| Abstract: |
| We give a lower bound for the number of positive, real eigenvalues of the Hamiltonian differential operator
\[
\begin{pmatrix}
0 & -L_- \ L_+ & 0
\end{pmatrix}
\]
on a compact interval with Dirichlet boundary conditions, where $L_\pm$ are arbitrary scalar-valued Schr\{o}dinger operators. Such an operator arises, for example, when linearising about a standing wave in the nonlinear Schr\{o}dinger (NLS) equation. Our lower bound follows from a straightforward application of the ``Maslov box``, and includes a contribution to the Maslov index from a degenerate crossing. We compute this contribution via a homotopy argument, analysing the local behaviour of the eigenvalue curves -- which represent the evolution of the eigenvalues as the domain is shrunk or expanded -- to do so. Applying our theory to standing wave solutions of the NLS equation leads to compact interval analogues of the Jones-Grillakis instability theorem and the Vakhitov-Kolokolov criterion. Comparison with existing lower bounds, which make use of constrained eigenvalue counts for $L_+$ and $L_-$, leads to some interesting connections between the objects appearing therein and the contribution from the degenerate crossing. |
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