A new treatment for some periodic
Instituto de Física Teórica, Universidade Estadual Paulista,
Barra Funda, 01140-070, São Paulo, SP, Brazil
We revise some aspects of the asymptotic solution for the eigenvalues for Schrödinger operators with periodic potential, from the perspective of the Floquet theory. In the context of classical Floquet theory, when the periodic potential can be treated as small perturbation we give a new method to compute the asymptotic spectrum. For elliptic potentials a generalized Floquet theory is needed. In order to produce other asymptotic solutions consistent with known results, new relations for the Floquet exponent and the monodromy of wave function are proposed. Many Schrödinger equations of this type, such as the Hill’s equation and the ellipsoidal wave equation, etc., can be treated by this method.
Mathematics Subject Classification (2010): 35P20, 33E10, 34E10.
Keywords: Schrödinger operator, periodic potential, elliptic potential, Floquet theory, asymptotic eigenvalue.
- 1 Introduction
- 2 Large eigenvalue perturbation
- 3 Some examples
- 4 On doubly-periodic Floquet theory
- 5 Conclusion
- A The contour integrals of and
- B Some formulae for elliptic functions
Consider the following 1-dimensional stationary Schrödinger equation with periodic potential, i.e. a second order periodic ordinary differential equation
It is applied in many areas, from celestial mechanics to accelerator physics and quantum mechanics. There is a large amount of literatures about the linear problem with periodic coefficient [1, 2, 3, 4, 5, 6, 7, 8]. In this paper we focus on the particular aspect about asymptotic solution for the spectrum . By “asymptotic solution” we mean a solution expanded as an asymptotic series controlled by a small parameter. The parameter space of equation (1) consists of and the coupling strength of collectively denoted by . A different asymptotic problem is the asymptotic series of eigenfunction for large complex .
The basic fact about the solution of (1) is the Floquet theory. There are two linearly independent basic solutions to (1), denoted as . As and also satisfy the equation, therefore they must be linear combinations of the basic solutions,
The nonsingular matrix does not depend on the base point , it is called the monodromy matrix. For Eq. (1) the Wronskian of are constant, so we have . Therefore the two eigenvalues of can be written as , they are called the Floquet multipliers. The Floquet exponent is a function of the eigenvalue and couplings of the potential, . In quantum physics is called the quasimomentum, and is (minus of) the energy, stable solution exists only for real . It is a principle problem to find the dispersion relation which is the spectral solution of (1). A commonly used method to determine the relation of and is Hill’s method using the infinite determinant. For most periodic potentials , when the parameters take generic value it is impossible to write down an explicit analytical solution. However, it is possible to obtain asymptotic solutions. If the leading order term and the small expansion parameter are known, we can derive the subleading terms from the relation obtained from the Hill’s determinant.
This problem has been a classical topic in differential equation and quantum theory. However, at least after reading the books [1, 2, 3, 4, 5, 6, 7, 8], it seems that there are some gaps on this topic. The Floquet theory introduced above can be refered as classical Floquet theory as it is a well understood topic for the case of real singly-periodic potential. If the potential is a periodic function of more general type, for example an elliptic function, is there an analogous theory? Generally speaking, not much is known about the Floquet theory for elliptic potential and its relation to the spectral problem. Consider the ellipsoidal wave equation for example. From the general consideration that, when the kinetic energy is very large the potential can be treated as small perturbation, i.e. where is the characteristic strength of the potential (which means the dominant one among all or certain “average” of all ), an asymptotic spectral solution should exist. Its existence can be inferred also from the relation of the ellipsoidal wave equation and the Lamé equation/Mathieu equation whose large spectrum are already known, see e.g. [6, 7, 8] and [9, 10]. However, it seems such large energy (weak coupling) asymptotic solution has not been given for the ellipsoidal wave equation. On the other hand, another asymptotic solution has been obtained sometime ago  which gives the spectrum of small perturbation at a stationary point of the potential, i.e. , with . But for the small energy (strong coupling) asymptotic solution its connection to the Floquet theory has not been clarified.
In this paper we provide some new results concerning the missing parts mentioned above. In the Section 2 we give a new method for computing the asymptotic solution for large , applicable to many periodic spectral problems. In the Section 3 we provide a few examples, including the Hill’s equation, the ellipsoidal wave equation and the Heun equation in the elliptic form, to demonstrate the method. We obtain the large energy asymptotic spectra for these equations. In the Section 4 we provide a relation between the small energy spectral solution and the doubly-periodic Floquet theory for elliptic potentials. We show that for a Schrödinger equation with elliptic potential the monodromy of the wave function along each period gives an asymptotic solution.
