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The Mathieu equation (https://en.wikipedia.org/wiki/Mathieu_function) is $y''+(a-2q\cos(2z))y=0$. The modified Mathieu equation is obtained by replacing $z$ with $\pm iz$: $$y''-(a-2q\cosh(2z))y=0.$$ In the study of the modified Mathieu equation, it is often assumed that $q$ is positive; see https://dlmf.nist.gov/28.20.

Here I am interested in the case $q<0$ in the above equation, and want to study normalizable solutions of the modified Mathieu equation. This is a well-defined problem. It is more clear to write the equation as $$-y''+2\tilde{q}\cosh(2z)y=\tilde{a}y,$$ where $\tilde{q}>0$ and $-\infty<z<\infty$. This equation is written in the form of a Schrödinger equation in which the potential is $2\tilde{q}\cosh(2z)$. Therefore, requiring the solution to be normalizable gives an eigenvalue problem. The eigenvalues of $\tilde{a}$ should be positive.

In the literature, most of the studies were on periodic solutions of the Mathieu equation. The special functions like $\text{ce}_\nu(z,q)$ and $\text{se}_\nu(z,q)$ are defined for this purpose. Non-periodic solutions were also studied, but they are usually non-normalizable in $-\infty<z<\infty$.

I think it is natural to study normalizable solutions of the modified Mathieu equation, and this is expected to be studied, but I cannot find useful references. So my question is: are there known results on normalizable solutions of the modified Mathieu equation?

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    $\begingroup$ Since the potential $V(x)=2q\cosh 2x$ satisfies $\lim_{x\to\pm\infty}V(x)=\infty$, the spectrum is indeed purely discrete and the eigenvalues are positive since $V>0$. I cannot see any reason to expect that these can be found explicitly. $\endgroup$
    – Christian Remling
    Commented Dec 5, 2023 at 19:40

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Solutions of this equation, normalized at $x\to\pm\infty$ are called Mathieu functions of the third kind. See, for example, D. Naylor, On a simplified asymptotic formula for the Mathieu function of the third kind, SIAM J. Math. Anal., 18, 6 (1987) 1616-1629.

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