Lamé spectral polynomials

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Polynomials

$n$

$P_n(h,m)$

h^3 - 2*h^2*m - 2*h^2 + h*m^2 + 3*h*m + h - m^2 - m

h^5 - 10*h^4*m - 10*h^4 + 33*h^3*m^2 + 87*h^3*m + 33*h^3 - 40*h^2*m^3 - 237*h^2*m^2 - 237*h^2*m - 40*h^2 + 16*h*m^4 + 208*h*m^3 + 492*h*m^2 + 208*h*m + 16*h - 48*m^4 - 252*m^3 - 252*m^2 - 48*m

h^7 - 28*h^6*m - 28*h^6 + 294*h^5*m^2 + 714*h^5*m + 294*h^5 - 1444*h^4*m^3 - 6654*h^4*m^2 - 6654*h^4*m - 1444*h^4 + 3409*h^3*m^4 + 27838*h^3*m^3 + 53043*h^3*m^2 + 27838*h^3*m + 3409*h^3 - 3528*h^2*m^5 - 52542*h^2*m^4 - 178056*h^2*m^3 - 178056*h^2*m^2 - 52542*h^2*m - 3528*h^2 + 1296*h*m^6 + 38448*h*m^5 + 242433*h*m^4 + 424062*h*m^3 + 242433*h*m^2 + 38448*h*m + 1296*h - 7776*m^6 - 98820*m^5 - 311580*m^4 - 311580*m^3 - 98820*m^2 - 7776*m

h^9 - 60*h^8*m - 60*h^8 + 1446*h^7*m^2 + 3354*h^7*m + 1446*h^7 - 18020*h^6*m^3 - 74190*h^6*m^2 - 74190*h^6*m - 18020*h^6 + 124881*h^5*m^4 + 831438*h^5*m^3 + 1476243*h^5*m^2 + 831438*h^5*m + 124881*h^5 - 482280*h^4*m^5 - 5038350*h^4*m^4 - 14462280*h^4*m^3 - 14462280*h^4*m^2 - 5038350*h^4*m - 482280*h^4 + 986896*h^3*m^6 + 16330512*h^3*m^5 + 73532001*h^3*m^4 + 118629678*h^3*m^3 + 73532001*h^3*m^2 + 16330512*h^3*m + 986896*h^3 - 944640*h^2*m^7 - 26300640*h^2*m^6 - 187600500*h^2*m^5 - 472392780*h^2*m^4 - 472392780*h^2*m^3 - 187600500*h^2*m^2 - 26300640*h^2*m - 944640*h^2 + 331776*h*m^8 + 17565696*h*m^7 + 213044160*h*m^6 + 837320784*h*m^5 + 1306136544*h*m^4 + 837320784*h*m^3 + 213044160*h*m^2 + 17565696*h*m + 331776*h - 3317760*m^8 - 77460480*m^7 - 500337280*m^6 - 1208058240*m^5 - 1208058240*m^4 - 500337280*m^3 - 77460480*m^2 - 3317760*m

Definition

The monic Lamé spectral polynomial is $P_n(h,m)=\prod_{j=0}^{2n}(h-h_{n,j}(m))$, where $h_{n,j}(m)$ are the finite band edges of $-d^2/dx^2+n(n+1)m\operatorname{sn}^2(x\mid m)$ for $0<m<1$ [1].

Parameters

$n$

— Order ($n$ is a positive integer.)

Formulas

(1)

$P_1(h,m)=(h-m)(h-1)(h-1-m)$ [1].

(2)

$P_n(h,0)=h\prod_{r=1}^{n}(h-r^2)^2$. At $m=0$ the operator is the free operator $-d^2/dx^2$ with limiting period $\pi$; the paired integer Fourier frequencies account for the double factors.

(3)

$P_n(h,m)=-P_n(n(n+1)-h,1-m)$. In particular, $P_n(n(n+1)/2,1/2)=0$.

Comments

(4)

Here $\operatorname{sn}$ is the Jacobi elliptic sine, $m=k^2$ is the squared modulus, and the energy is $h$ in the equation of [2]. The kinetic coefficient is $1$, and the potential has period $2K(m)$, where $K$ is the complete elliptic integral of the first kind. There is no period rescaling or subtraction of the mean potential. Each entry is expanded in the ordered polynomial variables $(h,m)$; values at $m=0$ and $m=1$ mean polynomial specializations.

(5)

The finite band edges are precisely the energies admitting a Lamé polynomial solution. These solutions have the form $\operatorname{sn}^a(x\mid m)\operatorname{cn}^b(x\mid m)\operatorname{dn}^c(x\mid m)p(\operatorname{sn}^2(x\mid m))$, where $a,b,c\in\{0,1\}$ and $p$ has degree $(n-a-b-c)/2\geq0$ [3]. The product includes every admissible sector. It does not include the higher closed-gap periodic and antiperiodic eigenvalues.

Programs

(P1)

Sage

# Download the attached generate.py into the current directory.
from generate import spectral_polynomial
spectral_polynomial(4)

References

[1]

Robert S. Maier, Lamé polynomials, hyperelliptic reductions and Lamé band structure, Philosophical Transactions of the Royal Society A 366 (2008), 1115–1153. See Table 3 and Definition 7.1. (arXiv) (doi)

Links

[2]

NIST DLMF, §29.2: Lamé differential equations

[3]

NIST DLMF, §29.12: Lamé polynomial solution types

Similar tables

Values of the Jacobi elliptic function $\operatorname{sn}(u\mid m)$ — The Jacobi elliptic sine determines the periodic potential.

Complete elliptic integral of the first kind $K(m)$ — The complete elliptic integral determines the period of the potential.

Characteristic values $a_n(q)$ of the Mathieu equation — Characteristic values for another second-order differential equation with a periodic potential.

Data properties

Entries are of type: rational polynomial

How well the digits are known: exact

Table is complete: no (it contains every positive order $1\leq n\leq4$, including all admissible solution sectors and the full symbolic dependence on $m$. This initial segment covers both parities and all eight solution types. The cutoff keeps expanded entries readable: $P_4$ has 843 characters, compared with 1462 for $P_5$ and 5039 for $P_8$)

How they were obtained:

Computed by exact arithmetic in $\mathbb Q[h,m]$, using tridiagonal characteristic polynomials of the finite invariant Lamé solution spaces. Every coefficient is an integer. No approximate eigenvalues enter the construction.

Each polynomial agrees coefficient by coefficient with the independently transcribed Weierstrass spectral factors in [1], Table 3, after the Jacobi energy conversion. The checker first reproduces the known factorizations at orders 1 and 2. It also checks degree, leading coefficient, integrality, the free-particle degeneration, the complementary-parameter identity and the lemniscatic central root. The attached checks are repeated on the values read back from the draft.