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Abstract. We consider the mass concentration phenomenon for the L 2 -critical nonlinear Schrödinger equations of higher orders. We show that any solution u to, which blows up in a finite time, satisfies a mass concentration phenomenon near the blow-up time. We verify that as α increases, the size of region capturing a mass concentration gets wider due to the stronger dispersive effect.

The electronic properties of two-dimensional double-quantum dots in the presence of external magnetic fields are investigated by a variational Monte Carlo method with s and s-p trial wavefunctions. We compute the exchange energy between two electrons as well as the two-electron total Coulomb energy for the singlet and triplet states in both strongly and weakly coupled quantum dots, and compare our data with the results of the numerically exact diagonalization of the Schrödinger Equation. In both systems, the singlet Coulomb energy decreases in magnetic fields as a consequence of magnetic localization, whereas the triplet Coulomb energy reaches a maximum value at intermediate magnetic fields before decreasing. Overall, good agreement between the two methods is obtained with s-p orbital trial wavefunction in strongly and weakly coupled quantum dots.