For finding solutions of a non-lin. Schroedinger eq. with a logarithmic non-linear term ("potential") the "RCP"-method (cf. https://www.researchgate.net/publication/296638302_Reconstruction_of_Potentials_for_Soliton_Solutions_of_the_Klein-Gordon-Maxwell_Field) for finding timeindependent soliton solutions AND appropriate potentials U(rho(x)), rho(x):=|psi(x)|^2,  of a non-lin. Klein-Gordon-Maxwell field might be helpful.

The basic idea of "Reconstruction of Potentials for Soliton Solutions of the Klein-Gordon-Maxwell Field" is to consider the Lagrangian to be NOT completely specified, instead L = ..... + U(rho(x)), and the "potential" U as a function of rho is the object that has to be determined by the field eqs. for a GIVEN wave packet psi(x).

Proceeding this way one obtains diff.eqs. for the e.m. vector potential A_mue(x) , mue=0,1,2,3 with given psi(x) and a linear system of eqs. for coefficients c_n where c_n are the expansion coefficients in a power series expansion U(rho)= c_0 rho^0 + c_1 rho^1 + c_2 rho^2 + c_3 rho^3 + ...

Numerical calculations showed that this "RCP-method" works at least for the non-linear KGM field, i.e. the resulting charge density and energy density of the KGM field has a solitary structure. Therefore, this method can be used to (re)construct potentials which allow for soliton solution of the field which is under consideration.

An exact solution ( Gaussian wave packet ) for the logarithmic potential rho + rho ln(rho) (up to some constants, see eq. 66) is compared with approximate solutions obtained using the RCP-method.

Thesis Reconstruction of Potentials for Soliton Solutions of the Kl...

Thank you for your answers. I have made progress. I have become aware of some Gausson solutions but I am looking for methods of solving e.g. a hydrogen-like radial equation with \Psi(r)*log(|\Psi|) term added to the (radial) Hamiltonian.  I have an exact analytical (Gausson) solution for the ground state but I am looking for solutions of excited states.

Much has been answered concerning the Log SE.  The most useful references were those of Iwo Bialynicki-Birula.  Surprises abound.  Yes, you can find bound states but these are NOT orthogonal etc.  We now have a body of results which I think will lead to a good paper.

Indeed the paper in question was published last July in J. Phys. Commun.

Article Solution of the logarithmic Schrödinger equation with a Coul...

DOI: 10.1088/2399-6528/aad302

It answers this question and others.

best wishes

Tony

The work done and published for the log SE in a Coulomb potential for zero-angular momentum has been generalized to the case for non-zero angular momentum. Surprises abound. I will update when it is published. In the meantime, we have demonstrated the absolute need for the logarithmic term in superfluids like Helium-4. See Article Resolving the puzzle of sound propagation in liquid helium a...

We have gone further in solving log SE for a Coulomb potential. We licked the 3D case of non-zero angular momentum. This was a computational tour-de-force by Janine Shertzer and her (very) precise Finite Element Methods (FEM). Article Solution of the 3D logarithmic Schrödinger equation with a c...