This paper is motivated by our previous works attempting to examine in detail a few simple examples of the Gauge/Bethe correspondence, proposed by Nekrasov and Shatashvili , where the infrared dynamics of some quantum gauge theories is related to the spectral problem of stationary Schrödinger equation with periodic potentials. Some results presented in this paper are still puzzling from the perspective of mathematical theory, albeit they are supported by solid computation and consistent with results already known. We hope the results presented in this paper be useful for further clarification.
2 Large eigenvalue perturbation
In this section our strategy is to use the Floquet theorem to compute for large . The eigenvalue relation is the reverse of the Floquet exponent , therefore if we can compute the monodromy of the wave function along the period then we obtain the eigenvalue expansion.
There is a general relation of the monodromy and the periodic potential. Write the wave function as , then according to the (2) the Floquet exponent is given by
We have picked the positive sign of , to obtain the result for the other sector we just change the sign of . Substitute the wave function into the Schrödinger equation (1), we get the relation
We use the notation , etc. Therefore in order to find all possible asymptotic spectral expansions we can first find all possible asymptotic solutions for from the equation (4) in the parameter space of , and then check if the integration (3) gives an asymptotic series. For some potentials it is possible to find many asymptotic solutions for from (4), but some solutions may not lead to an asymptotic series after performing the integration of (3).
Now we assume is large, therefore is a natural expansion parameter, then we can expand by
Substitute the expansion back to (4), we can solve order by order,
As and its derivatives are periodic, we can abandon all terms of total derivative in , and especially the even terms do not contribute, .
We may recognize are Hamiltonian densities of the KdV hierarchy, and relation (4) is the Miura transformation. Indeed, the procedure above is the same as in the KdV theory . This fact has been noticed in e.g. , but in their treatment this formalism was not really used. As we would show below this formalism is very useful for computation if the potential is periodic, because the integration in (3) is along the period. We emphasize that for general periodic potentials there is no known direct relation with the KdV theory, and the formal connection to the KdV theory is only helpful for computation. However, there are some potentials with the special choice of coupling strength which solve some higher order generalized stationary KdV hierarchy equations associated to the Hamiltonians given by . These special potentials include the Lamé potential and the Darboux-Treibich-Verdier potential with triangular number coupling constants, see e.g. [15, 16, 17]
For many periodic potentials it is very straightforward to explicitly compute because are polynomials of and its derivatives. In this way we obtain the asymptotic expansion of . Reverse the relation, we get the asymptotic expansion for the eigenvalue,
with . The large (therefore large ) expansion (8) is actually degenerate for .
Compared to the method of Hill’s determinant where the relation of and holds for generic value, in our treatment we have assumed the large asymptotic form for from the beginning of computation, in formula (5). Therefore, the procedure from (5) to (8) only applies to this asymptotic expansion region, however, it is very efficient from the computational point view. In fact, the method to derive eigenfunction from the solution of the relation (4) not only applies to the case of large energy solution, it also applies to the case of small energy solution discussed in the Section 4.
After obtaining the asymptotic eigenvalue, we can continue to find the corresponding asymptotic eigenfunction. The following are also standard results of the Floquet theory, see e.g.. If , then there are two independent stable solutions given by and , with a periodic function, . If , then there is only one stable solution given by , another independent solution is unstable. As we obtain the solution (5) for , the unnormalized asymptotic eigenfunction can be written in the exponential form which explicitly depends on , then we substitute the asymptotic eigenvalue (8) to obtain the corresponding eigenfunction with explicit dependence on .
In the following section we present a few examples to show how to use this method to obtain the asymptotic eigenvalue for some periodic Schrödinger equations.
3 Some examples
3.1 Hill’s equation
We start with an easy example to demonstrate the method. The Hill’s equation often refers to equation of the form (1) with a general real periodic potential. By the Fourier expansion the potential can be represented by a trigonometric polynomial,
the period is . The coupling constants are , in some cases they may be truncated to a finite subset if the approximation is valid. The Hill’s equation was used in celestial mechanics to achieve a high-accuracy description of the motion of moon under the influence of earth and sun.
