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The virial theorem is a nice property for the linear Schrodinger equation inatomic and molecular physics as it gives an elegant ratio between the kineticand potential energies and is useful in assessing the quality of numericallycomputed eigenvalues. If the governing equation is a nonlinear Schrodingerequation with power-law nonlinearity, then a similar ratio can be obtained butthere seems no way of getting any eigenvalue estimate. It is surprising as faras we are concerned that when the nonlinearity is either square-root orsaturable nonlinearity (not a power-law), one can develop a virial theorem andeigenvalue estimate of nonlinear Schrodinger (NLS) equations in R2 withsquare-root and saturable nonlinearity, respectively. Furthermore, we show herethat the eigenvalue estimate can be used to obtain the 2nd order term (which isof order $ln\Gamma$) of the lower bound of the ground state energy as thecoefficient $\Gamma$ of the nonlinear term tends to infinity.

The virial theorem and ground state energy estimate of nonlinear  Schr\"odinger equations in $\mathbb{R}^2$ with square root and saturable  nonlinearities in nonlinear optics