The quantum-mechanical problems of N 1-dimensional equal particles mass m interacting pairwise via quadratic (``harmonical'') and/or inversely (``centrifugal'') potentials is solved. In the first case, characterized by pair potential ¼mω2(xi − xj)2 + g(xi xj)−2, g > −ℏ2/(4m), complete energy spectrum (in center-of-mass frame) given formula E=ℏω(12N)12[12(N−1)+12N(N−1)(a+12)+ ∑ l=2Nlnl],with a = ½(1 4mgℏ−2)½. 1 quantum numbers nl are nonnegative integers; each set {nl; l 2, 3, ⋯, N}...

The problem of three equal particles interacting pairwise by inversecube forces (``centrifugal potential'') in addition to linear (``harmonical is solved one dimension.

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The problem of N quantum-mechanical equal particles interacting pairwise by inverse-cube forces (``centrifugal potential'') in addition to linear (``harmonical is considered a onedimensional space. An explicit expression for the ground-state energy and corresponding wavefunction exhibited. A class excited states similarly displayed.

The authors study the problem of wave modulation for a large and quite general class nonlinear evolution equations. They demonstrate that only very limited number 'universal' model equations, on relevant time space scales, describe phenomena interest under all circumstances. Classical among equations is course Schrodinger equation (NLS); however, certain conditions, modulations occur shorter scales than those NLS. On other hand, if NLS becomes linear by cancellation terms, then appropriate...

The exact solution for the ground state of quantum one-dimensional $N$-boson problem with attractive $\ensuremath{\delta}$-function two-body potentials is compared (exact) self-consistent corresponding variational Hartree problem.

For pt. I see ibid., vol.3, p.229-62, 1987. The authors continue their investigation of the model equations that govern wave modulations induced by weakly nonlinear effects, in context evolution 1+1 dimensions having a dispersive linear part. They study mainly with an even part (namely, features only derivatives order). identify several equations, including some they had found previously and new ones. also indicate how one can relate two classes odd parts.

The main purpose of this paper is to describe a technique reduction, whereby from the class evolution equations for matrices order N solvable via spectral transform associated (matrix) linear Schrödinger eigenvalue problem, one derives subclasses nonlinear involving less than N2 fields. To illustrate method, 2 two fields (rather 4) are obtained. first coincides, or rather includes, that generalized Zakharov–Shabat problem; further reduction single field reproduces number well-known...

The exact solution is presented of the scattering problem three equal particles interacting in one-dimension via two- and/or three-body inverse-square potentials. Both classical and quantal problems are treated. It shown that outcome this an extremely simple relation between initial final momenta, latter being univocally determined by former even case. solvability problem, results just mentioned, peculiar to particle