The single-body Schr\"odinger equation with complex potentials has been studied intensively. In this work, we investigate the dynamical process for Bose-Einstein condensed gases, based on the many-body Schr\"odinger equation with complex potentials. It is shown that a weakly imaginary periodic potential may induce perfect quantum coherence between different condensates, even without considering interatomic interactions. This imaginary-potential-induced quantum coherence may be tested with present experimental techniques. We also consider the quantum coherence in the presence of both the imaginary periodic potential and interatomic interactions.The single-body Schr\"odinger equation with complex potentials has been studied intensively. In this work, we investigate the dynamical process for Bose-Einstein condensed gases, based on the many-body Schr\"odinger equation with complex potentials. It is shown that a weakly imaginary periodic potential may induce perfect quantum coherence between different condensates, even without considering interatomic interactions. This imaginary-potential-induced quantum coherence may be tested with present experimental techniques. We also consider the quantum coherence in the presence of both the imaginary periodic potential and interatomic interactions.

arXiv:0912.4170Physical Review ANov 16, 201011 citations

The Schrödinger Wave Equation

JOHN C. MORRISON

Elsevier eBooksJan 1, 20100 citations

Localized states of the non-Hermitian Schr{ö}dinger equation with a complex $1/r^{2}$ potential: from Efimov states to bipolaronsLocalized states of the non-Hermitian Schr{ö}dinger equation with a complex $1/r^{2}$ potential: from Efimov states to bipolarons

Kyoka Tsunoda, Kazuki Yamamoto, Shimpei Endo

We study the non-Hermitian Schrödinger equation with a complex $1/r^{2}$ potential, which bridges Hermitian few-body and non-Hermitian many-body physics: a real attractive potential describes Efimov bound states, whereas a purely imaginary one describes the medium-induced interaction between two heavy polarons at large separations. We solve this equation analytically, obtaining both the localized and continuum states, and compare the solutions with numerical and WKB results. As the imaginary part of the potential is increased, the energies of the Efimov bound states acquire imaginary parts and rotate counterclockwise in the complex plane. On approaching the purely imaginary potential, the excited states disappear sequentially, while the Efimov ground state is smoothly connected to a bipolaron state localized at short distance. Such a bipolaron is localized solely by the medium-induced imaginary interaction, without any attractive real potential. Solving the equation by finite differences, we find that each bipolaron state is necessarily accompanied by another localized state that oscillates on the scale of the mesh size and hence has no counterpart in the continuum limit. Each such pair coalesces at an exceptional point as the strength of the purely imaginary potential is varied, whereas the coalescence is lifted once a finite real part is introduced. We show that this behavior originates from a symmetry of the discretized Schrödinger equation that holds for any purely imaginary potential.We study the non-Hermitian Schrödinger equation with a complex $1/r^{2}$ potential, which bridges Hermitian few-body and non-Hermitian many-body physics: a real attractive potential describes Efimov bound states, whereas a purely imaginary one describes the medium-induced interaction between two heavy polarons at large separations. We solve this equation analytically, obtaining both the localized and continuum states, and compare the solutions with numerical and WKB results. As the imaginary part of the potential is increased, the energies of the Efimov bound states acquire imaginary parts and rotate counterclockwise in the complex plane. On approaching the purely imaginary potential, the excited states disappear sequentially, while the Efimov ground state is smoothly connected to a bipolaron state localized at short distance. Such a bipolaron is localized solely by the medium-induced imaginary interaction, without any attractive real potential. Solving the equation by finite differences, we find that each bipolaron state is necessarily accompanied by another localized state that oscillates on the scale of the mesh size and hence has no counterpart in the continuum limit. Each such pair coalesces at an exceptional point as the strength of the purely imaginary potential is varied, whereas the coalescence is lifted once a finite real part is introduced. We show that this behavior originates from a symmetry of the discretized Schrödinger equation that holds for any purely imaginary potential.

cond-mat.quant-gasCross-listedarXiv:2610.21155Oct 16, 20260 citations

The Schrödinger Wave Equation

International geophysics/International geophysics seriesJan 1, 20020 citations

Time fractional Schrödinger equation: Fox's H-functions and the effective potential

Selçuk Ş. Bayın

After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.After introducing the formalism of the general space and time fractional Schrödinger equation, we concentrate on the time fractional Schrödinger equation and present new results via the elegant language of Fox's H-functions. We show that the general time dependent part of the wave function for the separable solutions of the time-fractional Schrödinger equation is the Mittag-Leffler function with an imaginary argument by two different methods. After separating the Mittag-Leffler function into its real and imaginary parts, in contrast to existing works, we show that the total probability is ⩽1 and decays with time. Introducing the effective potential approach, we also write the Mittag-Leffler function with an imaginary argument as the product of its purely decaying and purely oscillating parts. In the light of these, we reconsider the simple box problem.

arXiv:1103.3295Journal of Mathematical PhysicsJan 1, 201355 citations

Localized states of the non-Hermitian Schr{ö}dinger equation with a complex

1/r^{2}

potential: from Efimov states to bipolarons

We study the non-Hermitian Schrödinger equation with a complex

1/r^{2}

potential, which bridges Hermitian few-body and non-Hermitian many-body physics: a real attractive potential describes Efimov bound states, whereas a purely imaginary one describes the medium-induced interaction between two heavy polarons at large separations. We solve this equation analytically, obtaining both the localized and continuum states, and compare the solutions with numerical and WKB results. As the imaginary part of the potential is increased, the energies of the Efimov bound states acquire imaginary parts and rotate counterclockwise in the complex plane. On approaching the purely imaginary potential, the excited states disappear sequentially, while the Efimov ground state is smoothly connected to a bipolaron state localized at short distance. Such a bipolaron is localized solely by the medium-induced imaginary interaction, without any attractive real potential. Solving the equation by finite differences, we find that each bipolaron state is necessarily accompanied by another localized state that oscillates on the scale of the mesh size and hence has no counterpart in the continuum limit. Each such pair coalesces at an exceptional point as the strength of the purely imaginary potential is varied, whereas the coalescence is lifted once a finite real part is introduced. We show that this behavior originates from a symmetry of the discretized Schrödinger equation that holds for any purely imaginary potential.