Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics

Four Lectures on Weierstrass Elliptic Function and Applications in Classical and Quantum Mechanics

Georgios Pastras
NCSR “Demokritos”, Institute of Nuclear and Particle Physics
15310 Aghia Paraskevi, Attiki, Greece
pastras@inp.demokritos.gr
Abstract

In these four lectures, aiming at senior undergraduate and junior graduate Physics and Mathematics students, basic elements of the theory of elliptic functions are presented. Simple applications in classical mechanics are discussed, including a point particle in a cubic, sinusoidal or hyperbolic potential, as well as simple applications in quantum mechanics, namely the Lamé potential. These lectures were given at the School of Applied Mathematics and Physical Sciences of the National Technical University of Athens in May 2017.


Lecture 1: Weierstrass Elliptic Function

1.1 Prologue

In the present lectures, basic elements of the theory of elliptic functions are presented and simple applications in classical and quantum mechanics are discussed. The lectures target an audience of senior undergraduate or junior graduate Physics and Mathematics students. For lectures 1 and 2, the audience is required to know basic complex calculus including Cauchy’s residue theorem. For lecture 3, basic knowledge on classical mechanics is required. For lecture 4, the audience is required to be familiar with basic quantum mechanics, including Schrödinger ’s equation and Bloch’s theorem.

The original constructions of elliptic functions are due to Weierstrass [1] and Jacobi [2]. In these lectures, we focus on the former. Excellent pedagogical texts on the subject of elliptic functions are the classic text by Watson and Whittaker[3] and the more specialized text by Akhiezer [4]. Useful reference handbooks with many details on transcendental functions including those used in these lectures are provided by Bateman and Erdélyi, [5], which is freely available online, as well as the classical reference by Abramowitz and Stegun [6].

On the applications in quantum mechanics we will meet the Lamé equation. Historically, this equation was studied by Lamé towards completely different applications [7]. An excellent treatment of this class of ordinary differential equations in given by Ince in [8]

The presenter took advantage of the experience acquired during his recent research on classical string solutions and minimal surfaces to prepare these lectures. The applications of elliptic functions in physics extend to many, much more interesting directions.

1.2 Elliptic Functions

Basic Definitions

Assume a complex function of one complex variable obeying the property

(1.1)

for two complex numbers and , whose ratio is not purely real (thus, they correspond to different directions on the complex plane). Then, the function is called doubly-periodic with periods and . A meromorphic, doubly-periodic function is called an elliptic function.

The complex numbers , , and define a parallelogram on the complex plane. Knowing the values of the elliptic function within this parallelogram completely determines the elliptic function, as a consequence of (1.1). However, instead of and , one could use any pair of linear combinations with integer coefficients of the latter, provided that their ratio is not real. In the general case the aforementioned parallelogram can be divided to several identical cells. If and have been selected to be “minimal”, or in other words if there is no within the parallelogram (boundaries included, vertices excepted), such that , then the parallelogram is called a fundamental period parallelogram. Two points and on the complex plane whose difference is an integer multiple of the periods

(1.2)

are called congruent to each other. For such points we will use the notation

(1.3)

Obviously, by definition, the elliptic function at congruent points takes the same value,

(1.4)

A parallelogram defined by the points , , and , for any , is called a “cell”. It is often useful to use the boundary of an arbitrary cell instead of the fundamental period parallelogram to perform contour integrals, when poles appear at the boundary of the latter.

ReIm0
Figure 1: The fundamental period parallelogram is shaded with blue and an arbitrary cell is shaded with red. The dashed lines define the period parallelograms.

Knowing the roots and poles of an elliptic function within a cell suffices to describe all roots and poles of the elliptic function, as all other roots and poles are congruent to the former. As such, a set of roots and poles congruent to those within a cell is called an irreducible set of roots or poles, respectively.

Modular Transformations

Given two fundamental periods and , one can define two different fundamental periods as,

(1.5)
(1.6)

where . Any period in the lattice defined by and , is obviously a period of the old lattice, but is the opposite also true? In order for the opposite statement to hold, the area of the fundamental period parallelogram defined by the new periods and has to be equal to the area of the fundamental period parallelogram defined by the original ones and . The area of the parallelogram defined by two complex numbers and is given by

(1.7)
ReIm
Figure 2: The green periods and can generate the whole original lattice. The red periods and generate only half of the original lattice. For example, there is no way to reach the red-dotted vertex by a linear combination of and with integer coefficients. The area of the parallelogram defined by and is equal to the area of the parallelogram defined by the original periods, whereas the one defined by and has double this area.

