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He Wei. A New Treatment for Some Periodic Schrödinger Operators II: The Wave Function. Communications in Theoretical Physics, 2018, 69(6): 645  
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A New Treatment for Some Periodic Schrödinger Operators II: The Wave Function
He Wei *
School of Electronic Engineering, Chengdu Technological University, Chengdu 611730, China

 

† Corresponding author. E-mail: weihephys@foxmail.com

Abstract
Abstract

Following the approach of our previous paper we continue to study the asymptotic solution of periodic Schrödinger operators. Using the eigenvalues obtained earlier the corresponding asymptotic wave functions are derived. This gives further evidence in favor of the monodromy relations for the Floquet exponent proposed in the previous paper. In particular, the large energy asymptotic wave functions are related to the instanton partition function of N = 2 supersymmetric gauge theory with surface operator. A relevant number theoretic dessert is appended.

Keyword:spectral theory;periodic differential operators;Seiberg-Witten duality;instanton partition function
1 Introduction

In the previous paper Ref. [1] we study how the Floquet theory manifests in the multiple asymptotic spectral solutions of some periodic Schrödinger operators. We have only studied the eigenvalue aspect of these solutions. Following this method, it is very convenient to derive the corresponding asymptotic wave functions, we present the results in this paper. We focus on our canonical examples: the Mathieu equation and the Lamé equation, which are the most widely used in periodic spectral problem. Our main conclusion are made for elliptic potentials, but as we have explained, to collect evidences for the proposals a crucial consistent requirement is that the solutions of the Lamé equation must reduce to corresponding solutions of the Mathieu equation. Therefore the Mathieu equation, which is much better understood, is included here as a reference example. Unlike in Ref. [1], in this paper we do not use the ellipsoidal wave equation as the example for the elliptic potential as it would lead to lengthly formulae for wave functions, instead the wave functions of Lamé equation are already enough for our purpose.

In Sec. 2 we derive the wave functions for large energy (weak coupling) perturbation. In Sec. 3 we derive the wave functions for small energy (strong coupling) perturbation. Some wave functions have been studied before, we briefly comment the old materials where earlier treatments can be found. In Sec. 4 we explain how the eigenfunctions are related to supersymmetric gauge theory, in the context of Gaueg/Bethe correspondence.[2]

The conclusion of this paper is that the wave functions give further evidence for the relations between multiple asymptotic solutions and the Floquet property associated with multiple periods. Among the wave functions, the eigenfunction (43) is a new solution.

2 Large Energy Wave Function
2.1 The Floquet Wave Function

As we have shown previously, the large energy perturbation can be carried out using a method from KdV theory,[3–4] the wave function is given by

where v(x) satisfies vx + v2 = u + λ and has the WKB expansion

vℓ(x) are given by the KdV Hamiltonian densities.[4]

In fact there is another sector of solution for the v(x), it is expanded in the same form of Eq. (2) with changed to . Therefore we get the unnormalized asymptotic expansion of the corresponding wave functions,

The dispersion relation λ(ν), where ν is the Floquet exponent, is obtained by the classical Floquet theory,

The relations (3) and (4) give a complete perturbative solution for large energy.

2.2 Mathieu Equation

The potential for the Mathieu equation is

from Eq. (3) we get the large energy asymptotic wave functions

It is clear that with λ as the expansion parameter, the coefficients of λ− ℓ/2 are not necessary periodic functions. However, the wave functions must satisfy the Floquet property, this is made clear by the following parameter change using the eigenvalue expansion.

The large energy dispersion relation is a classic result, see for example Refs. [5–8]. The relation (4) can be used to compute, it is

It has a solution

Another solution −λ1/2 does not lead to new wave function. Substitute λ1/2 into Eq. (6) we get the wave functions in the form

Now the coefficients of ν−l, with l ⩾ 1, are periodic functions, the wave functions take the form ψ±(x) = e± iνxϕ(± x) with ϕ(x) a periodic function. This wave function is related to the N = 2 pure Yang-Mills gauge theory with surface operator, see the discussion in Sec. 4.

