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1. CJM 2008 (vol 60 pp. 1168)
Short Time Behavior of Solutions to Linear and Nonlinear Schr{Ã¶dinger Equations We examine the fine structure of the short time behavior
of solutions to various linear and nonlinear Schr{\"o}dinger equations
$u_t=i\Delta u+q(u)$ on $I\times\RR^n$, with initial data $u(0,x)=f(x)$.
Particular attention is paid to cases where $f$ is piecewise smooth,
with jump across an $(n-1)$-dimensional surface. We give detailed
analyses of Gibbs-like phenomena and also focusing effects, including
analogues of the Pinsky phenomenon. We give results for general $n$
in the linear case. We also have detailed analyses for a broad class of
nonlinear equations when $n=1$ and $2$, with emphasis on the analysis of
the first order correction to the solution of the corresponding linear
equation. This work complements estimates on the error in this approximation.
Categories:35Q55, 35Q40 |
2. CJM 2008 (vol 60 pp. 572)
Non-Selfadjoint Perturbations of Selfadjoint Operators in Two Dimensions IIIa. One Branching Point This is the third in a series of works devoted to spectral
asymptotics for non-selfadjoint
perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2, having a
periodic classical flow. Assuming that the strength $\epsilon$
of the perturbation is in the range $h^2\ll \epsilon \ll h^{1/2}$
(and may sometimes reach even smaller values), we
get an asymptotic description of the eigenvalues in rectangles
$[-1/C,1/C]+i\epsilon [F_0-1/C,F_0+1/C]$, $C\gg 1$, when $\epsilon F_0$ is a saddle point
value of the flow average of the leading perturbation.
Keywords:non-selfadjoint, eigenvalue, periodic flow, branching singularity Categories:31C10, 35P20, 35Q40, 37J35, 37J45, 53D22, 58J40 |
3. CJM 1998 (vol 50 pp. 1298)
Imprimitively generated Lie-algebraic Hamiltonians and separation of variables Turbiner's conjecture posits that a Lie-algebraic Hamiltonian
operator whose domain is a subset of the Euclidean plane admits a
separation of variables. A proof of this conjecture is given in
those cases where the generating Lie-algebra acts imprimitively.
The general form of the conjecture is false. A counter-example is
given based on the trigonometric Olshanetsky-Perelomov potential
corresponding to the $A_2$ root system.
Categories:35Q40, 53C30, 81R05 |