It is a matter of simple algebra to show that

Thus, the new periods can generate the original lattice if

(1.8)

or in other words if

(1.9)

It is a direct consequence that an elliptic function necessarily obeys

(1.10)

when

(1.11)

where .

Basic Properties of Elliptic Functions

Theorem 1.1.

The sum of residues over an irreducible set of poles of an elliptic function vanishes.

To demonstrate this, we use Cauchy’s residue theorem over the boundary of a cell.

Shifting by in the second integral and by in the third, we yield

which vanishes as a consequence of being a doubly periodic function. Therefore,

(1.12)
Theorem 1.2.

An elliptic function with an empty irreducible set of poles is a constant function.

An elliptic function with no poles in a cell, necessarily has no poles at all, as a pole outside a cell necessarily would have a congruent pole within the cell. Consequently, such a function is not just meromorphic, but rather it is analytic. Furthermore, an analytic function in a cell is necessarily bounded within the cell. A direct consequence of property (1.1) is that an analytic elliptic function is bounded everywhere. But a bounded analytic function is necessarily a constant function.

The number of roots of the equation

(1.13)

is the same for all . Before demonstrating this, we will review some properties of Cauchy’s integral.

Consider a meromorphic function , with a number of poles and roots with multiplicities and respectively, within a region bounded by a closed contour . The Laurent series of the function at the regime of a pole or a root is

with . In the case is a root , then , while in the case is a pole , then . The derivative of in the regime of a pole or root is

Furthermore, assume an analytic function . Its Laurent series at the regime of a pole or root of is trivially

It is a matter of simple algebra to show that the Laurent series of the function at the region of is

Thus, at any root or pole of the function , the function has a first order pole with residue . It is a direct consequence of Cauchy’s residue theorem that

(1.14)

Let’s now return to the case of an elliptic function . We would like to calculate the contour integral of formula (1.14) with , and being the boundary of a cell, namely

The function is trivially elliptic, while differentiating equations (1.1), one yields

(1.15)

implying that is also an elliptic function with the same periods as . In an obvious manner, the function is an elliptic function with the same periods as . A direct application of the fact that the sum of the residues of an elliptic function over a cell vanishes (1.12) is

Finally, using equation (1.14), we get

(1.16)

Therefore,

Theorem 1.3.

the number of roots of the equation in a cell is equal to the number of poles of in a cell (weighted by their multiplicity), independently of the value of .

This number is called the order of the elliptic function .

Theorem 1.4.

The order of a non-constant elliptic function cannot be equal to 1.

A non-constant elliptic function has necessarily at least one pole in a cell as a consequence of theorem 1.2, thus its order is at least 1. However, an elliptic function of order 1 necessarily has only a single first order pole in a cell. In such a case though, the sum of the residues of the elliptic function in a cell equals to the residue at this single pole, and, thus, it cannot vanish. This contradicts (1.12) and therefore an elliptic function cannot be of order one. The lowest order elliptic functions are of order 2. Such a function can have either a single second order pole in a cell or two first order poles with opposite residues.

Theorem 1.5.

The sum of the locations of an irreducible set of poles (weighted by their multiplicity) is congruent to the sum of the locations of an irreducible set of roots (also weighted by their multiplicity).

To demonstrate this, we will calculate Cauchy’s integral with , and as contour of integration the boundary of a cell. The left hand side of (1.14) equals

We shift by in the second integral and by in the third one to yield

Using the periodicity properties of , (1.1) and (1.15), we yield

Although , due to the branch cut of the logarithmic function, in general we have that and , with . Thus,

Applying property (1.14) we get

implying

(1.17)

which is the proof of theorem 1.5.

1.3 The Weierstrass Elliptic Function

Definition

As we showed in previous section, the lowest possible order of an elliptic function is 2. One possibility for such an elliptic function is a function having a single second order pole in each cell. It is actually quite easy to construct such a function. It suffices to sum an infinite set of copies of the function each one shifted by for all . The usual convention includes the addition of a constant cancelling the contributions of all these functions at (except for the term with ), so that the Laurent series of the constructed function at the region of has a vanishing zeroth order term. Following these directions, we define,

(1.18)

By construction, this function is doubly periodic with fundamental periods equal to and .

(1.19)

This function is called Weierstrass elliptic function.