Another bases of the asymptotic wave functions, commonly used in many literatures, are cem(x) and sem(x). Up to a constant, their relation to ψ±(x) is

2.3 Lamé Equation

The potential for the Lamé equation is an elliptic function, for the large energy perturbation we should use the Weierstrass form to obtain compact formulae. In this paper we use the potential

which is defined by a shifted elliptic function with ζ1 a constant given by the Weierstrass zeta function ζ1 = ζ(ω1)/ω1. The coupling constant α is often represented as n(n − 1), the nome of the elliptic function is q = exp(2 π i (ω2/ω1)). The use of shifted potential is more convenient for us to see the relation to N = 2 supersymmetric gauge theory in Sec. 4, it does not change essential aspects of the spectral solution. As we have demonstrated in Ref. [1] the large energy asymptotic solution is related to the period 2ω1. Then the corresponding wave functions are

where g2, g3 are the modular invariants of ℘(x). We use another shifted function defined by , it is 2ω1-periodic and satisfies the relation . Again, some coefficients of λ−n/2 are not periodic functions.

The corresponding dispersion relation was derived by Langmann,[9] expressed as a q-series, the same expression also appears in the context of its relation to gauge theory,[2] we examined this relation in Ref. [10]. Another way to derive the dispersion relation is to use the formula (4), then we get an expression involving quasi-modular functions,[11]

It has a solution

Substitute λ1/2 into Eq. (12) we get the wave functions in the Floquet form,

The wave functions also satisfy the property ψ±(−x) = ψ∓(x). In Sec. 4 we would show the connection of this wave function and the partition function of the N = 2* supersymmetric gauge theory with surface operator.

There is a comment about the polynomials of elliptic functions that appear in Eqs. (12) and (15). Recall that any elliptic function can be expressed as a linear combination of zeta functions and their derivatives. In fact, the Hamiltonian densities vℓ(x) for the elliptic potential have no pole of order one at x = 0, therefore, they are linear combinations of with k ⩾ 0. Then the integrated Hamiltonians appearing in Eq. (12) are linear combinations of x and with k ⩾ 0. In the wave function (15) the phase e±iνx contains the linear term of x, the coefficients of ν−l are linear combinations of with k ⩾ 0, probably include an x-independent constant term. The constant terms can be absorbed into the normalization constant, then the expressions are linear polynomials of with k ⩾ 0. Or equivalently, because , they are linear polynomials of with k ⩾ 1. This point is important when we connect the wave function to the instanton partition function in Sec. 4, especially for higher order terms which we do not explicitly give in Eq. (15).

When take the limit q → 0, α → ∞ with α q1/2 → − h/4 fixed, all the results obtained here reduce to the case of the Mathieu equation. Some details in the limit process need further explanation. We first examine how the elliptic potential is reduced to the trigonometric potential. Instead of taking the limit for , we have to shift the argument and take the limit for . From the leading order expansion of given below in Eq. (18), the resultant potential takes the form

with period 2ω1, its eigenvalue denoted by can be obtained from the limit of Eq. (13). If we use the rescaled coordinate χ = πx/2ω1 and eigenvalue , the limit of the Lamé equation could be written in the standard form of the Mathieu equation,

The corresponding large energy asymptotic eigenvalue λ and wave functions ψ±(χ,ν) take the same functional form as the eigenvalue (7) and the wave functions (9), but with the coordinate variable substituted by χ.

