Justify the nonvanishing denominator in the spectral-evolution equation

Establish a formal justification that the denominator \(\int_D q_t(x)\varphi_t(x)^2\,dx\) does not vanish, thereby validating the division used to derive the eigenvalue spectral-evolution equation for the homotopy \(q_t=(1-t)\|q\|_\infty\mathbf{1}_D+tq_{\rm rad}\).

Background

The paper derives an evolution equation for eigenvalue branches $\lambda(t)$ of the compact operator $k^2V_{q_t}(k)$ along a homotopy from a constant reference medium to a radial medium. The derivation divides by the bilinear quantity $\int_D q_t\varphi_t^2\,dx$, where $\varphi_t$ is an associated eigenfunction.

The authors explicitly state that they lack a formal justification for this division. Their subsequent WKB-based spectral propagation assumes that the denominator is well-defined and relies on numerical examples to assess whether the resulting procedure is practically applicable. A proof that the denominator cannot vanish, or a characterization of conditions under which it remains nonzero, would provide the missing analytical foundation for this step.