The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase...The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations(PDEs)that can be derived using the standard rules of standard derivatives.However,for the space fractional Schrödinger equation(FSE),the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations,because not only the problem becomes non-local,but also the rules for the standard derivatives generally do not hold for the fractional derivatives.In this work,we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)form for the space FSE based on the quantum Riesz fractional operators.We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local,the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE,and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order.We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes.Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.展开更多
A semiclassical method based on the closed-orbit theory is applied to analysing the dynamics of photodetached electron of H- in the parallel electric and magnetic fields. By simply varying the magnetic field we reveal...A semiclassical method based on the closed-orbit theory is applied to analysing the dynamics of photodetached electron of H- in the parallel electric and magnetic fields. By simply varying the magnetic field we reveal spatial bifurcations of electron orbits at a fixed emission energy, which is referred to as the fold caustic in classical motion. The quantum manifestations of these singularities display a series of intermittent divergences in electronic flux distributions. We introduce semiclassical uniform approximation to repair the electron wavefunctions locally in a mixed phase space and obtain reasonable results. The approximation provides a better treatment of the problem.展开更多
We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valu...We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.展开更多
基金support was received from the Simons Foundation 714376.
文摘The semiclassical approximation is an efficient approach for studying the standard Schrödinger equation(SE)both analytically and numerically,where the wavefunction is approximated by an ansatz such that its phase and amplitude are determined through Hamilton-Jacobi type partial differential equations(PDEs)that can be derived using the standard rules of standard derivatives.However,for the space fractional Schrödinger equation(FSE),the introduction of the fractional differential operators makes it challenging to derive relevant semiclassical approximations,because not only the problem becomes non-local,but also the rules for the standard derivatives generally do not hold for the fractional derivatives.In this work,we first attempt to derive the semiclassical approximation in the Wentzel-Kramers-Brillouin-Jeffreys(WKBJ)form for the space FSE based on the quantum Riesz fractional operators.We find that the phase and amplitude can also be determined by local Hamilton-Jacobi type PDEs even though the space FSE is non-local,the Hamiltonian for the phase is consistent with that in the classical Hamilton-Jacobi approach for the space FSE,and the semiclassical approximation reduces to that for the standard SE when the fractional order becomes integer order.We then compute the numerical solutions for the space FSE through the semiclassical approximation by solving the local Hamilton-Jacobi type PDEs with well-established numerical schemes.Numerical experiments are presented to verify the accuracy and efficiency of the derived semiclassical formulations.
基金Project supported by the National Natural Science Foundation of China （Grant Nos 10374061 and 90403028）. We thank Professor Du Meng-Li for some useful suggestions.
文摘A semiclassical method based on the closed-orbit theory is applied to analysing the dynamics of photodetached electron of H- in the parallel electric and magnetic fields. By simply varying the magnetic field we reveal spatial bifurcations of electron orbits at a fixed emission energy, which is referred to as the fold caustic in classical motion. The quantum manifestations of these singularities display a series of intermittent divergences in electronic flux distributions. We introduce semiclassical uniform approximation to repair the electron wavefunctions locally in a mixed phase space and obtain reasonable results. The approximation provides a better treatment of the problem.
基金Acknowledgements This work was supported by the National Natural Science Foundation of China （Grant Nos. 11371218, 91630205）.
文摘We extend the theory of global geometrical optics method, proposed originally for the linear scalar high-frequency wave-like equations in [Commun. Math. Sci., 2013, 11（1）： 105-140], to the more general vector- valued Schrodinger problems in the semi-classical regime. The key ingredient in the global geometrical optics method is a moving frame technique in the phase space. The governing equation is transformed into a new equation but of the same type when expressed in any moving frame induced by the underlying Hamiltonian flow. The classical Wentzel-Kramers-Brillouin （WKB） analysis benefits from this treatment as it maintains valid for arbitrary but fixed evolutionary time. It turns out that a WKB-type function defined merely on the underlying Lagrangian submanifold can be obtained with the help of this moving frame technique, and from which a uniform first-order approximation of the wave field can be derived, even around caustics. The general theory is exemplified by two specific instances. One is the two-level SchrSdinger system and the other is the periodic SchrSdinger equation. Numerical tests validate the theoretical results.