Let us inspect more carefully the limit for eigenvalue and wave functions of the Lamé equation. For the wave functions (15), we shift the argument by x → x + ω2, then take the limit q → 0 (i.e. with ω1 fixed, ω2 → i∞) for ψ±(x + ω2). The following expansions are needed,

The Lamé wave functions (15) indeed reduce to the Mathieu wave functions, if we further substitute the Floquet exponent by . The modification of the exponent can be understood as follows. In this subsection, the exponent ν conjugates to the variable χ of the π-periodic potential cos 2χ, it is different from the exponent that conjugates to the variable x of the 2ω1-periodic potential cos(πx/2ω1) which we should have denoted by another letter . According to the Floquet theorem their wave functions produce phases under periodic shift by ψ±(χ+ π, ν) = exp(± iνπ)ψ±(χ,ν) and . But actually the potentials are the same, so the phases must be the same, which means the exponents are related by the relation . It is easy to check that the limit for the eigenvalue (13) which is associated to the potential cos(πx/2ω1), hence with ν substituted by , equals (π/2ω1)2λ where λ is the Mathieu eigenvalue for the potential cos2χ.

3 Small Energy Wave Function
3.1 Location of Small Energy Perturbation

Besides the large energy solution, there exists other solutions which are small energy excitations around local minima, i.e. the critical points of potential. We notice for some periodic potentials at each local minimum there is an asymptotic solution, and all known asymptotic solutions are located at a local minimum.[1,10]

For example, the potential u(x) = 2h cos 2x has local minima at x* = 0 and x* = π/2 modulo periods. At the minima u(x*) = ± 2h, therefore the eigenvalues take the form

where δ is the energy of small excitations. The small energy perturbation is also the strong coupling solution for the potential, h≫1, see Refs. [6] (Chapter V). In a similar way, the elliptic potential has local minima at x* = ωi, where the potential u(x*) = α(ei + ζ1), i = 1,2,3. The first minima at x* = ω1 is associated to the large energy excitations already analyzed in the Subsec. 2.3, the leading order energy comes from the quasimomentum λ ∼ − ν2 + ··· The other two minima are associated to small energy perturbative solutions, nevertheless, in order to get compact formulae we should use the Jacobian form of the Lamé equation to compute.

In this section we derive the corresponding strong coupling wave function, they have the Floquet form, and for elliptic potential their monodromies along periods 2 K and 2 K + 2 K′ indeed satisfy the relations we proposed in the previous paper.[1]

3.2 Mathieu Equation

The first small energy perturbation

Around the minimum x* = 0, λ = −2h + δ, the potential strength h ≫ 1 is large compared to the energy δ, therefore the expansion parameter is h1/2. The relation vx + v2 = u + λ has an asymptotic solution in the form[1]

with

Another solution can be obtained by changing the signs of odd terms v2ℓ + 1 to −v2ℓ + 1; or with vℓ unaltered but changing to in Eq. (21). This leads to the following asymptotic wave functions,

We can change the parameter δ to the Floquet exponent ν, by the following strong coupling expansion of the dispersion relation which is well known,[5–8]

Then we get the wave functions in the form

Because the wave functions are unnormalized, the terms in the exponent might appear in slightly different form, nevertheless the differences are constants and can be absorbed into the normalization constant. This comment applies to all of the asymptotic wave functions in this paper. When the exponent ν takes real value the asymptotic wave functions have the property . This solution seems often not recorded in the mathematical literature, however, it was analyzed in a paper by Stoner and Reeve.[12] The book Ref. [13] contains a discussion about this solution in the context of quantum mechanics.

The second small energy perturbation

Around the minimum x* = π/2, λ = 2h + δ, the potential strength h1/2 again serves as the expansion parameter. The relation vx + v2 = u + λ has an asymptotic solution in the form (21), with

As in the previous solution, the other solution is obtained by changing to . The corresponding wave functions are

The dispersion relation at this local minimum[5–8]

allows us to change the parameter δ to the Floquet exponent ν. Then we get the corresponding asymptotic wave functions

The study of this solution dates back to the work of Ince and the work of Goldstein in the 1920s. Some recent materials easier to access include the paper by Dingle and Müller,[14] the books by McLachlan,[5] by Arscott[6] and by Müller-Kirsten.[13]

This asymptotic solution is related to the large h limit of the standard Mathieu functions by

with m takes either even or odd integers.[6] As we have ψ±(−x) = ψ∓(x), then cem(x) is an even function and sem(x) is an odd function, as desired.