Basic Properties

A direct consequence of the definition (1.18) is the fact that the Weierstrass elliptic function is an even function

(1.20)

Let’s acquire the Laurent series of the Weierstrass elliptic function at the regime of . It is easy to show that

Consequently,

where

The fact that is even implies that only the even indexed coefficients do not vanish,

(1.21)
(1.22)

For reasons that will become apparent later, we define and so that

(1.23)

implying that

(1.24)
(1.25)

Weierstrass Differential Equation

Directly differentiating equation (1.18), the derivative of Weierstrass function can be expressed as

(1.26)

It follows that the derivative of Weierstrass elliptic function is an odd function

(1.27)

The Laurent series of , and at the regime of are

It is not difficult to show that there is a linear combination of the above, which is not singular at and furthermore it vanishes there. One can eliminate the sixth order pole by taking an appropriate combination of and . This leaves a function with a second order pole. Taking an appropriate combination of the latter combination and allows to write down a function with no poles at . Trivially, adding an appropriate constant results in a non-singular function vanishing at . The appropriate combination turns out to be

But, the derivative, as well as powers of an elliptic function are elliptic functions with the same periods. Therefore, the function is an elliptic function with the same periods as . Since the latter has no pole at , it does not have any pole at all, and, thus, it is an elliptic function with no poles. According to theorem 1.2, elliptic functions with no poles are necessarily constants and since vanishes at the origin, it vanishes everywhere. This implies that Weierstrass elliptic function obeys the differential equation,

(1.28)

This differential equation is of great importance in the applications of Weierstrass elliptic function in physics. For a physicist it is sometimes useful to even conceive this differential equation as the definition of Weierstrass elliptic function.

It turns out that the Weierstrass elliptic function is the general solution of the differential equation

(1.29)

Performing the substitution , the equation (1.29) assumes the form

which obviously has the solutions, . This implies that and since the Weierstrass elliptic function is even, the general solution of Weierstrass equation (1.29) can be written in the form

(1.30)

In the following, we will deduce an integral formula for the inverse function of . In order to do so, we define the function as

(1.31)

Differentiating with respect to one gets

We just showed that the general solution of this equation is

Since the integral in (1.31) converges, it should vanish at the limit , or equivalently, . This implies that is the position of a pole, or in other words it is congruent to . This means that

Substituting the above into the equation (1.31) yields the integral formula for Weierstrass elliptic function,

(1.32)

Once should wonder, how the above formula is consistent with the fact that is an elliptic function, and, thus, all numbers congruent to each other should be mapped to the same value of . The answer to this question is that the integrable quantity in (1.32) has branch cuts. Depending on the selection of the path from to infinity and more specifically depending on how many times the path encircles each branch cut, one may result in any number congruent to or . A more precise expression of the integral formula is

(1.33)

The Roots of the Cubic Polynomial

We define the values of the Weierstrass elliptic function at the half-periods , and as

(1.34)

The permutation between the indices of ’s and ’s is introduced for notational reasons that will become apparent later. The periodicity properties of combined with the fact that the latter is an even function, imply that is stationary at the half periods. For example,

implying that . Similarly one can show that

(1.35)

Substituting a half-period into Weierstrass equation (1.29), we yield

(1.36)

The derivative of , as shown in equation (1.26) have a single third order pole in each cell, congruent to . Thus, is an elliptic function of order 3 and therefore it has exactly three roots in each cell. Since , and all lie within the fundamental period parallelogram, they cannot be congruent to each other, and, thus, there is no other root within the latter. This also implies that , and are necessarily first order roots of . All other roots of are congruent to those. Finally, when equation (1.36) has a double root, the solution of the differential equation (1.29) cannot be an elliptic function.

An implication of the above is the fact that the locations , and are the only locations within the fundamental period parallelogram, where the Laurent series of the function have a vanishing first order term at the region of . Consequently the equation has a double root only when equals any of the three roots , or . Since is an order two elliptic function, the complex numbers , and are the only ones appearing only once in a cell, whereas all other complex numbers appear twice.

Finally, equation (1.36) implies that are the three roots of the polynomial appearing in the right hand side of Weierstrass equation, namely

(1.37)

This directly implies that obey

(1.38)
(1.39)
(1.40)

Other Properties

The Weierstrass elliptic function obeys the homogeneity relation

(1.41)

For the specific case , the above relation assumes the form

(1.42)

Finally, when two of the roots , and coincide, the Weierstrass elliptic function degenerates to a simply periodic function. Assuming that the moduli and are real, then the existence of a double root implies that all roots are real. When the double root is larger than the simple root, the Weierstrass elliptic function takes the form

(1.43)

whereas when the double root is smaller than the simple root, it takes the form

(1.44)

If there is only one triple root, then it must be vanishing, since the three roots sum to zero. In this case, the Weierstrass elliptic function degenerates to a function that is not periodic at all, namely

(1.45)

The proofs of the homogeneity relation, as well as the double root limits of the Weierstrass elliptic function are left as an exercise for the audience.

