if and only if λ = λ j , j ∈ N.  (9) coexist for λ = λ 2n−1 = λ 2n or λ = λ 2n−1 = λ 2n , respectively.
Ince [12] showed that coexistence occurs if and only if m ∈ Z and that, in such a case, the number of gaps equals at most and Erdélyi [8] established that not fewer than gaps remain. We shall prove that all points of coexistence are located in the infinite spectral band. For future reference, we summarise these results in the following theorem. A detailed account of Ince's result can be found in the book by Magnus and Winkler [14]. Following their line of reasoning, we recall some key constructions and results that we need to complete the proof of Theorem 1. Since (8)- (9) are invariant under m → −m − 1, we may and shall restrict attention to m ∈ Z ≥0 .
For convenience, we transform the Lamé equation (9) into an equation of Ince's type (11) (1 + a cos 2φ) where a, b, c, d are real parameters with |a| < 1. More precisely, by the substitutions where the Jacobi amplitude am x = am(k; x) is determined by and ψ(x) = y(am x), we arrive at (11) with Note that x = 2K corresponds to φ = π, which is reflected in the fact that the coefficients in (11) are periodic with period π.
If Ince's equation (11) has two linearly independent π-periodic or π-antiperiodic solutions, then we can find two solutions y 1 , y 2 such that (13) Substituting the periodic series into (11) with (12) in force, we obtain the recurrence relations where we have introduced Similarly, the coefficients in the antiperiodic series should satisfy the recurrence relations At this point, we find it convenient to consider even and odd values of m separately. Letting m = 2ν, ν ∈ Z ≥0 , we have P (ν) = P * (−ν) = 0, so that the infinite tridiagonal matrices corresponding to the recurrence relations (15)- (16) and (19)- (20) become reducible.
In the periodic case, the finite dimensional parts of these matrices are which correspond to sequences terminating at n = ν in the sense that A 2n = B 2n = 0 for n > ν.
Introducing the functions be the sets of corresponding eigenvalues of the Lamé operator (8). Recalling (17), it is clear that all off-diagonal entries of the matrices K ν,1 and K ν,2 are positive. From the general theory of tridiagonal (Jacobi) matrices (see e.g. [20]), it follows that S ν,1 and S ν,2 consist of ν + 1 respectively ν real and distinct eigenvalues. By specializing Theorem 7.3 in [14] to the Lamé case, we find that S ν := S ν,1 ∪ S ν,2 contains the simple part of the periodic spectrum. More precisely, we have the following lemma.
If λ belongs to the periodic spectrum of the Lamé operator (8) and has multiplicity 1, then λ ∈ S ν .
Turning to the the antiperiodic case, we note that the recurrence relations (19)- (20) have no solutions given by terminating sequences. Instead, the relevant finite dimensional matrices . Just as in the periodic case, we see that each of the two sets S * ν,j consists of ν real and distinct eigenvalues. Moreover, we infer from Theorem 7.4 in [14] that S * ν := S * ν,1 ∪ S * ν,2 contains the simple part of the antiperiodic spectrum. Lemma 2. Let m = 2ν, ν ∈ Z ≥0 . If λ belongs to the antiperiodic spectrum of the Lamé operator (8) and has multiplicity 1, then λ ∈ S * ν . Since the spectrum of the Lamé operator (8) has precisely m = 2ν gaps, the ends of the gaps must be given by the m largest eigenvalues in S ν and all m eigenvalues in S * ν . (Recall that (−∞, λ 0 ] is not counted as a gap.) We are now ready to complete the proof of Theorem 1 for even m. From (17)- (18) and (23)-(24) it easily follows that (21)-(22) and (25) This implies that the m gaps close at the points λ = j 2 , j = 1, . . . , m, as k → 0.
On the other hand, taking k → 0 in (9), we obtain the free Schrödinger equation −ψ = λψ, and (excluding the constant solution ψ = 1) the m smallest values of λ for which it has π-periodic or π-antiperiodic solutions are precisely λ = j 2 , j = 1, . . . , m. Using that the eigenvalues λ = λ(k) ∈ S ν ∪ S * ν have multiplicity one, a standard argument shows that they depend analytically on k, see e.g. Chapter XII in [18]. Hence there exists k 1 ∈ (0, 1) such that all points of coexistence are indeed contained in the infinite spectral band for k ∈ (0, k 1 ).
There remains to show that neither of the following two scenarios can occur for k ≥ k 1 : first, a point of coexistence crosses one or more open gaps; and, second, an open gap closes while a point of coexistence turns into an open gap. The first scenario is immediately ruled out by the interlacing property (10), and the second by the analyticity of the eigenvalues.
