| Abstract: |
| The virial theorem is a nice property for the linear Schrodinger equation in atomic and
molecular physics as it gives an elegant ratio between the kinetic and potential energies
and is useful in assessing the quality of numerically computed eigenvalues. If the governing
equation is a nonlinear Schrodinger equation with power-law nonlinearity, then a similar
ratio can be obtained but there seems no way of getting any eigenvalue estimate. It is
surprising as far as we are concerned that when the nonlinearity is either square root or
saturable nonlinearity (not a power-law), one can develop a virial theorem and eigenvalue
estimate of nonlinear Schrodinger equations in R2 with square root and saturable
nonlinearity, respectively. Furthermore, the eigenvalue estimate can
be used to prove the 2nd order term (which is of order $\ln\Gamma$) of the lower bound of the
ground state energy as the coefficient $\Gamma$ of the nonlinear term tends to infinity. |
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