The Laplace-Beltrami operator on the embedded torus
Introduction
In this paper we discuss eigenvalues and eigenfunctions of the Laplace-Beltrami operator [6] on the torus embedded in three dimensional euclidean space. Most results concern eigenvalues but eigenfunctions are also mentioned in Sections 3 and 10.
The eigenvalues −Λ of the Laplace-Beltrami operator on the embedded torus can be reduced to the eigenvalues λ of the Sturm-Liouville equation (which we may call the torus equation) subject to the condition that it admits a nontrivial solution with period 2π. More precisely, if denote the inner and outer radius of the torus, respectively, then the eigenvalues are connected by the equation . The parameter is the quotient of these radii. Therefore, we always assume that except in Section 7 where we allow complex p. The parameter m is a nonnegative integer. The simplest case refers to eigenvalues of the Laplace-Beltrami operator corresponding to rotationally invariant eigenfunctions. The eigenvalue problem for (1) is a special case of a Sturm-Liouville problem with periodic boundary conditions [1], [2], [3], [17] and we collect the basic results in Section 3.
In Section 4 we show that the torus equation can be transformed to an Ince equation, also called Magnus-Winkler-Ince equation in [11, page 21]. This equation has the form containing four parameters . We notice that equation (1) (with ) is a special case of (2) when but it requires a transformation to bring (1) into the form (2) when . Usually, are assumed to be real numbers and c plays the role of the eigenvalue parameter. However, in the Ince form of the torus equation (1) complex parameters appear when , and both c and d involve the eigenvalue parameter. These observations indicate that, although the torus equation (1) is closely related to known equations, results on these related equations cannot be applied to it.
In Sections 5 and 6, using the Ince form of equation (1), we show that the eigenvalue problem for (1) is equivalent to an eigenvalue problem for infinite tridiagonal matrices similar to corresponding results [12] for the Mathieu, Lamé and Ince equations. However, we obtain a generalized eigenvalue problem of the form with both A and B tridiagonal. In the corresponding results for Mathieu, Lamé and Ince equations, B is the identity matrix. By considering finite submatrices of A and B we obtain computable upper and lower bounds for the eigenvalues of (1). This is the simplest method to compute the eigenvalues because the work is reduced to standard procedures contained in every linear algebra package of mathematical software (we used the LinearAlgebra package of Maple).
In Sections 7 and 8 we investigate the behavior of the eigenvalues when . It turns out that the eigenvalues are analytic functions at and we derive their power series expansions at together with lower bounds for the convergence radii.
In Section 9 we discuss the behavior of the eigenvalues when . This is an example of singular perturbation which is more difficult to treat than the regular perturbation . We use the Prüfer transformation to obtain results.
In the final Section 10 we derive the asymptotic behavior of eigenvalues for fixed and related sum formulas. Finally, we use the amplitude theorem to prove some properties of the graphs of eigenfunctions.
Section snippets
The Laplace-Beltrami operator
We consider a torus with outer radius R and inner radius r, where We represent the torus in by using coordinates . A function can be identified with a function which has period 2π in both variables separately. The tangent vectors are The metric tensor is given by Since , the Laplace-Beltrami operator Δ is
A Sturm-Liouville eigenvalue problem
We consider the differential equation (4) with , , and : We call λ an eigenvalue if (6) admits a nontrivial solution with period 2π. This poses a well-known eigenvalue problem for a periodic differential equation; see [2], [3], [17]. If is a solution of (6) then also is a solution. Therefore, our eigenvalue problem splits into two eigenvalue problems, one for even eigenfunctions and one for odd eigenfunctions; see [3,
Other forms of the Sturm-Liouville equation
We transform the differential equation (6) into Liouville normal form by setting We obtain the differential equation where The boundary conditions (7), (8) become and , respectively.
We note that
Therefore, we obtain the following estimate for eigenvalues [15, §27.IX]. Lemma 2 For we have The same
Fourier expansions and coexistence
Suppose that λ is one of the eigenvalues, and let be a corresponding eigenfunction satisfying (6). Then is an even function with period 2π that we can expand in a Fourier cosine series. By substituting this series in differential equation (6) we obtain a recursion formula for the coefficients of the Fourier series. However, it is of advantage to use differential equation (16) in place of (6). The function defined by (15) is a nontrivial even solution of (16) with period 2π.
Matrix representation
We consider the Hilbert sequence space endowed with its standard inner product and norm . For we define a linear operator on the domain by where, for , and all other . In other words, is represented by the infinite tridiagonal matrix , . This matrix is formally hermitian.
Similarly,
Complex eigenvalues
In the theory of the Mathieu equation , power series expansions of the eigenvalue λ in powers of the parameter q play an important role [9, Section 2.25]. These expansions arise when we consider the Mathieu equation with as a perturbation of the simple equation . Similarly, equation (13) with can be treated as a perturbation of the equation . Since we want to apply results from complex analysis, it is natural to permit complex values of p in (13) (with
The behavior of eigenvalues as
According to Theorem 10, the eigenvalues and are analytic functions of p in a disk centered at the origin. Therefore, they admit power series expansions in powers of p; see [13], [14] for corresponding results for the Lamé and Ince equation. If we replace p by −p and t by then equation (6) remains the same. It follows from this observation that these power series expansions of and contain only even powers of p. Let us calculate the coefficient of .
Theorem 11 Let and .
The behavior of eigenvalues as
In this section we apply the Prüfer transform [15, §27.IV] to the Sturm-Liouville equation (13). In (13) we set where and are smooth real-valued functions. Then we obtain where is defined by (14). We recall a few basis facts about solutions of (56). If the solution θ is determined by an initial condition at (independent of λ), then exists for all and is a smooth strictly increasing function
Further results
We give the asymptotic behavior of the eigenvalues and as .
Theorem 16 For every , we have where The asymptotic expansion (69) is also true with in place of . Proof The asymptotic behavior of the eigenvalues of equation (13) subject to boundary conditions is known; see [4, (4.11)]. We obtain where Then (69) follows
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