Problems

Problem 1.1.

Show, using the integral formula for the Weierstrass elliptic function (1.32) that when and are real and all roots , and are also real, the half-period corresponding to the largest root is congruent to a real number, whereas the half-period corresponding to the smallest root is congruent to a purely imaginary number.

Then, show that when there is one real root and two complex ones, the half-period corresponding to the real root is congruent to both a real and a purely imaginary number.

Problem 1.2.

Show that at the limit two of the roots , and coincide, the Weierstrass elliptic function degenerates to a simply periodic function and can be expressed in terms of trigonometric or hyperbolic functions as described by formulae (1.43) and (1.44). Find the value of the unique period in terms of the double root. Also show that at the limit all three roots , and coincide, the Weierstrass elliptic function degenerates to the non-periodic function of equation (1.45).

Problem 1.3.

Use the definition (1.18) to deduce the homogeneity property of the Weierstrass elliptic function (1.41).

Lecture 2: Weierstrass Quasi-periodic Functions

In the previous lecture, we used several times the fact that the derivative of an elliptic function is also an elliptic function with the same periods. However, the opposite statement is not correct; the indefinite integral of an elliptic function is not necessarily an elliptic function. Such non-elliptic functions typically expose other interesting quasi-periodicity properties. In the following we will study two such quasi-periodic functions that are derived from Weierstrass elliptic function and have numerous important applications.

2.1 Quasi-periodic Weierstrass Functions

The Weierstrass Function

The Weierstrass function is defined as

(2.1)

furthermore satisfying the condition

(2.2)

which fixes the integration constant.

Using the definition of Weierstrass function, we acquire

(2.3)

Weierstrass function is an odd function.

(2.4)

Notice that the definition (2.1) coupled with the fact that is an even function implies that is an odd function up to a constant. The condition (2.2) fixes this constant to zero, so that is an odd function.

Quasi-periodicity of the Function

Equation (2.3) implies that in each cell defined by the periods of the corresponding function, the function has only a first order pole with residue equal to one. As such, it cannot be an elliptic function with the same periods as . Actually, it cannot be an elliptic function with any periods, since it is not possible to define a cell where the sum of the residues would vanish. The Weierstrass function is quasi-periodic. Its quasi-periodicity properties can be deduced from the periodicity of Weierstrass function. More specifically, integrating the relation , we find

The above relation for yields . Since is an odd function, the above implies that , which in turn yields

(2.5)

One can easily show inductively that

(2.6)

The quantities and are related with an interesting property. Consider the contour integral

Since has only a first order pole with residue equal to one within a cell, Cauchy residue theorem implies that

Performing the contour integral along the boundary of a cell, we get

Shifting by in the second integral and by in the third, we yield

As a result, and are related as

(2.7)

The Weierstrass Function

Since Weierstrass elliptic function has a single second order pole in each cell, integrating it once resulted in a function (the Weierstrass function) with a single first order pole in each cell. Integrating once more would lead to a logarithmic singularity in each cell. To avoid this, we define the next quasi-periodic function as the exponential of the integral of the Weierstrass function.

The Weierstrass function is defined as

(2.8)

together with the condition

(2.9)

which fixes the integration constant.

Integrating equation (2.3) term by term results in the following expression for function.

(2.10)

This implies that is analytic. At the locations of the poles of , it has first order roots.

Similarly to the definition of function, definition (2.8) implies that is odd up to a constant, which is set to zero by the defining condition (2.9), so that is an odd function,

(2.11)

Quasi-periodicity of the Function

The Weierstrass function is a quasi-periodic function. Its quasi-periodicity properties follow from the corresponding properties of the Weierstrass function. Integrating the equation (2.5), one yields

or

where . Substituting , we get

implying, . This means that the quasi-periodicity of the function is

(2.12)

Equation (2.12) can be used to prove inductively that

(2.13)

2.2 Expression of any Elliptic Function in Terms of Weierstrass Functions

The functions , or can be used to express any elliptic function with the same periods. In the following, we will derive such expressions and deduce interesting properties of a general elliptic function.

Expression of any Elliptic Function in Terms of and

Assume an elliptic function . Then, it can be written as

Both functions and are even. Thus, in order to express an arbitrary elliptic function in terms of and , it suffices to find an expression of an arbitrary even elliptic function in terms of and .

Assume an even elliptic function . As it is even, any irreducible set of poles of can be divided to a set of points with multiplicity and another set of points congruent to with equal multiplicities. In a similar manner, an irreducible set of roots of can be divided to a set of points with multiplicities and another set of points congruent to with equal multiplicities. Now consider the function