Sandro Graffi
Convergence of a quantum normal form and an exact quantization formula.
Mon, 02/07/2011 - 16:01 — SomNambulAuthors: Sandro Graffi, Thierry Paul
Subjects: Dynamical Systems
AbstractWe consider the Schr\"odinger operator defined by the quantization of the
linear flow of diophantine frequencies over the l-dimensional torus, perturbed
by a holomorphic potential which depends on the actions only through their
particular linear combination defining the Hamiltonian of the linear flow.Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schr\"odinger propagator.
Thu, 06/10/2010 - 15:01 — SomNambulAuthors: Sandro Graffi, Lorenzo Zanelli
Subjects: Mathematical Physics
AbstractWe construct a family of Fourier Integral Operators, defined for arbitrary
large times, representing a global parametrix for the Schr\"odinger propagator
when the potential is quadratic at infinity. This construction is based on the
geometric approach to the corresponding Hamilton-Jacobi equation and thus
sidesteps the problem of the caustics generated by the classical flow.
Moreover, a detailed study of the real phase function allows us to recover a
WKB semiclassical approximation which necessarily involves the multivaluedness
of the graph of the Hamiltonian flow past the caustics.PT Symmetric Schr\"odinger Operators: Reality of the Perturbed Eigenvalues.
Thu, 01/21/2010 - 16:01 — SomNambulAuthors: Francesco Cannata, Emanuela Caliceti, Sandro Graffi
Subjects: Mathematical Physics
AbstractWe prove the reality of the perturbed eigenvalues of some PT symmetric
Hamiltonians of physical interest by means of stability methods. In particular
we study 2-dimensional generalized harmonic oscillators with polynomial
perturbation and the one-dimensional $x^2(ix)^{\epsilon}$ for $-1<\epsilon<0$.
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