Function $\phi_n$ satisfies the differential equation of this form
$$\mathcal{L}_z\phi_n=-\nu \frac{\mathrm{d}^4 \phi_n}{\mathrm{d} z^4}-\frac{\mathrm{d}^2 \phi_n}{\mathrm{d} z^2}=\mu_n \phi_n, \quad n=1,2,...,\infty \quad (1)$$
where the parameter $\nu>0$ and $z\in[-\pi,\pi]$, $\phi_n$ denotes an eigenfunction and $\mu_n$ denotes an eigenvalue, subject to $4$ periodic boundary conditions
$$\frac{\mathrm{d}^j \phi_n}{\mathrm{d} z^j}(-\pi)=\frac{\mathrm{d}^j \phi_n}{\mathrm{d} z^j}(\pi), \quad j=0,1,2,3 \quad (2)$$
I can solve a simpler version of the eigenvalue problem, that is,
$$\frac{\mathrm{d}^2 \phi_n}{\mathrm{d} z^2}=\mu_n \phi_n, \quad n=1,2,...,\infty \quad (3)$$
subject to 2 boundary conditions
$$\frac{\mathrm{d}^j \phi_n}{\mathrm{d} z^j}(-\pi)=\frac{\mathrm{d}^j \phi_n}{\mathrm{d} z^j}(\pi), \quad j=0,1 \quad (4)$$
where the eigenvalue $\mu_n$ is a real constant.
To solve the eigenvalue problem (3) and (4), first, letting $\mu_n=-\lambda_n^2$, the characteristic equation of (3) is $r^2+\lambda_n^2=0$, whose roots are $r=\pm \mathrm{i}\lambda_n$. Then (3) has the general solution $\phi_n(z)=a \sin\lambda_n z+b\cos\lambda_n z$ with constant $a$ and $b$. By applying the boundary conditions (4), the pair of algebraic equations for $a$ and $b$ can be written as
$$ \left[ \begin{array}{cc} \sin\lambda_n\pi&0\\ 0&\lambda_n\sin\lambda_n\pi \end{array} \right]\left[ \begin{array}{c} a\\ b \end{array} \right]=\left[ \begin{array}{c} 0\\ 0 \end{array} \right] $$
For there to be nontrivial solutions to this set of equations for $a$ and $b$, the determinant of the coefficient matrix must be zero, that is, $\lambda_n\sin^2\lambda_n\pi=0$, which determines the values of $\lambda_n$. It is $\lambda_n=0$ or $\sin\lambda_n\pi=0$. The roots of the latter one is $\lambda_n=n$, where $n=0, \pm1, \pm2, ...$. The resulting eigenfunctions are $\phi_0=b$ corresponding to the eigenvalue $\mu_0=0$ and $\phi_n(z)=a \sin n z+b\cos n z$ corresponding to the eigenvalue $\mu_n=-n^2=-1,-4,-9,...$, in which $\mu_0=0$ has multiplicity two.
Question: I have difficulty in solving the problem with a higher-order differential operator, say, 4th-order in my problem of Eqs.(1) and (2), in which the eigenvalue $\mu_n$ should be a complex number.
Can anybody give me some suggestions for solving the eigenvalue problem of Eqs.(1) and (2)? Thank you in advance!
Here is the answer for your reference:
The eigenvalues: $\mu_0=0$ and $\mu_n=-\nu n^4+n^2$ ($\mu_n$ has multiplicity two)
The eigenfunctions $\psi_0=\frac{1}{\sqrt{2\pi}}$, $\phi_n(z)=\frac{1}{\sqrt{\pi}}\sin{(nz)}$ and $\psi_n(z)=\frac{1}{\sqrt{\pi}}\cos{(nz)}$ with $n=1,2,...,\infty$.