Eigenvalue spectrum of the spheroidal harmonics: A uniform asymptotic analysis

The spheroidal harmonics $S_{lm}(\theta;c)$ have attracted the attention of both physicists and mathematicians over the years. These special functions play a central role in the mathematical description of diverse physical phenomena, including black-hole perturbation theory and wave scattering by nonspherical objects. The asymptotic eigenvalues $\{A_{lm}(c)\}$ of these functions have been determined by many authors. However, it should be emphasized that all previous asymptotic analyzes were restricted either to the regime $m\to\infty$ with a fixed value of $c$, or to the complementary regime $|c|\to\infty$ with a fixed value of $m$. A fuller understanding of the asymptotic behavior of the eigenvalue spectrum requires an analysis which is asymptotically uniform in both $m$ and $c$. In this paper we analyze the asymptotic eigenvalue spectrum of these important functions in the double limit $m\to\infty$ and $|c|\to\infty$ with a fixed $m/c$ ratio.

The characteristic angular equation (1) for the spheroidal harmonic functions is supplemented by a regularity requirement for the corresponding eigenfunctions S(θ; c) at the two boundaries θ = 0 and θ = π. These boundary conditions single out a discrete set of eigenvalues {A lm } which are labeled by the discrete spheroidal harmonic index l (where l − |m| = {0, 1, 2, ...}). For the special case c = 0 the spheroidal harmonic functions S(θ; c) reduce to the spherical harmonic functions Y (θ), which are characterized by the familiar eigenvalue spectrum A lm = l(l + 1).
The various asymptotic spectrums of the spheroidal harmonics with c 2 ∈ R (when c ∈ R the corresponding eigenfunctions are called oblate, while for ic ∈ R the eigenfunctions are called prolate) were explored by many authors, see [1, [12][13][14][15][16][17] and references therein. In particular, in the asymptotic regime m 2 ≫ |c| 2 the eigenvalue spectrum is given by [12,13] while in the opposite limit, |c| 2 ≫ m 2 with ic ∈ R, the asymptotic spectrum is given by [1, [13][14][15]17] The asymptotic regime c 2 ≫ m 2 (with c ∈ R) was studied in [1, [13][14][15][16][17][18], where it was found that the eigenvalues are given by: Note that the spectrum (4) is doubly degenerate. It should be emphasized that all previous asymptotic analyzes of the eigenvalue spectrum were restricted either to the regime m → ∞ with a fixed value of c [12,13], or to the complementary regime |c| → ∞ with a fixed value of m [1, [13][14][15][16]. A complete understanding of the asymptotic eigenvalue spectrum requires an analysis which is uniform in both m and c [that is, a uniform asymptotic analysis which is valid for a fixed (non-negligible) m/c ratio as both m and |c| tend to infinity].
The main goal of the present paper is to present a uniform asymptotic analysis for the spheroidal harmonic eigenvalues in the double asymptotic limit m → ∞ and |c| → ∞ (5) with a fixed m/c ratio.

II. A TRANSFORMATION INTO THE SCHRÖDINGER-TYPE WAVE EQUATION
For the analysis of the asymptotic eigenvalue spectrum, it is convenient to use the coordinate x defined by [12,17] x ≡ ln tan in terms of which the angular equation (1) for the spheroidal harmonic eigenfunctions takes the form of a onedimensional Schrödinger-like wave equation [19] where the effective radial potential is given by Note that the transformation (6) The effective potential U (θ) is invariant under the transformation θ → π − θ. It is characterized by two qualitatively different spatial behaviors depending on the relative magnitudes of A and c 2 . We shall now study the asymptotic behaviors of the spheroidal eigenvalues in the two distinct cases: A/c 2 > 1 and A/c 2 < 1 [20].
If A > c 2 then the effective radial potential U (x(θ)) is in the form of a symmetric potential well whose local minimum is located at [Note that θ min = π 2 corresponds to x min = 0.] Spatial regions in which U (x) < 0 (the 'classically allowed regions') are characterized by an oscillatory behavior of the corresponding wave function S, whereas spatial regions in which U (x) > 0 are characterized by an exponentially decaying wave function (these are the 'classically forbidden regions'). The effective radial potential U (x) is characterized by two 'classical turning points' {x − , x + } (or equivalently, {θ − , θ + }) for which U (x) = 0 [21].
The one-dimensional Schrödinger-like wave equation (7) is in a form that is amenable to a standard WKB analysis. In particular, a standard textbook second-order WKB approximation yields the well-known quantization condition [22][23][24][25][26] x + for the bound-state 'energies' (eigenvalues) of the Schrödinger-like wave equation (7), where N is a non-negative integer. The characteristic WKB quantization condition (10) determines the eigenvalues {A} of the spheroidal harmonic functions in the double limit {|c|, m} → ∞. The relation so obtained between the angular eigenvalues and the parameters m, c, and N is rather complex and involves elliptic integrals. However, if we restrict ourselves to the fundamental (low-lying) modes which have support in a small interval around the potential minimum x min [27], then we can use the expansion (10) to obtain the WKB quantization condition [25] |U min | where a prime denotes differentiation with respect to x. The subscript "min" means that the quantity is evaluated at the minimum x min of U (x(θ)). Substituting (8) with x min = 0 into the WKB quantization condition (11), one finds the asymptotic eigenvalue spectrum A(c, m, N ) = m 2 + (2N + 1) m 2 − c 2 + O(1) ; N = {0, 1, 2, ...} (12) in the N ≪ √ m 2 − c 2 regime [27]. The resonance parameter N = {0, 1, 2, ...} corresponds to l − |m| = {0, 1, 2, ...}, where l is known as the spheroidal harmonic index.
It is worth noting that the eigenvalue spectrum (12), which was derived in the double asymptotic limit {|c|, m} → ∞, reduces to (2) in the special case m ≫ |c| and reduces to (3) in the opposite special case |c| ≫ m with ic ∈ R. The fact that our uniform eigenvalue spectrum (12) reduces to (2) and (3)  If A < c 2 then the effective radial potential U (x(θ)) is in the form of a symmetric double-well potential: it has a local maximum at and two local minima at [29] with Thus, the two potential wells are separated by a large potential-barrier of height The fact that the two potential wells are separated by an infinite potential-barrier in the c → ∞ limit (with c 2 > m 2 ) [30] implies that the coupling between the wells (the 'quantum tunneling' through the potential barrier) is negligible in the c → ∞ limit. The two potential wells can therefore be treated as independent of each other in the c → ∞ limit [22, 31]. Thus, the two spectra of eigenvalues (which correspond to the two identical potential wells) are degenerate in the c → ∞ limit [32]. Substituting (8) with θ min = 1 2 arccos(−A/c 2 ) into the WKB quantization condition (11), one finds the asymptotic eigenvalue spectrum It is worth noting that the eigenvalue spectrum (17), which was derived in the double asymptotic limit {|c|, m} → ∞, reduces to (4) in the special case c 2 ≫ m 2 . The fact that our uniform eigenvalue spectrum (17) reduces to (4) in the appropriate special limit provides a consistency check for our analysis [36].

ACKNOWLEDGMENTS
This research is supported by the Carmel Science Foundation. I thank Yael Oren, Arbel M. Ongo, Ayelet B. Lata, and Alona B. Tea for stimulating discussions.