3.3 Lamé Equation

Now we turn to the more interesting case of elliptic potential where the advantage of our method becomes more transparent. As we have shown in Refs. [1, 10], for small energy perturbative solution the Jacobian form of the elliptic function is more suitable, therefore we rewrite the potential as u(z) = αk2 sn2(z|k2), and the Lamé equation is

To transform the elliptic functions from to sn2z we use the following relations of the coordinates, eigenvalues, the elliptic modulus and the nome,

We also use μ to denote the Floquet exponent throughout of this subsection, it is different from the Floquet exponent ν used for the Weierstrass form.[10]

The locations of the small energy perturbations are given by two solutions of the condition ∂zsn2z = 0 at z* = 0 and z* = K which correspond to u(z*) = 0 and u(z*) = αk2.

The first small energy expansion

Around z* = 0 we have Λ = 0 + small correction, therefore in this case Λ itself is a small quantity compared with the potential strength αk2. Therefore serves as the expansion parameter. The relation vz + v2 = u + Λ has an asymptotic solution that can be expanded as

with

The minus sector is obtained by changing to . Then by a straightforward integration we get the following asymptotic wave functions,

In order to change the parameter Λ to the Floquet exponent μ, we use the widely known strong coupling expansion of the dispersion relation[6,8,13]

It leads to the wave functions in the form

Only in the case when all quantities, including the elliptic modulus k, take real values we have . In particular, up to the first two leading order the wave functions can be written as

This asymptotic solution can be compared to the earlier results about the asymptotic Lamé function obtained by Malurkar in the 1930s, and results by Müller in the 1960s.[13,15]

In the limit α → ∞, k → 0, μ → ν with α1/2k → 2i h1/2 finite, we recover the unnormalized wave functions, which differ some constant terms in the exponent from the asymptotic Mathieu wave functions (25).

The second small energy expansion

Around z* = K we have Λ = − αk2 + small correction, we set where is a small quantity compared with the potential strength αk2. Here serves as the expansion parameter. The relation has an asymptotic solution in the form

with

where is the complementary modulus. After performing integration about the Jacobian elliptic functions we get the wave functions, now including both sectors,

The corresponding dispersion relation has been missed for a long time in the literature, motivated by some ideas from quantum gauge theory[2] recently we have derived it by the WKB analysis and a duality argument,[10] then we rederive it using the method adopted in this paper.[1] It is

It gives us new asymptotic wave functions,

They satisfy the property ψ±(−z) = ψ∓(z). In companion with the expression (38) we could write the first two leading order results of the wave functions as

Taking the limit to the Mathieu wave functions (29), we would again encounter the difference of some constant terms, which can be absorbed into the normalization constant.

Up to now, everything about the small energy expansions for the Lamé equation is consistent with the known results, although the monodromy relations, formulae (26) and (32) in Ref. [1], used to derive the corresponding dispersion relation remain a physics induced conjecture.

4 A Connection to N = 2 Gauge Theory

Now we come back to the original motivation which inspired our study to the spectral problem of periodic Schrödinger operators, especially for the elliptic potentials. As we have shown in Refs. [1, 10], the asymptotic eigenvalues of the Mathieu and the Lamé equations are related to the solution of some deformed N = 2 supersymmetric Yang-Mills gauge theories in the Nekrasov-Shatashvili limit (NS).[2] The three asymptotic spectral solutions are precisely i n accordance with three different dual descriptions of the low energy effective physics of gauge theory, i.e. the Seiberg-Witten duality,[16–17] in particular the large energy solution is related to the Nekrasov instanton partition function.[18]

The large energy asymptotic wave functions are related to the instanton partition function of gauge theory with surface operator inserted. The partition function with surface operator extends Nekrasov’s localization formula, it is introduced and developed in Refs. [19–20]. The computation can be carried out by the characters developed in Ref. [21]. The paper by Alday and Tachikawa gives a detailed study about the relations between the SU(2) gauge theory with surface operator, the SL(2) conformal block and the two-body quantum Calogero-Moser model.[22] In the following, we briefly explain the relation between the gauge theory partition function with surface operator and the asymptotic wave functions (9) and (15).