This concludes the proof in the case of even m. It is straightforward to adapt the arguments above to handle the case of odd m. In particular, the roles of the periodic and antiperiodic solutions will then be interchanged, and analogues of Lemmas 1-2 can once more be obtained from Theorems 7.3-7.4 in [14]. Example 1. Consider the simplest nontrivial even case m = 2, which corresponds to the Lamé operator . The edges of its two spectral gaps [λ 1 , λ 2 ] and [λ 1 , λ 2 ], which correspond to the first two eigenvalues λ 1 , λ 2 from its antiperiodic spectrum and the second and third eigenvalues λ 1 , λ 2 from its periodic spectrum, are easily computed explicitly.
Indeed, observing that , we find that the edges of the first gap are given by (22). To determine the edges of the second gap, we should find the roots λ of the polynomials Recalling (17)- (18), we deduce that the two roots of the former polynomial are whereas the latter has the single root The relevant eigenvalues are plotted together with the ground state eigenvalue from the periodic spectrum in Figure 3.
In the limit k → 1 the real period goes to infinity and the Lamé potential The potential u(x) = −6sech 2 x decays exponentially at infinity and is a special example of a 2-soliton potential, see e.g. [6]. The corresponding operator x is known to be reflectionless. Its spectrum has continuous part [0, ∞) and discrete part consisting of the two eigenvalues −1 and −4. Note that for k → 1 the formulae (28)-(30) show that the two finite spectral bands degenerate to the eigenvalues λ 0 = λ 1 = 2 and λ 2 = λ 1 = 5, while λ 2 → 6, in agreement with Figure 3.

The spectrum of the complex Lamé operator
Let E = C/L be a general elliptic curve, where L is a period lattice. To begin with, we do not impose any reality conditions. Let ℘(z) be the corresponding Weierstrass' elliptic function, which satisfies the equation see e.g. [22]. Consider the complex Lamé operator L = L(E, ω, m, z 0 ) in L 2 (R) given by , with m ∈ N, ω a half-period and z 0 ∈ C chosen such that the line z = ωx+z 0 , x ∈ R, does not contain any points of L. Note that the potential m(m + 1)ω 2 ℘(ωx + z 0 ) is regular and periodic with period 2, but is in general complex-valued.
From the work of Rofe-Beketov [19] on the spectral theory of non-self-adjoint differential operators in L 2 (R) with complex-valued periodic coefficients, it follows that the spectrum of L can be characterised as the set of λ ∈ C such that the associated Lamé equation To prove this, we consider the Lamé equation in the complex domain: (33) − d 2 ψ dz 2 + m(m + 1)℘(z)ψ = λψ, z ∈ C. Its solutions were described by Hermite [11] in the following way, see e.g. Section 23.7 in [22]. First, one notes that the product X of any pair of solutions of (33) satisfies the third order equation Changing variable to ξ = ℘(z) yields the algebraic form of this equation: Making a power series ansatz in ξ − e 2 (say), it is straightforward to verify that the Lamé equation (33) has two solutions ψ ± whose product X is a polynomial in ξ = ℘(z) of the form with some coefficients c r ∈ C. This polynomial can be written in the factorised form where the k j ∈ C are determined up to a sign by λ. Let us now assume that ψ ± are linearly independent, so that their Wronskian (Otherwise ψ ± are one of the 2m + 1 Lamé functions.) The sign of k j can then be fixed by requiring dX dξ Finally, by solving (34) and d(ψ + ψ − )/dz = dX/dz for d log ψ ± /dz and integrating, one finds that (33) is satisfied by the two functions Restricting (33) to the line z = ωx + z 0 , we see that (32) has the solutions The quasi-periodicity of the σ-and ζ-function yield the Floquet property with Floquet multiplier Since f is manifestly independent of z 0 , it follows that the spectrum of the complex Lamé operator (31) is indeed independent of the value of z 0 . Let us now focus on the simplest non-trivial case m = 1. We fix generators 2ω 1 , 2ω 3 ∈ C of the period lattice L such that Im(ω 3 /ω 1 ) > 0, and set ω 2 = ω 1 +ω 3 . The solutions (36) of (32) take the simple form Letting η = ζ(ω), the corresponding Floquet multiplier is given by so that φ ± (x, −ω 2 ℘(k)) are bounded for x ∈ R if and only if  Moreover, the index of u(k) at k = ±k * is equal to one.
