Mathieu function: Difference between revisions

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For the even and odd periodic Mathieu functions <math> ce, se</math> and the associated characteristic numbers <math>a</math> one can also derive asymptotic expansions for large <math>q</math>.<ref>McLachlan (1947), p. 237; Dingle and Müller (1962); Müller (1962); Dingle and Müller(1964)</ref> For the characteristic numbers in particular, one has with <math>N</math> approximately an odd integer, i.e. <math>N\approx N_0= 2n+1, n =1,2,3,...,</math>

:<math>

\begin{align}

a(N) = -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{1}{2^7q^{1/2}}N(N^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9)</math>

:<math>\;\;\;\;\;\;\;\;a(N) ={}& -2q + 2q^{1/2}N -\frac{1}{2^3}(N^2+1) -\frac{171}q{2^7q^{31/2}}N(33N^4 +410NN^2 +4053) -\frac{1}{2^{2012}q^2}(63N^6 + 1260N5N^4 + 2943N34N^2 +418079) +\\

& -\frac{1}{2^{17}q^{3/2}}N(33N^4 +410N^2 +405) -\frac{1}{2^{20}q^2}(63N^6 + 1260N^4 + 2943N^2 +41807) +

\mathcal{O}(q^{-5/2})

\end{align}

</math>

Observe the symmetry here in replacing <math> q^{1/2}</math> and <math>N</math> by <math>-q^{1/2}</math> and <math>- N</math>, which is a significant feature of the expansion. Terms of this expansion have been obtained explicitly up to and including the term of order <math>|q|^{-7/2}</math>.<ref name="Dingle and Müller 1962">Dingle and Müller (1962)</ref> Here <math>N</math> is only approximately an odd integer because in the limit of <math>q\rightarrowto \infty</math> all minimum segments of the periodic potential <math> \cos 2x</math> become effectively independent harmonic oscillators (hence <math>N_0</math> an odd integer). By decreasing <math>q</math>, tunneling through the barriers becomes possible (in physical language), leading to a splitting of the characteristic numbers <math> a \rightarrowto a_{\mp}</math> (in quantum mechanics called eigenvalues) corresponding to even and odd periodic Mathieu functions. This splitting is obtained with boundary conditions<ref name="Dingle and Müller 1962"/> (in quantum mechanics this provides the splitting of the eigenvalues into energy bands).<ref name="Müller-Kirsten 2012">Müller-Kirsten (2012)</ref> The boundary conditions are:

:<math> \biggleft(\frac{dce_{N_0-1}}{dx}\biggright)_{\pi/2}=0,\;\; ce_{N_0}(\pi/2)=0, \;\; \biggleft(\frac{dse_{N_0}}{dx}\biggright)_{\pi/2}=0, \;\; se_{N_0+1}(\pi/2)=0.</math>

Imposing these boundary conditions on the asymptotic periodic Mathieu functions associated with the above expansion for <math>a</math> one obtains

:<math> N-N_0 = \mp 2\biggleft(\frac{2}{\pi}\biggright)^{1/2} \frac{(16q^{1/2})^{N_0/2}e^{-4q^{1/2}}}{[\frac{1}{2}(N_0-1)]!}\biggleft[1-\frac{3(N_0^2+1)}{2^6q^{1/2}} + \frac{1}{2^{13}q}(9N_0^4 -40N_0^3 +18N_0^2 -136N_0 + 9) + ...\dots \biggright].</math>

The corresponding characteristic numbers or eigenvalues then follow by expansion, i.e.

:<math> a(N) = a(N_0) + (N-N_0) \biggleft(\frac{\partial a}{\partial N}\biggright)_{N_0} + ...\cdots . </math>

Insertion of the appropriate expressions above yields the result

:<math>\begin{align}

:<math>a(N)\rightarrow a_{\mp}(N_0)= -2q + 2q^{1/2}N_0 -\frac{1}{2^3}(N^2_0+1) - \frac{1}{2^7q^{1/2}}N_0(N_0^2+3) -

a(N)\to a_{\mp}(N_0) = {} & -2q + 2q^{1/2}NN_0 -\frac{1}{2^3}(N^22_0+1) - \frac{1}{2^7q^{1/2}}NN_0(NN_0^2 +3) -\frac{1}{2^{12}q}(5N^4 + 34N^2 +9)</math>

\frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - ... </math>

:<math>\;\;\;\;\;\;frac{1}{2^{12}q}(5N_0^4+34N_0^2 +9) - \;\;\;\;\;\;cdots \;\;
& \mp \frac{(16q^{1/2})^{N_0/2+1}e^{-4q^{1/2}}}{(8\pi)^{1/2}[\frac{1}{2}(N_0-1)]!}\bigg[1-\frac{N_0}{2^6q^{1/2}}(3N_0^2+8N_0+3) + ...\cdots\bigg].

\end{align}

</math>

For <math>N_0=1,3,5,...\dots</math> these are the eigenvalues associated with the even Mathieu eigenfunctions <math> ce_{N_0} </math> or <math>ce_{N_0-1}</math> (i.e. with upper, minus sign) and odd Mathieu eigenfunctions <math>se_{N_0+1}</math> or

<math>se_{N_0}</math> (i.e. with lower, plus sign). The explicit and normalised expansions of the eigenfunctions can be found in <ref name="Dingle and Müller 1962"/> or.<ref name="Müller-Kirsten 2012"/>