We study the l1-error of a Hamiltonian-preserving scheme, developed in [11], for the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through barriers, and of the high frequency waves through interfaces. We use the l1-error estimates established in ...

We study the l1-error of a Hamiltonian-preserving scheme, developed in [11], to the Liouville equation with a piecewise constant potential in one space dimension. This problem has important applications in computations of the semiclassical limit of the linear Schrödinger equation through barriers, and of the high frequency waves through interfaces. We use the l1-error estimates established in [...

We develop a fourth order numerical method for the computation of multivalued physical observables (density, momentum, etc.) in the semiclassical limit of the one dimensional linear Schrödinger equation in the case of discontinuous potentials. We adopt the level set framework developed in [21] which allows one to compute the multivalued physical observables via solving the classical Liouville e...

The phase flow method was originally introduced in [28] which can efficiently solve autonomous ordinary differential equations. In [13], the method was generalized to solve Hamiltonian system where the Hamiltonian function was discontinuous. However, both these methods require phase flow map constructed on an invariant manifold. This can increase computational cost when the invariant domain is ...

This paper develops a spectral expansion approach to the valuation of contingent claims when the underlying state variable follows a one-dimensional diffusion with the infinitesimal variance a(x), drift b(x) and instantaneous discount (killing) rate r(x). The Spectral Theorem for self-adjoint operators in Hilbert space yields the spectral decomposition of the contingent claim value function. Ba...

We explore the role of a special class of optimization problems involving controlled Liouville equations. We present some new results on the controllability of the Liouville equation and discuss the optimal control of its moments for some important special cases. The problems treated suggest a new domain of applicability for optimal control, one which can include constraints more appropriate fo...

We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to BV ; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data t...

The zeros of the eigenfunctions of self-adjoint Sturm–Liouville eigenvalue problems interlace. For these problems interlacing is crucial for completeness. For the complex Sturm–Liouville problem associated with the Schrödinger equation for a non-Hermitian PT-symmetric Hamiltonian, completeness and interlacing of zeros have never been examined. This paper reports a numerical study of the Sturm– ...

In this paper we apply the Homotopy perturbation method to derive the higher-order asymptotic distribution of the eigenvalues and eigenfunctions associated with the linear real second order equation of Sturm-liouville type on $[0,pi]$ with Neumann conditions $(y'(0)=y'(pi)=0)$ where $q$ is a real-valued Sign-indefinite number of $C^{1}[0,pi]$ and $lambda$ is a real parameter.

The derivation of the Navier-Stokes equation starting from the Liouville equation using projection techniques yields a friction term which is nonlinear in the velocity. Using the results of multilinear mode-coupling technique for calculating equlibrium correlation functions, it is shown that the secondorder part of the term is non-zero, thus leading to an incorrect formula for the equation.