Solution of the 3D logarithmic Schrödinger equation with a central potential

Abstract

Some form of the time-independent logarithmic Schrödinger equation (log SE) arises in almost every branch of physics. Nevertheless, little progress has been made in obtaining analytical or numerical solutions due to the nonlinearity of the logarithmic term in the Hamiltonian. Even for a central potential, the Hamiltonian does not commute with L 2 or L ; z the Hamiltonian is invariant under the parity operation only if the wave function is an eigenstate of the parity operator. We show that the solutions with well-defined parity can be expressed as a linear combination of eigenstates of L 2 and L , z where the parity restrictions on l and m determine the nodal structure of the wave function. The dominant contribution in the sum is designated as l˜and m; these serve as approximate quantum numbers. Using an iterative finite element approach, we also carry out fully converged numerical calculations in 1D, 2D and 3D for the special case of a Coulomb potential. The nodal structure of the wave functions and the expectation values á ñ L 2 and á ñ L z 2 are consistent with the analytical predictions. Values for the energy and expectations values are tabulated for the low-lying states. The methods developed for this problem are applicable across many areas of physics.

An Alternative to Dark Matter and Dark Energy: Scale-Dependent Gravity in Superfluid Vacuum Theory

“…which is also confirmed by analytical and numerical studies of differential equations with logarithmic nonlinearity of various types [23][24][25]28,31,35,42].…”

“…Some special cases of Equation ( 9), for example when b → b 0 = const, were extensively studied in the past, although not for reasons related to quantum liquids [22,23]. There were also extensive mathematical studies of these equations, to mention just some very recent literature [24][25][26][27][28][29][30][31][32][33][34][35][36][37].…”

Superfluid stars and Q-balls in curved spacetime

“…This equation is a Lorentz-covariant analogue of the logarithmic Schrödinger equation, which was extensively studied: to mention only a few very recent works [17][18][19][20][21][22]. Relativistic wave equations with logarithmic nonlinearity have also been extensively studied in the past, assuming fixed Minkowski spacetime [10,11,16,23,27].…”

On the convergence of the Crank-Nicolson method for the logarithmic Schrödinger equation

“…We would like to fill out our references list, by mentioning the work [21] for the use of the Crank-Nicolson method to approximate the solution to the logarithmic Schrödinger equation with artificial boundary conditions without providing rigorous error estimates for the numerical approximation error, and the works [24] and [25] for the numerical approximation of the time independent logarithmic Schrödinger equation.…”