Spectra of operator pencils with small đť’«đť’Ż-symmetric periodic perturbation
ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 21.

We study the spectrum of a quadratic operator pencil with a small đť’«đť’Ż-symmetric periodic potential and a fixed localized potential. We show that the continuous spectrum has a band structure with bands on the imaginary axis separated by usual gaps, while on the real axis, there are no gaps but at certain points, the bands bifurcate into small parabolas in the complex plane. We study the isolated eigenvalues converging to the continuous spectrum. We show that they can emerge only in the aforementioned gaps or in the vicinities of the small parabolas, at most two isolated eigenvalues in each case. We establish sufficient conditions for the existence and absence of such eigenvalues. In the case of the existence, we prove that these eigenvalues depend analytically on a small parameter and we find the leading terms of their Taylor expansions. It is shown that the mechanism of the eigenvalue emergence is different from that for small localized perturbations studied in many previous works.

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DOI : 10.1051/cocv/2019070
Classification : 34B07, 34K27
Mots-clés : Quadratic operator pencil, PT-symmetric periodic perturbation, band spectrum, emerging eigenvalue, asymptotics 
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     author = {Borisov, Denis and Cardone, Giuseppe},
     title = {Spectra of operator pencils with small {\ensuremath{\mathscr{P}}\ensuremath{\mathscr{T}}-symmetric} periodic perturbation},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {26},
     year = {2020},
     doi = {10.1051/cocv/2019070},
     mrnumber = {4068302},
     zbl = {1444.47023},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2019070/}
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Borisov, Denis; Cardone, Giuseppe. Spectra of operator pencils with small đť’«đť’Ż-symmetric periodic perturbation. ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 21. doi : 10.1051/cocv/2019070. http://www.numdam.org/articles/10.1051/cocv/2019070/

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