Some properties of higher-order Darboux transformations

We introduce the operator of the Darboux transformation of order n as an nth-order differential operator transforming the solution of the one-dimensional Schrödinger equation with a given potential into the solution corresponding to a different potential. For n=1 we obtain the usual Darboux transformation. It is shown that when n1 this transformation can always be represented as a product of n first-order Darboux transformations. We discuss the relationship between higher-order Darboux transformations and the supersymmetric approach to quantum mechanics and also the algebra of the dynamical symmetry of the system.Tomsk State University. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 7, pp. 3–8, July, 1995.     

The reaction of tetrahydrochromeno[3,4-c]pyridines with activated alkynes. The first synthesis of tetrahydrochromeno[4,3-d]azocines

The first synthesis of tetrahydrochromeno[4,3-d]azocines via an alkyne-induced expansion reaction is described.     

Osmium dimethyl sulfoxide complexes: Synthesis and properties of [H(dmso)2][OsIII(dmso)2Br4]

The reaction of H₂[OsBr₆] with DMSO in ethanol solution resulted in DMSO complex [H(dmso-O)₂][Os^III(dmso-S)₂Br₄] (1) described previously as an intermediate product in the reaction of K₂[OsBr₆] with DMSO and characterized by EAS and ESR spectra. The coordination of DMSO molecules was established by IR and ¹H and ¹³C NMR spectroscopy. The oxidation state of osmium and trans arrangement of DMSO molecules in the anion were established by ESR. The behavior of complex 1 in solutions was studied by EAS, ESR, and mass-spectrometry: a displacement of Br^? ions accompanied by the reduction of osmium to oxidation state +2 occurs in DMSO, a solvation with displacement of DMSO molecules is observed at the first stage in water and methanol (rate constants 2.3 × 10^?4 and 1.7 × 10^?3 s^?1, respectively), the sequential substitution of DMSO molecules and osmium oxidation to form [Os^IVBr₆]_2? ions takes place in 4 mol/L HBr.     

Compact (correlated) coherent states of an electron moving in the field of a plane electromagnetic wave

Compact coherent states of the relativistic electron which satisfy the conditions of completeness and orthogonality are constructed in the field of a plane electromagnetic wave. The physical sense of the quantum parameters introduced is defined, as well as their connection with the indeterminacy of coordinates and momentum. The breakdown of the coherent state is found with respect to all possible states which correspond to various classical trajectories of a particle, and the association of the derived results with Volkov's solutions is established. Tomsk Pedagogical Institute. Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 45–49, February, 1993.