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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.

A review on spectral and differential-geometric properties of Delsarte transmutation operators in multidimension is given. Their differential geometrical and topological structure in multidimension is analyzed, the relationships with De Rham-Hodge-Skrypnik theory of generalized differential complexes are stated. Some applications to integrable dynamical systems theory in multidimension are presented.

We consider a class of Hankel operators $H$ realized in the space $L^2 ({\Bbb R}_{+}) $ as integral operators with kernels $h(t+s)$ where $h(t)=P (\ln t) t ^{-1}$ and $P(X)= X^n+p_{n-1} X^{n-1}+\cdots$ is an arbitrary real polynomial of degree $n$. This class contains the classical Carleman operator when $n =0$. We show that a Hankel operator $H$ in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator $A = v Q(D) v$ in the space $L^2 ({\Bbb R}) $. Here $Q(X)= X^n+ q_{n-1} X^{n-1}+\cdots$ is a polynomial determined by $P(X)$ and $v(\xi)=\pi^{1/2} (\cosh(\pi\xi))^{-1/2} $ is the universal function. Then the operator $A = v Q(D) v$ reduces by the generalized Liouville transform to the standard differential operator $B = D^n+ b_{n-1} (x)D^{n-1}+\cdots+ b_{0} (x)$ with the coefficients $b_{m}(x)$, $m=0,\ldots, n-1$, decaying sufficiently rapidly as $|x|\to \infty$. This allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized Carleman operators. In particular, we show that the absolutely continuous spectrum of $H$ is simple and coincides with $\Bbb R$ if $n$ is odd, and it has multiplicity $2$ and coincides with $[0,\infty)$ if $n\geq 2$ is even. The singular continuous spectrum of $H$ is empty, and its eigenvalues may accumulate to the point $0$ only. As a by-product of our considerations, we develop spectral theory of a new class of {\it degenerate} differential operators $A = v Q(D) v$ where $Q(X)$ is an arbitrary real polynomial and $v(\xi)$ is a sufficiently arbitrary real function decaying at infinity.

We consider a class of Hankel operators $H$ realized in the space $L^2 ({\Bbb R}_{+}) $ as integral operators with kernels $h(t+s)$ where $h(t)=P (\ln t) t ^{-1}$ and $P(X)= X^n+p_{n-1} X^{n-1}+\cdots$ is an arbitrary real polynomial of degree $n$. This class contains the classical Carleman operator when $n =0$. We show that a Hankel operator $H$ in this class can be reduced by an {\it explicit} unitary transformation (essentially by the Mellin transform) to a differential operator $A = v Q(D) v$ in the space $L^2 ({\Bbb R}) $. Here $Q(X)= X^n+ q_{n-1} X^{n-1}+\cdots$ is a polynomial determined by $P(X)$ and $v(\xi)=\pi^{1/2} (\cosh(\pi\xi))^{-1/2} $ is the universal function. Then the operator $A = v Q(D) v$ reduces by the generalized Liouville transform to the standard differential operator $B = D^n+ b_{n-1} (x)D^{n-1}+\cdots+ b_{0} (x)$ with the coefficients $b_{m}(x)$, $m=0,\ldots, n-1$, decaying sufficiently rapidly as $|x|\to \infty$. This allows us to use the results of spectral theory of differential operators for the study of spectral properties of generalized Carleman operators. In particular, we show that the absolutely continuous spectrum of $H$ is simple and coincides with $\Bbb R$ if $n$ is odd, and it has multiplicity $2$ and coincides with $[0,\infty)$ if $n\geq 2$ is even. The singular continuous spectrum of $H$ is empty, and its eigenvalues may accumulate to the point $0$ only. As a by-product of our considerations, we develop spectral theory of a new class of {\it degenerate} differential operators $A = v Q(D) v$ where $Q(X)$ is an arbitrary real polynomial and $v(\xi)$ is a sufficiently arbitrary real function decaying at infinity.

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills eqautions. The soliton like solutions to the related set of nonlinear dynamical systemare discussed.

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills eqautions. The soliton like solutions to the related set of nonlinear dynamical systemare discussed.

The differential-geometric and topological structure of Delsarte transmutation operators and associated with them Gelfand-Levitan-Marchenko type eqautions are studied making use of the De Rham-Hodge-Skrypnik differential complex. The relationships with spectral theory and special Berezansky type congruence properties of Delsarte transmuted operators are stated. Some applications to multidimensional differential operators are done including three-dimensional Laplace operator, two-dimensional classical Dirac operator and its multidimensional affine extension, related with self-dual Yang-Mills eqautions. The soliton like solutions to the related set of nonlinear dynamical systemare discussed.

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