Let us start from the SU(2) N = 2* gauge theory with surface operator, whose partition function takes the following form,

where a is the scalar v.e.v, m is the mass of adjoint matter, ϵ1, ϵ2 are the Ω-deformation and x1, x2 are the counting parameters. Written in the exponential form, its pole structure in the limit ϵ1 → 0, ϵ2 → 0 is

The functions F and G are x1, x2-asymptotic series, their coefficients are functions of a,m, ϵ1, ϵ2 and in particular are regular with respect to ϵ1, ϵ2. There remains an ambiguity of distributing terms, because a term like f(a,m,x1,2)/ϵ1 can be put in either F/ϵ1ϵ2 or G/ϵ1. To fix the ambiguity we require that the function F contains only terms with expansion parameters of the form (x1x2)n, while G does not contain such terms. The presence of the surface operator beaks the symmetry between ϵ1 and ϵ2 in F and G, therefore F is not exactly the deformed prepotential obtained from the Nekrasov partition function, nevertheless, in the NS limit ϵ2 → 0 they give the same eigenvalue as in Eq. (50).

In order to relate gauge theory and the quantum mechanics spectral problem, some manipulations on the function Z are needed. The spectral solution of the Lamé operator is related to the large-a-expansion of instanton partition function (46), in accordance with the large-ν-expansions of the eigenvalue (13) and the eigenfunctions (15). Both F/ϵ1ϵ2 and G/ϵ1 contain a-independent terms when expanded as large-a-series, which deserve special attention. These terms are polynomials of x1,x2, and can be represented by the Dedekind eta function and the elliptic theta function,

To see the connections of functions F, G and the eigenvalue, eigenfunction, we first need to identify the parameters by

The elliptic nome q is the instanton parameter of gauge theory, therefore, the function F is an x-independent q-series which gives the eigenvalue, the function G is a q-series depending on the coordinate x which gives the wave function. The eigenvalue λ in (13) is related to the function F in the limit ϵ2 → 0 by

In gauge theory the term −ν2 is perturbative, hence not included in the instanton partition function. This relation is examined in detail in Ref. [10] (see formula (34) in that paper), there is a difference of on the right hand side because here we use the shifted potential .

On the other hand, the wave functions (15) is related to the function G in the limit ϵ2 → 0 by

In the expression we emphasize the parameters used on both sides, and use the property of the large energy wave functions ψ−(−x − ω2) = ψ+(x + ω2). For example, up to the order a−2 we have

Using the relation of parameters given in Eq. (49), the first three pieces are summed into three elliptic functions,

which precisely match the left hand side of Eq. (51). The coefficients of the term ν−l, with l ⩾ 1, are linear polynomials of with k ⩾ 1. As the argument of wave functions in Eq. (51) is x + ω2, we have , therefore, the coefficients of the term ν−l can be rewritten as linear polynomials of with k ⩾ 0, in accordance with the discussion in Subsec. 2.3. In another form, these coefficients can be rewritten as linear polynomials of the Jacobi zeta function and its derivatives, because ∂x ln ϑ4(πx/2ω1,q) = (K/ω1)zn((Kx/ω1)|k2).

The eigenvalue (13) and the eigenfunction (15) provide an elliptic modular representation for the gauge theory partition function when ϵ2 = 0. In fact, we observe evidence that even for the case when both deformation parameters are turned on, ϵ1 ≠ 0, ϵ2 ≠ 0, the instanton partition function with surface operator can be expressed in terms of theta functions. This property indicates the instanton partition function secretly records relations to the elliptic curve. Indeed, this connection can be seen from the point of view of either integrable system[2] or conformal field theory.[22–23]

In the decoupling limit, the N = 2* gauge theory becomes the pure gauge theory. The corresponding partition function with surface operator can be found in Ref. [24], it is related to the asymptotic Mathieu wave functions (9).

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