By the quasi-periodicity of ζ(z), we have Hence the Legendre relation η 1 ω 3 − η 3 ω 1 = iπ/2 ensures that the right-hand side equals zero, so that u(k) is a well-defined function on E × . Writing k = s + it, the critical points of u(k) are the solutions of u s = u t = 0, which are equivalent to Since ℘(k) is an even function of order two, this equation has precisely two solutions k = ±k * . Letting v(k) = Im[f (k)], we infer from the Cauchy-Riemann equations that the Hessian matrix

We recall that u(k) is a Morse function if H(u)
is non-singular at all of its critical points, with the index being equal to the number of negative eigenvalues. Clearly, H(u) is singular if and only if u ss = v tt = 0 or, equivalently, Recalling that the only zeros of ℘ (k) on E × are k = ω j , j = 1, 2, 3, we see that H(u) is singular only if η + ωe j = 0 for some j = 1, 2, 3, which is excluded by the assumption (40).
Using Morse theory, we can thus determine the topological nature of the zero set (39). We assume that (40) holds true, so that the critical points of u(k) are non-degenerate, and that the level set u(k) = 0 is non-singular in the sense that it does not contain the critical points k = ±k * . We note that the lemniscatic case yields an example of a singular level set and the rhombic cases contain singular examples as well as a degenerate level set, see Figures 5 and 7.
Proposition 2. Under our non-degeneracy and non-singularity assumptions, the closureZ u ⊂ E of the zero set consists of two simple closed curves in E representing the same (non-trivial) homology class.
Proof. By our non-singularity assumption, we can fix the sign of k * such that For c ∈ R, we let E c = E × c ⊂ E be the closure of the set E × c := {k ∈ E × : −∞ < u(k) ≤ c} (where, as we shall see below, taking the closure amounts to adding the point k = 0). First, we prove that each E c with c < −c * is diffeomorphic to a disc. Since the meromorphic function f (k) = ηk −ωζ(k) is odd and has a simple pole at k = 0 with residue −ω, we can find neighbourhoods U, V 0 and a biholomorphic mapping g : U → V such that (1) g(0) = 0 and g (0) = 1, the set Re[−ω/z] ≤ c, z = 0, is given by where (x, y) = (0, 0). By choosing c < −c * sufficiently small, we can ensure that the disc (43) is contained within V and therefore that it is diffeomorphic to E c . For c < c < −c * , we infer from Theorem 3.1 in [15] that E c is diffeomorphic to E c , and the assertion follows. (To be precise, the theorem does not apply as it stands, since u(k) is not defined at k = 0. However, < grad u, grad u > −1 grad u extends to a smooth vector field on a neighbourhood of E c , c < −c * , and the construction of the diffeomorphisms ϕ t : E → E is readily adapted to the present case.) Letting 0 < < c * , the region u −1 ([−c * − , −c * + ]) contains no critical point of u other than −k * . Since the index of u(k) at k = −k * is one, it follows that E −c * + is diffeomorphic to a disc with a thickened 1-cell attached, c.f. Theorem 3.2 in [15]. Moreover, increasing c from −c * + to 0 does not alter the diffeomorphism type of E c . Considering the boundary of E 0 , we thus arrive at the statement.
Recalling that the spectrum of the m = 1 operator (31) consists of all λ = −ω 2 ℘(k), k ∈ Z u , we are now ready to state and prove the first of the main results in this section.
Theorem 2. Let m = 1. Under our non-degeneracy and non-singularity assumptions, the spectrum of the complex Lamé operator (31) consists of two regular analytic arcs. Moreover, precisely one arc extends to infinity and the remaining endpoints are −ω 2 e j , j = 1, 2, 3.
Proof. By Proposition 2, there exist simple closed curves γ 1 , γ 2 such that For each j = 1, 2, 3, we deduce from the Legendre relation that ω j ∈ Z u . In addition,Z u contains k = 0 and is invariant under the reflection k → −k, which implies that each curve γ j contains precisely two fixed points. Since the quotient map E → E/(k → −k) = CP 1 is given by k → ℘(k), the image ofZ u in CP 1 under the map k → λ = −ω 2 ℘(k) consists of two regular analytic arcs connecting two pairs of fixed points.
In the next proposition we look at the behaviour of the infinite spectral arc at infinity.
Proposition 3. For m = 1, the spectral arc of the complex Lamé operator (31) that extends to infinity is asymptotic to the positive real line.
Proof. Recalling from the proof of Proposition 2 the biholomorphic mapping g : U → V , we observe that any z ∈ V satisfying the spectral condition u(g −1 (z)) = Re[−ω/z] = 0 is of the form z = iωs, s = 0.
