6,036 research outputs found
Integral formulas for wave functions of quantum many-body problems and representations of gl(n)
We derive explicit integral formulas for eigenfunctions of quantum integrals
of the Calogero-Sutherland-Moser operator with trigonometric interaction
potential. In particular, we derive explicit formulas for Jack's symmetric
functions. To obtain such formulas, we use the representation of these
eigenfunctions by means of traces of intertwining operators between certain
modules over the Lie algebra , and the realization of these modules
on functions of many variables.Comment: 6 pages. One reference ([FF]) has been corrected. New references and
an introduction have been adde
Small divisor problem in the theory of three-dimensional water gravity waves
We consider doubly-periodic travelling waves at the surface of an infinitely
deep perfect fluid, only subjected to gravity and resulting from the
nonlinear interaction of two simply periodic travelling waves making an angle
between them. \newline Denoting by the dimensionless
bifurcation parameter ( is the wave length along the direction of the
travelling wave and is the velocity of the wave), bifurcation occurs for
. For non-resonant cases, we first give a large family of
formal three-dimensional gravity travelling waves, in the form of an expansion
in powers of the amplitudes of two basic travelling waves. "Diamond waves" are
a particular case of such waves, when they are symmetric with respect to the
direction of propagation.\newline \emph{The main object of the paper is the
proof of existence} of such symmetric waves having the above mentioned
asymptotic expansion. Due to the \emph{occurence of small divisors}, the main
difficulty is the inversion of the linearized operator at a non trivial point,
for applying the Nash Moser theorem. This operator is the sum of a second order
differentiation along a certain direction, and an integro-differential operator
of first order, both depending periodically of coordinates. It is shown that
for almost all angles , the 3-dimensional travelling waves bifurcate
for a set of "good" values of the bifurcation parameter having asymptotically a
full measure near the bifurcation curve in the parameter plane Comment: 119
Quantum integrable systems and representations of Lie algebras
In this paper the quantum integrals of the Hamiltonian of the quantum
many-body problem with the interaction potential K/sinh^2(x) (Sutherland
operator) are constructed as images of higher Casimirs of the Lie algebra gl(N)
under a certain homomorphism from the center of U(gl(N)) to the algebra of
differential operators in N variables. A similar construction applied to the
affine gl(N) at the critical level k=-N defines a correspondence between higher
Sugawara operators and quantum integrals of the Hamiltonian of the quantum
many-body problem with the potential equal to constant times the Weierstrass
function. This allows one to give a new proof of the Olshanetsky-Perelomov
theorem stating that this Hamiltonian defines a completely integrable quantum
system. We also give a new expression for eigenfunctions of the quantum
integrals of the Sutherland operator as traces of intertwining operators
between certain representations of gl(N).Comment: 17 pages, no figure
- …