Explicit solutions to hyper-Bessel integral equations of second kind

AbstractIn earlier papers, the authors have used the transmutation method to find solutions to Volterra integral or differ-integral equations of second kind, involving Erdélyi-Kober fractional integration operators (see [1,2]), as well as to dual integral equations and some Bessel-type differential equations (see [3,4]). Here we consider the so-called hyper-Bessel integral equations whose kernel-function is a rather general special function (a Meijer's G-function). Such an equation can be written also in a form involving a product of arbitrary number of Erdélyi-Kober integrals. By means of a Poisson-type transmutation, we reduce its solution to the well-known solution of a simpler Volterra equation involving Riemann-Liouville integration only. In the general case, the solution is found as a series of integrals of G-functions, easily reducible to series of G-functions. For particular nonhomogeneous (right-hand side) parts, this solution reduces to some known special functions. The main techniques are based on the generalized fractional calculus

The pressure broadening of spectral lines A unified theory for the Van Der Waals interaction

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Comments on employing the Riesz-Feller derivative in the Schrödinger equation

In this paper, we deal with a fractional Schrödinger equation that contains the quantum Riesz-Feller derivative instead of the Laplace operator in the case of a particle moving in a potential field. In particular, this equation is solved for a free particle in terms of the Fox H-function. On the other hand, we show that from physical viewpoint, the fractional Schrödinger equation with the quantum Riesz-Feller derivative of order α, 0 < α ≤ 2 and skewness θ makes sense only if it reduces to the Laplace operator (α = 2) or to the quantum Riesz fractional derivative (θ = 0). The reason is that the quantum Riesz-Feller derivative is a Hermitian operator and possesses real eigenvalues only when α = 2 or θ = 0. We then focus on the time-independent one-dimensional fractional Schrödinger equation with the quantum Riesz derivative in the case of a particle moving in an infinite potential well. In particular, we show that the explicit formulas for the eigenvalues and eigenfunctions of the time-independent fractional Schrödinger equation that some authors recently claimed to receive cannot be valid. The problem to find right formulas is still open