The expectation value of the vacuum energy density tied to the cosmological constant is derived from the non-relativistic quantum deion of the reheating process using the unitary transformation method. Our research is performed for two special cases, i.e. under the assumption of two different duration terms for the radiation-dominated era. We suppose that the radiation-dominated era lasted for 3,000 years in the first case, and for a very short time so that it can be ignored in the other case. For the former case, the resulting present cosmological constant is much higher than the previously known one. However, for the latter case, it is in good agreement with that obtained from recent cosmic microwave background radiation measurements. Hence, we cannot exclude the possibility that the change of the universe from the radiation-dominated era to the matter-dominated era might have progressed very swiftly after the beginning of reheating. The analysis shows that the vacuum energy density has been dissipated by the nonconservative force acting on the coherent oscillations of the scalar field according to the expansion of the universe.

The quantum states with continuous spectrum for the time-dependent harmonic oscillator perturbed by a singularity are investigated. This system does not oscillate while the system that has discrete energy eigenvalue does. Exact wave functions satisfying the Schrödinger equation for the system are derived using invariant operator and unitary operator together.

The SU(1,1) generators are considered in order to investigate the uncertainty relation in the Barut–Girardello coherent state for the time-dependent singular oscillator. The corresponding uncertainty product is obtained by evaluating the expectation values of the canonical variables and their squares.

The problem of a charged particle in the presence of a variable magnetic field is considered. Using the linear and the quadratic invariants as a tool, the wave functions in Fock state as well as in coherent state are obtained. The corresponding propagators which propagate the wave functions in the space–time are derived. Using numerical computations we have managed to draw some plots for the real, imaginary, and absolute values of the propagators. This has been used to analyze the properties of the propagators associated with both of the linear and the quadratic invariants. It has been shown that there is no essential difference between the behavior of the absolute value of the propagators in both of the linear and the quadratic invariants.

For the continuum and discrete elastic equations, we derive exact artificial boundary conditions (ABCs), often referred to as transparent boundary conditions, that can be applied at a planar interface below which there are no forces. Solution of the elasticity equations can then be performed using this interface as an artificial boundary, often with greatly reduced computational effort, but without loss of accuracy. A general solvability requirement is presented for the existence of an artificial boundary operator for discrete systems (such as discrete elasticity) on an unbounded (semi-infinite) domain. The solvability requirement is validated by introducing a sum-of-exponentials ansatz for the solution below the artificial boundary. We also derive a new expression for the total energy for the system, involving only the region above the artificial boundary. Numerical examples are provided to confirm and illustrate the accuracy and effectiveness of the results.

Abstract. Using the vector field method, we find a more general condition than finite type that implies a local regularity result for the ∂-Neumann operator. In our result, it is possible for an analytic disk to be on the part of the boundary where we have local regularity. 1.