Application of the nonlinear Schrödinger equation with a logarithmic inhomogeneous term to nuclear physics

Solution of the logarithmic Schrödinger equation with a Coulomb potential

“…The logarithmic Schrödinger equation (log SE) is one of the nonlinear modifications of Schrödinger's equation, with applications in quantum optics [1,2], nuclear physics [3,4], diffusion phenomena [5], stochastic quantum mechanics [6], effective quantum gravity [7] and Bose-Einstein condensation [8,9]. A relativistic version in the form of a Klein-Gordon type equation displaying dilatation/conformal covariance was first proposed by G. Rosen [10].…”

Schrödinger Equations with Logarithmic Self-Interactions: From Antilinear PT-Symmetry to the Nonlinear Coupling of Channels

“…Firstly, although the experimental tests as performed for conserved systems in atomic physics excluded any quantitatively predictive implementation of the "effective nonlinearity" hypothesis of the logarithmic type, the reality-mimicking situation appeared perceivably more encouraging in nuclear-physics phenomenology where the spatial separation of the individual fermions may be expected to be reduced [61,62], as well as in a theory of superfluidity, where many-body interactions become strongly nonlinear with an increase of density [69][70][71][72]. The appealing possibility of making quantum theory slightly nonlinear survived as a challenging theoretical option.…”

“…One of the best known examples is provided by [16,17] in which the toy models are considered in the form of the nonlinear logarithmic Schrödinger Equation (5) with the wave-function solutions ψ ∈ L 2 ( d ) studied in an interval of time t ∈ (t 0 , t 1 ). This equation, along with its relativistic analogue, finds multiple applications in the physics of quantum fields and particles [49][50][51][52][53][54][55], extensions of quantum mechanics [16,56], optics and transport or diffusion phenomena [57][58][59][60], nuclear physics [61,62], the theory of dissipative systems and quantum information [63][64][65][66][67][68], the theory of superfluidity [69][70][71][72] and the effective models of the physical vacuum and classical and quantum gravity [73][74][75][76], where one can utilize the well-known fluid/gravity analogy between inviscid fluids and pseudo-Riemannian manifolds [77][78][79][80][81]. The relativistic analogue of Equation (5) is obtained by replacing the derivative part with the d'Alembert operator and is not considered here.…”

On the linearity of the Schrödinger equation

“…The logarithmic Schrödinger equation outlived these defeats but not as a fundamental theory. Owing to its unique properties it has been used as an exactly soluble model of nonlinear phenomena in nonlinear optics [16,17], in nuclear physics [14], in the study of dissipative systems [15], in geophysics [18], and even in computer science [19]. The logarithmic nonlinearity is also theoretically appealing due to its connection with stochastic dynamics [22][23][24].…”