Universality in an Information-theoretic Motivated Nonlinear Schrodinger Equation

Using perturbative methods, we analyse a nonlinear generalisation of Schrodinger’s equation that had previously been obtained through information-theoretic arguments. We first compute numerically the leading correction, in terms of the nonlinearity scale, to the energy eigenvalues of the linear Schrodinger equation in the presence of some common external potentials and parametrise the results in a simple form. We then study the problem analytically so as to explain the generic features that are observed. In one space dimension these are: (i) For nodeless ground states, the energy shifts are subleading in the nonlinearity parameter compared to the shifts for the excited states, (ii) the shifts for the excited states are due predominantly to contribution from the nodes of the unperturbed wavefunctions and (iii) the energy shifts for excited states are positive for small values of a regulating parameter and negative at large values, vanishing at a universal critical value that is not manifest in the equation. Some of these features hold true for higher dimensional problems. We also study two exactly solved nonlinear Schrodinger equations so as to contrast our observations. Finally, we comment on the possible significance of our results if the nonlinearity is physically realised. Email: [email protected]