Numerical Methods For The Delta-potential Schr(o|")dinger Equations

The Schrodinger equation is a basic equation of quantum physics. It has a lot of applications not only in the fields of modern physics, but also in the regimes of pure and applied mathematics. The Schrodinger equations with distribution poten-tials, e.g.δ(x)’s,δ’(x)’s or linear combinations of them, can be found in modeling Boson or Fermi gas, some semiconductor heterostructures, or condensate problems of sufficiently dilute ultracold atomic gas which is confined by singular external poten-tials, etc. In this thesis, we study numerical methods for the stationary and dynamical Schrodinger equations with these distribution potentials. The main results are stated as follows.The first part concerns interface methods for the stationary Schrodinger equa-tions (SSEs). Explicit Jump Immersed Interface Method (EJIIM) and Peskin’s im-mersed boundary Method (IBM) are explored for solving the SSEs with single-and double-delta potentials. By using the EJIIM and Peskin’s IBM, we obtain the gen-eralized and standard algebraic eigenvalue problems (AEPs) respectively, and then prove that the former can be transformed into the corresponding standard ones. The bound-state problems, moreover, are extensively investigated from various numeri-cal aspects by utilizing the classical shifted inverse power method and QR method as well. In addition, EJIIM and IIM are applied to the ID SSEs, in which the poten-tial is - aδ(x)+bδ’(x) and the mass is a constant or has a jump at the origin. The bound-state problems for the SSEs, are discussed in detail from some theoretical and numerical aspects. The "implicit" AEPs, which arise in the stationary discretizations for SSEs with - aδ(x)+bδ’(x) potential by IIM, can be solved by an novelly designed method, i.e. the classical shifted inverse power method combining with a consistent fixed-point iteration. Extensive numerical experimentations show that, IBM, EJIIM and IIM are all computationally efficient and numerically stable and convergent for these differential eigenvalue equations. Moreover, the numerical accuracies for the computational bound-state energy levels can reach the corresponding theoretical ones obtained via the local Taylor-expansion truncation analyses.The second part focuses on multisymplectic geometric numerical integration for an important model in condensed matter physics, i.e. the dynamical nonlinear Schrodinger equations (DNLSEs) with an inclusion of delta potentials. Due to the inclusion, the model can not be reformulated as a multisymplectic Hamiltonian par-tial differential equation (MHPDE) as the normal DNLSEs. With some functional settings, a weak multisymplectic geometric reformulation is proposed for the model, some local and global conservation laws are addressed in the same weak sense. It is pointed out that, since the space translation invariance is broken, the system does not possess the momentum conservation laws even in the weak sense. Based on the weak reformulation, the novel concatenating Runge-Kutta (RK) and Runge-Kutta-Nystrom (RKN) methods are elaborately constructed and the discrete multisymplecticity for them is revealed. Under multisymplectic Runge-Kutta (MSRK) and MSRKN dis-cretizations, we prove that the normalization conservation laws can be preserved ex-actly. Numerical experiments validate our theoretical analyses. Comparisons with non-MS schemes show that, a remarkable advantage of the MS ones applied to the DNLSEs is the precise preservation of the normalization conservation laws. And the energy and parity symmetry, can be preserved stably to some accuracies over the long-time evolutions by the MS and also non-MS schemes.