BIRS Workshop Lecture Videos

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BIRS Workshop Lecture Videos

Nodal Sets and Eigenvalues for Small Radial Pertrubations of the Harmonic Oscillator Hanin, Boris


In this talk, I will present some recent results (joint with Tom Beck) about the behavior at infinity of nodal sets of eigenfunctions for small radial perturbations of the harmonic oscillator. For the unperturbed oscillator, separation of variables eigenfunctions are products of Laguerre functions and spherical harmonics. The angular momenta for a given \hbar that are present at fixed energy E = \hbar (n + d/2) are the spherical harmonics of frequency up to \hbar^{-1}. After a radial perturbation, the energy E eigenspace will break up into the different energies for different angular momenta. The radial rate of growth for the eigenfunctions is an increasing function of their energy E (namely r^{E-d/2}\exp{-r^2/2} in dimension d). Our results give precise information about which angular momenta give the biggest energies after perturbation and hence a good understanding of the size of the nodal set deep into the forbidden region.

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