Let us specify to the simple case with for ,
The resulting equation is called the Whittaker-Hill equation. It arises when we rewrite the 3-dimensional wave equation in the paraboloidal coordinates and apply the separation of variables method, the wave equation reduces to three identical equations of Whittaker-Hill type, see . The integration results for are
and then by (8) we obtain
3.2 Ellipsoidal wave equation
The more interesting examples are the equation (1) with elliptic potentials. The elliptic potentials have two independent periods, from which we can make the third period. It is questionable whether the classical Floquet theory can be directly applied to all periods, up to now very limited results on this problem has been obtained [3, 4, 5]. As we would show in the rest of the paper, the classical Floquet theory is still valid for one period, but a generalization is needed for other periods.
If we rewrite the 3-dimensional wave equation in the ellipsoidal coordinates, apply the separation of variables method, then the three identical equations are ellipsoidal wave equation, see . Written in the Jacobian form it is
where , and is the Jacobian elliptic function with the elliptic modulus , its quarter periods are the complete elliptic integrals and .
The Weierstrass form is also useful for our purpose,
where is the Weierstrass elliptic function, are the two independent half periods, and also plays a role in this study. The following relations between and are used,
where and they satisfy . The relation between periods is . The nome of the -function and is , related to the elliptic modulus by
The parameters () are related to () by
There is a technical, nevertheless useful fact about the equations with elliptic potential, that the Jacobian form and the Weierstrass form each has an advantage against the other in deriving different asymptotic solutions. The Weierstrass form is more suitable for driving the large perturbation given in this section, and the Jacobian form is more suitable for other two perturbations given in the next section.
The large energy asymptotic solution
In the definition of the elliptic function, the two periods and are on equal footing. Does the Floquet theorem apply to each period equally? Do we get two asymptotic spectral solutions from periods and , respectively, using the same method in the Section 2? Based on the computational results, given in the Section 4, the answer is not. The classical Floquet theory is valid only for the period and the computation method for large solution indeed works. But for the periods and we need a generalized Floquet theory, the corresponding asymptotic spectral solution has a different nature, because the large assumption fails. In this section we only deal with the large solution, and in the Section 4 we deal with cases associated to periods and .
The integrands contain higher powers of and , where the prime denotes , they can be simplified using relations derived from the basic relation . The simplified integrands, after discarding total derivative terms, take the form which is ready for integration, where are polynomial functions with arguments . The integration results for are
where is defined by the Weierstrass zeta function , the modular invariants are given by . They also can be rewritten in terms of the Eisenstein series , or in terms of the theta constants . We denote the Floquet exponent of wave function in Eq. (14) as , i.e. , then the asymptotical expansion for is
The eigenvalue for the equation in Jacobian form can be transformed from . However, the definition for the Floquet exponent differs. We use to denote the Floquet exponent of wave function in Jacobian form (13), i.e. . Shifting by is the same as shifting by , therefore the phases should be the same, therefore we have . Taking into account the relation in (17), the relation of and , we obtain
Taking the limit we recover the results for the Lamé equation, already treated in , see also  and references in. The Lamé equation comes from the same procedure of solving the Laplace equation in the ellipsoidal coordinates. Taking the limit while keeping we recover the result for the Whittaker-Hill equation. Taking a further limit we recover the result for the Mathieu equation.
3.3 Heun equation in elliptic form
A generalization of the Lamé equation is the Heun equation in the elliptic form. The term Heun equation often refers to its normal form where the Fushian property of the equation becomes apparent [20, 8]. In the Jacobian form given by Darboux  the equation is
In the Weierstrass form it is
where . The multi-component potential in (22) is the so-called Treibich-Verdier potential, known for its role in the theory of “elliptic soliton” for KdV hierarchy . To relate (21) and (22), the coordinates transformation is the same as in (15), and are related by
As in the previous example, the equation in the form (22) is more suitable for the large expansion. In the process of computing we need to simplify the integrands by repeatedly using relations about the -function with the same argument, already used in the previous subsection, and some other relations such as,
in the above index and . The integrands are finally simplified to the form , ready for integration. Then we obtain
In the above expressions is related to , and are related to . They generalizes the results for the Lamé equation presented in .