For the remainder of this section, we restrict attention to real elliptic curves E given by There are two types of such curves, depending on the sign of the discriminant ∆ = g 3 2 − 27g 2 3 . If ∆ > 0, the roots e 1 , e 2 , e 3 of the polynomial p(x) = 4x 3 − g 2 x − g 3 are all real, the corresponding real curve consists of two ovals (M -curve) and the period lattice L is rectangular.
If ∆ < 0, we have one real root e 1 and a pair e 2 , e 3 of complex conjugate roots, e 3 = e 2 . The corresponding real curve consists of one oval and the period lattice L is rhombic.
We first consider the rectangular case, with basic half-periods The simplest instance of the corresponding Lamé operator (31) that is truly complex is obtained by choosing ω = ω 2 = ω 1 + ω 3 .
For this choice of ω, we have used the software R to numerically compute the spectrum of (31) for g 3 = 1 and four different values of g 2 . The results are displayed in Figure 4. We note that the (anti-)periodic solutions (37) of (32) are given by k ∈ Z u satisfying By the Legendre relation, v(ω 3 ) = −v(ω 1 ) = π/2 and v(ω 2 ) = 0. Since λ = −ω 2 ℘(ω j ), j = 1, 2, 3, at the endpoints of the spectral arcs, the remaining values of p must correspond to points contained in one of the two spectral arcs. Following standard terminology for the real Lamé operator, we call such a point in the spectrum a closed spectral gap. In analogy with Theorem 1, we have the following result.
Theorem 3. For m = 1, ω = ω 2 and ω 1 , ω 3 of the form (44), all closed spectral gaps of the complex Lamé operator (31) are located on the spectral arc extending to infinity. Our proof of this result relies on a detailed analysis of the lemniscatic case. The corresponding setZ u and spectrum of (31) are shown in Figure 5. The curve k = (1 − i)ω 1 s, −1 ≤ s ≤ 1, and spectral arc [0, ∞) are deduced analytically in the proof of Proposition 4 below, whereas the remaining curve and spectral arc were obtained numerically using the software R. are Proof. For the lemniscatic ζ-and ℘-function, we recall the identities , e 2 = 0, see e.g. Section 23.5(iii) in [5]. We note that the spectrum is given by which clearly implies invariance of the spectrum under complex conjugation. Using the first identity in (48), we find that Since e 2 = 0 andλ = λ whenever k = (1 − i)ω 1 s, 0 < |s| ≤ 1, Proposition 3 ensures that [0, ∞) coincides with the spectral arc extending to infinity. The expression (46) for the endpoints of the finite spectral arc follows directly from (49) and Theorem 2. Following the same line of reasoning as in the proof of Proposition 3, and using standard power series expansions of ζ(z) and ℘(z) around z = ω j , we find parameterisations such that (50) λ(t) = −ω 2 2 e j + (3e 2 j − g 2 /4) η 2 ω 2 + e j |ω 2 | 2 2 t + O(t 2 ), t → +0. (In fact, this requires no reality assumptions on L and any half-period ω and corresponding η can be substituted for ω 2 and η 2 , respectively.) Specialising to the lemniscatic case, we readily deduce (47).
We turn now to the proof of Theorem 3. SinceZ u is invariant under the reflection k → −k, we may and shall restrict attention to that part of the fundamental periodparalellogram given by Re k ≥ Im k. From (45) and the Legendre relation, we infer that Given that k = ±k * are the only critical points of f (k), the function v(k) must be monotone on each component of Z u . Moreover, since f (k * ) = 0 and f (k * ) = 0, we can find neighbourhoods U k * , V 0 and a biholomorphic mapping g : Hence, in the neighbourhood U of k = k * , the zero setZ u is given by x = ±y. From these observations and (51), we infer that v(k) is strictly decreasing and increasing as k → k * along the curve connecting 0,ω 2 and ω 1 , −ω 3 , respectively, and that 0 < v(k * ) < π/2. In other words, all remaining solutions of (45) are located on the component extending from k = k * towards k = 0.
This result is readily generalised to all τ := ω 3 /ω 1 ∈ i(0, ∞). Indeed, when we alter the value of τ from τ = i two things can happen: either the two curves remain intersecting or they break up into two disconnected curves. In the former case we can follow the same line of reasoning as in the lemniscatic case, and in the latter case we need only note that the closed spectral gaps cannot move from one curve to the other, since their locations depend continuously on τ . Applying the mapping k → λ = −ω 2 ℘(k), we thus conclude the proof of Theorem 3.
