On some non-Gaussian wave packets

We expand on a previous study by offering a generalized wave function associated with the parabolic cylinder function and a connection with a two-particle position-space wave function. We also provide an explicit formula for a wave function associated with a recent work by the present author and M. Wolf.

Inviscid linear stability analysis of two vertical columns of different densities in a gravitational acceleration field

We study the inviscid linear stability of a vertical interface separating two fluids of different densities and subject to a gravitational acceleration field parallel to the interface. In this arrangement, the two free streams are constantly accelerated, which means that the linear stability analysis is not amenable to Fourier or Laplace solution in time. Instead, we derive the equations analytically by the initial-value problem method and express the solution in terms of the well-known parabolic cylinder function. The results, which can be classified as an accelerating Kelvin–Helmholtz configuration, show that even in the presence of surface tension, the interface is unconditionally unstable at all wavemodes. This is a consequence of the ever increasing momentum of the free streams, as gravity accelerates them indefinitely. The instability can be shown to grow as the exponential of a quadratic function of time.

UNITARY TRANSFORMATION APPROACH FOR THE PHASE OF THE DAMPED DRIVEN HARMONIC OSCILLATOR

Using the invariant operator method and the unitary transformation method together, we obtained discrete and continuous solutions of the quantum damped driven harmonic oscillator. The wave function of the underdamped harmonic oscillator is expressed in terms of the Hermite polynomial while that of the overdamped harmonic oscillator is expressed in terms of the parabolic cylinder function. The eigenvalues of the underdamped harmonic oscillator are discrete while that of the critically damped and the overdamped harmonic oscillators are continuous. We derived the exact phases of the wave function for the underdamped, critically damped and overdamped driven harmonic oscillator. They are described in terms of the particular solutions of the classical equation of motion.

The asymptotic behaviour of Pearcey’s integral for complex variables

A deeper understanding of the rich structure of the canonical form of the oscillatory integral describing the cusp diffraction catastrophe, generally known as Pearcey’s integral P '( X , Y ), can be obtained by considering its analytic continuation to arbitrary complex variables X and Y . A new integral representation for P '( X , Y ) is given in the form of a contour integral involving a Weber parabolic cylinder function whose order is the variable of integration. It is shown how the asymptotics of P '( X , Y ) may be obtained from this representation for complex X and Y when either | X | or | Y | → ∞, without reference to the usual stationary points of the integrand. For the case | X | → ∞, Y finite the full asymptotic expansion of P '( X , Y ) is derived and its asymptotic character is found to be either exponentially large or algebraic in certain sectors of the X - plane. The case | Y | → ∞ , X finite is complicated by the presence of exponentially small subdominant terms in certain sectors of the Y - plane, and only the first terms in the expansion are given. The asymptotic behaviour of P '( X , Y ) on the caustic Y 2 + (2/3 X ) 3 = 0 is also obtained from the new representation and is shown to agree with recent results of D. Kaminski. The various properties of the Weber parabolic cylinder function required in this paper are collected together in the Appendix.