What is the maximum number of eigenvalues $\lambda < 0$ for the trigonometric problems?: $$ \begin{array}{c} -\frac{d^{2}f}{dx^{2}}=\lambda f,\\ \cos\alpha f(0) + \sin\alpha f'(0) = 0,\\ \cos\beta f(L)+\sin\beta f'(L)=0, \end{array} $$ where $0 \le \alpha,\beta < \pi$ are allowed to vary.
Example: If $\alpha = \beta \in (0,\pi/2)\cup(\pi/2,\pi)$, there is one negative eigenvalue $\lambda=-\cot^{2}(\alpha)$ with non-normalized eigenfunction $e^{-(\cot(\alpha))x}$. This gives rise to an unusual complete orthogonal basis of eigenfunctions for $L^{2}[0,L]$: $$ \exp\left\{-x\cot(\alpha)\right\},\sin\alpha \cos\left(\frac{n\pi x}{L}\right) -\cos\alpha\frac{L}{n\pi}\sin\left(\frac{n\pi x}{L}\right),\;\; n =1,2,3,\cdots\;. $$ When properly normalized, these eigenfunctions form a complete orthonormal basis of $[0,L]$. In other cases, there can be more than one negative eigenvalue.