It is straightforward to compute (and ) from , here we do not give its explicit expression. Using these results we can verify the relation between the spectrum of Schrödinger operator and the effective action of the deformed N=2 supersymmetric gauge theory model, in the spirit of Gauge/Bethe correspondence , up to arbitrary higher order expansion. This computation completes the attempts in  where the leading order expansion was examined 111We remind readers the issue of notation. For the Jacobian elliptic functions in this paper we use to denote the elliptic modulus, while in  we used . For the Weierstrass elliptic functions in this paper we use to denote the nome, while in  we used . The reason for this change is that in  we tried to respect the convention of gauge theory literature while in this paper we adopt convention of mathematical literature. In the computation the following relation is used, in the notation of this paper, , where is the deformed gauge theory prepotential and is the instanton counting parameter for SU(2) super-QCD model discussed in ..
4 On doubly-periodic Floquet theory
4.1 Spectral problem for elliptic potentials
We have shown that for elliptic potentials the large asymptotic solution is always related to the monodromy along the periodic . So what is the role for and ? This question is related to the generalized Floquet theory for elliptic function, the so-called doubly-periodic Floquet theory, which only has been occasionally discussed during the past, e.g. in [23, 24, 25, 26, 27]. Some new features due to the complex nature of the elliptic function arise, make the extension nontrivial. Among the limited results that already exist on this topic, it seems that there is not an explicit statement about the relation of monodromy along , and the spectrum of the equation. In this section we give a few examples to show that the monodromy of along and indeed play a role in the spectral problem, they are related to two other asymptotic solutions that differ from (20) given above.
Therefore the problem we are trying to answer is related to the complete characterization of all asymptotic spectra for elliptic potentials. For such a Schrödinger operator the spectral solution is controlled by the characteristic coupling strength of potential , or more precisely by the ratio , often it has no analytical expression. How does the relation vary when we turn the value of ? When can be represented by an asymptotic series? The answer is not obvious at all. In the literature it is even not systematically studied how many asymptotic solutions there are for an elliptic potential.
It is necessary to explain the meaning of “spectrum” for a complex potential. The elliptic potentials are meromorphic function defined on the complex plane, therefore, they are not the most suitable examples for quantum mechanics. Instead their appearance in quantum field theory looks more natural, where the complex valued spectrum of Schrödinger operator is explained in a very different way. Indeed, in the context of Gauge/Bethe correspondence  the spectral solution of elliptic potentials nicely fits into the theory of 4-dimensional quantum gauge theory. For the Lamé potential, due to its connection with a typical Seiberg-Witten gauge theory model [28, 29], the idea of using elliptic curve is very helpful for the analysis, there is a one-to-one correspondence between the asymptotic solutions and the monodromy of wave function along . Upon a careful examination, the complete spectral solutions are precisely related to nonperturbative and duality properties of the low energy effective gauge theory. Another related context for the elliptic potential is the algebraic integrable theory, see e.g. , albeit neither the questions mentioned above have been seriously addressed there.
4.2 Lamé equation
The Lamé potential is the first example that motivates us to revise the doubly-periodic Floquet theory from a new perspective. It is in the Weierstrass form, or in the Jacobian form. We briefly review the result to give a general picture about the answer, the details are already given in .
The first fact is about the stationary points of the potential. There are three stationary points for the potential given by the solutions of , they are at where we have . In the Jacobian form the three stationary points are given by the solutions of , they are at , and . The information about these stationary points does not tell us what are the possible asymptotic solutions, the following facts entirely come from computation  (see also ).
It turns out that each stationary point is associated to an asymptotic expansion for . The stationary point at (i.e. at ) is associated to the large solution. The equation in the Weierstrass form is better for computation. The leading order energy comes from the quasimomentum, , the potential can be treated as small perturbation, therefore we have . The relation is given by the monodromy along the period as in the formula (3). This is well described by the classical Floquet theory, the asymptotic solution can be treated by the method given in the Section 2, see .
The other two stationary points are related to two other asymptotic solutions, to study them we need the generalized Floquet theory. The equation in the Jacobian form is better for computation. In these cases the quasimomentum is small compared to the scale of potential which means . The solution (i.e. ) is a perturbation at (i.e. at ), here we have . The relation is given by the monodromy of wave function along the period (i.e. ). A key point is that the naive definition of the Floquet exponent is not right. If we want to produce the correct asymptotic expansion that is already derived by other method in , then a modification is needed, the right relation is . The solution (i.e ) is a perturbation at (i.e. at ). The subleading terms are denoted by , i.e. with . Then the relation is given by the monodromy along the period (i.e. ). Again the classical Floquet theory fails and the correct definition of Floquet exponent is given by , with the complementary module satisfying the relation .
