Bifurcations of thresholds in essential spectra of elliptic operators under localized non?Hermitian perturbations
نویسندگان
چکیده
We consider the operator $${\cal H} = {\cal H}' -\frac{\partial^2\ }{\partial x_d^2} \quad\text{on}\quad\omega\times\mathbb{R}$$ subject to Dirichlet or Robin condition, where a domain $\omega\subseteq\mathbb{R}^{d-1}$ is bounded unbounded. The symbol ${\cal H}'$ stands for second order self-adjoint differential on $\omega$ such that spectrum of contains several discrete eigenvalues $\Lambda_{j}$, $j=1,\ldots, m$. These are thresholds in essential H}$. study how these bifurcate once we add small localized perturbation $\epsilon{\cal L}(\epsilon)$ H}$, $\epsilon$ positive parameter and an abstract, not necessarily symmetric operator. show into resonances H}$ vicinity $\Lambda_j$ sufficiently $\epsilon$. prove effective simple conditions determining existence find leading terms their asymptotic expansions. Our analysis applies generic non-self-adjoint perturbations and, particular, characterized by parity-time ($PT$) symmetry. Potential applications our result embrace broad class physical systems governed dispersive diffractive effects. use findings develop scheme controllable generation non-Hermitian optical states with normalizable power real part complex-valued propagation constant lying continuum. corresponding eigenfunctions can be interpreted as generalization bound embedded For particular example, persistence expansions confirmed direct numerical evaluation perturbed spectrum.
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ژورنال
عنوان ژورنال: Studies in Applied Mathematics
سال: 2021
ISSN: ['0022-2526', '1467-9590']
DOI: https://doi.org/10.1111/sapm.12367