Consider now the case ∆ < 0, corresponding to a rhombic period lattice L. We choose basic half-periods ω 1 , ω 3 of the form Figure 6.
Proposition 5. In the rhombic case (52) the spectrum of the complex Lamé operator (31) with ω = ω 1 is invariant under λ →λ, and the spectral arc extending to infinity coincides with [−ω 2 1 e 1 , ∞). Moreover, all closed spectral gaps are located on the spectral arc extending to infinity.
Using the software R, we have in Figure 7 plotted the spectrum of the complex Lamé operator (31) in four different rhombic cases, see Figure 6 for the corresponding setsZ u and period lattices L. These four cases exemplify the different types of spectra that occur for rhombic period lattices. In the lower left plot, we have the pseudo-lemniscatic case; and the upper left and lower right plots correspond to the two real forms of the complex equianharmonic curve. It would be interesting to study the properties of the corresponding elliptic curve E * . Note that it differs from both the lemniscatic and the equianharmonic curve, which correspond to j = 1728 and j = 0, respectively.
We note here only that the angles between the corresponding spectral arcs are all equal to 2π/3. More generally, we have the following general result. Proposition 6. The angle between intersecting spectral arcs is always a multiple of π/k, where k ∈ N.
Indeed, λ is contained in the spectrum if and only if τ (λ) = trM (λ) ∈ [−2, 2] ⊂ C, where M (λ) is the monodromy matrix. Since M (λ) depends analytically on λ, we have locally for some k ∈ N. If a 0 = ±2, which corresponds to λ 0 being an end point of the spectral arc, we have an k-pod with angles 2π/k, see e.g. tripod in Figure  7. If a 0 ∈ (−2, 2), then we have k arcs intersecting at λ 0 with angles between neighbouring arcs being π/k, see e.g. the k = 2 cases in the bottom two plots in We restrict the attention to the rhombic case when ∆ = g 3 2 − 27g 2 3 < 0. Setting g 3 = 1, this corresponds to g 2 ∈ (−∞, 3).
For g 2 ∈ (0, 3), the roots of the first factor in the right-hand side of (55) z 2 − 3g 2 = 0 yield two real endpoints λ = ω 2 z = ±ω 2 √ 3g 2 , whereas the second factor 4z 3 − 9g 2 z − 27g 3 = 0 gives one real positive endpoint as well as a complex conjugate pair of endpoints. Hence we expect that for g 2 ∈ (0, 3) the spectrum of (31) consists of two real intervals, one of which extends to infinity, and one arc connecting the two complex endpoints. Moreover, as g 2 → 0, the finite interval collapses to a double point. We can see this in the first two plots in Figure 8, produced numerically using the software R. Note that in the special case g 2 = 0, the spectrum consists of only two arcs.
For g 2 ∈ (−∞, 0), the first factor yields two purely imaginary endpoints λ = ±iω 2 √ −3g 2 , which suggests a spectrum consisting of two arcs each connecting two endpoints in the upper-and lower-half-plane, respectively, and a real interval extending to infinity (see the last plot in Figure 8).  Figure 8. Spectra of the complex Lamé operator (54) for rhombic period lattices L and ω = ω 1 .

Concluding remarks
The spectral theory of non-self-adjoint operators, in particular Schrödinger operators with complex potentials, has attracted significant attention in the last two decades, see e.g. [3,4,21]. Nevertheless, it is fair to say that it is still a far less complete theory than in the self-adjoint case. In particular, we feel there is a need for more examples, which can be studied explicitly (like the complex harmonic oscillator in [4]).
In our paper we used the example of the classical Lamé equation, whose general solution was found by Hermite in the 19th century. However, the explicit formulae quickly become quite complicated as m grows. This makes the question about the geometry of the corresponding spectra highly nontrivial. Our results give a general picture for m = 1 and to some extent also for m = 2, but we have some open questions left even in these simplest cases.
To understand the structure of the spectrum of the complex Lamé operator (31) for larger m, we probably need to combine the explicit formulae with numerical computations. A relation with the quantum top [10] could be useful as well.
A closely related interesting problem is to study the spectrum of the difference version of the Lamé operator where T η is the shift operator defined by T η ψ(x) = ψ(x+η), and θ 1 (x, τ ) is the odd Jacobi theta function [22] (see [23] and references therein). When η = p q is rational we have an operator with periodic coefficients, in general complex. We expect that the geometry of the corresponding spectrum (in particular, the band structure in the real case) depends on the arithmetic of p and q.