Resonances of 4th order differential operators

Article type: Research Article

Authors: Badanin, Andrey | Korotyaev, Evgeny L.^{; *}

Affiliations: Saint-Petersburg State University, Universitetskaya nab. 7/9, St. Petersburg 199034, Russia. E-mails: [email protected], [email protected], [email protected], [email protected]

Correspondence: [*] Corresponding author. E-mail: [email protected].

Abstract: We consider fourth order ordinary differential operator with compactly supported coefficients on the line. We define resonances as zeros of the Fredholm determinant which is analytic on a four sheeted Riemann surface. We determine estimates of the number of resonances in complex discs at large radius. We consider resonances of an Euler–Bernoulli operator on the real line with the positive coefficients which are constants outside some finite interval. We show that the Euler–Bernoulli operator has no eigenvalues and resonances iff the positive coefficients are constants on the whole axis.

Keywords: Fourth order operators, resonances, scattering, trace formula

DOI: 10.3233/ASY-181489

Journal: Asymptotic Analysis, vol. 111, no. 3-4, pp. 137-177, 2019