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.
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