Spectral and scattering theory of fourth order differential operators

Abstract

Abstract. An ordinary differential operator of the fourth order with coefficients converging at infinity sufficiently rapidly to constant limits is considered. Scattering theory for this operator is developed in terms of special solutions of the corresponding differential equation. In contrast to equations of second order "scattering" solutions contain exponentially decaying terms. A relation between the scattering matrix and a matrix of coefficients at exponentially decaying modes is found. In the second part of the paper the operator D 4 on the half-axis with different boundary conditions at the point zero is studied. Explicit formulas for basic objects of the scattering theory are found. In particular, a classification of different types of zero-energy resonances is given.

Spectral and scattering theory for perturbations of the Carleman operator

“…It turns out that the link between these two "bases" is given by the elements of scattering matrix (5.16). The proof of the following assertion is very similar to the corresponding result of [11].…”

“…In the first case one can develop the theory relying exclusively on Volterra integral equations while such a possibility is of course lacking in the second case. In this respect, the theory of Hankel operators is closer to the theory of one-dimensional differential operators of order higher than two where Fredholm integral equations occur naturally (see [11]). Note that for the operator D n in L 2 (R) the eigenfunctions e ±ikx are the same as those for the operator D 2 for all n = 1, 2, .…”

Spectral and scattering theory for differential and Hankel operators

“…Finally, we discuss part (iv) of Theorem 2.1. It can be easily checked (see, e.g., [5,24]) that for every λ ∈ R \ {0} the differential equation Bf = λf has solutions f j , j = 1, . .…”

“…This is the classical stuff for n = 2; we refer to the paper [7] by L. D. Faddeed or the book [26], Chapters 4 and 5. For all n ≥ 3, the construction below is probably not explicitly written in the literature, but it is essentially the same as for the particular case n = 4 discussed in [24]. We note that in contrast to the case n = 2 when one can use Volterra integral equations, for n ≥ 3 one is obliged to work with Fredholm equations.…”

“…In particular, the operator B does not have the singular continuous spectrum. For the proof, one can use the scheme of S. Agmon [1] which significantly simplifies (see [24]) in the one-dimensional case. Thus we obtain statements (i) and (ii) of Theorem 2.1.…”