While we do not have a mathematical theory to explain why the monodromies along three periods are in one-to-one correspond with three asymptotic solutions, nevertheless a physical explanation was given . Viewed from the Gauge/Bethe correspondence , the spectral problem of the Lamé operator is roughly the same problem about the low energy effective theory of gauge theory model. The monodromies along different periods are related by electro-magnetic duality of the effective gauge theory, in the spirit of Seiberg-Witten theory [28, 29]. For the gauge theory model there is an asymptotic solution in each duality frame, hence for the Lamé operator there is an asymptotic solution related to the monodromy along each period.
4.3 Ellipsoidal wave equation
Now we turn to the next example. It seems that this method found for the Lamé equation also applies to other equations with elliptic coefficient. The ellipsoidal wave equation is non-Fushian, has no known relation to gauge theory, nevertheless, it can be treated in a similar way.
The stationary points of the Lamé potential are also the stationary points of the potential . The monodromy along (i.e. ) gives the large asymptotic solution, this is the result given in (20). In the following we give the computation details to demonstrate that the monodromies along and give other asymptotic eigenvalues, one of them was already obtained by another method .
The first small energy asymptotic solution
The equation in the Jacobian form (13) is more suitable for this asymptotic solution. We assume is the dominant term of the potential, i.e. , the other term is a small perturbation. At the point , we have , therefore is the perturbative energy. The parameters satisfy . We shall find the asymptotic expansion for the integrand from the relation . Now the expansion parameter should be , or equivalently , with expanded as
then can be recursively solved. The even terms are total derivatives, they do not contribute in the final periodic integration (28). The nonzero contributions come from , the first few are
Then we come to the issue of relating the monodromy of wave function along period to the Floquet exponent . According to the classical Floquet theory, the relation should be . Indeed we can use this relation as the definition of the Floquet exponent. However, the corresponding asymptotic spectral solution has already been obtained by a different method in , the result suggests that the classical Floquet theory cannot be directly applied to the period . We find the correct relation between the period integral and the Floquet exponent is
This modified relation is the same as that for the Lamé equation, it leads to the asymptotic solution given in (30) which is the one obtained in . We believe this is how the classical Floquet theory should be generalized for elliptic potentials, albeit at the moment we do not have a rigorous proof.
The integration formulae for can be found in . If we denote , then we have for , and the remaining non-vanishing are
The second small energy asymptotic solution
The second small energy expansion is a perturbation at where , therefore . Similar to the treatment in  we set , where is the perturbative energy around the local minimum of potential. The equation becomes
The parameters satisfy , therefore similar to the case of the previous solution, we choose as the expansion parameter and expand the integrand as
It includes the factor because the potential in this case is . From the relation we obtain the expressions for . The even terms are again total derivatives and do not contribute to the final integration of (34). Other contribute non-vanishing integrations, the first few are
Concerned the issue of relating the monodromy of the wave function along period and the Floquet exponent , similar to the case of the first small energy asymptotic expansion, the classical Floquet theory is invalid. Although the corresponding asymptotic expansion presented below in (36) has not been given in other literature, there is the requirement of consistent with other known results. We find the correct relation is given by
This relation gives the asymptotic expansion (36) consistent with all known results, especially in the limits of (the Lamé potential), and in the limit , with fixed (the Mathieu potential).
The integration formulae for we need in this case are denoted by , we have for , and
they have been used in  too. After getting the asymptotic series we reverse it, the final asymptotic expansion for the spectral relation is
We write the expansion in a form easy to see its connection to the eigenvalue of Lamé equation, in the limit . The -independent part in expansion (36) has an interesting relation to the expansion (30), which is already explained in .
The potential of ellipsoidal wave equation actually has more stationary points given by the solutions of . Are they associated to new asymptotic spectral solutions? We do not have a definite answer to this question. However, even new asymptotic solutions exist they are unlikely given by the monodromy along a period, therefore not in the scope of Floquet theory.
4.4 Darboux-Treibich-Verdier potential
The Heun equation is also a Fushian equation, in the Section 3 using its elliptic form we have shown the classical Floquet theory indeed applies to the period