Schrödinger type equations with real-time variable and complex spatial variables
In recent work, heat and Laplace equations, (un)damped wave equations, the Burgers and the Black-Merton-Scholes equations with real-time variable and complex spatial variable were studied. The purpose of this article is to make a similar study for the Schrödinger equation with real-time variable and complex spatial variable. The complexification of the spatial variable in the case of the Schrödinger equation is made by two different methods which produce different equations: first, one complexifies the spatial variable in the corresponding convolution formula by replacing the usual sum of variables (translation) by an exponential product (rotation) and second, one complexifies the spatial variable in the corresponding evolution equation and then one searches for non-analytic and for analytic solutions. By both methods, new kinds of evolution equations (or systems of equations) in two-dimensional spatial variables are generated and their solutions are constructed. It is of interest to note that in the case of the first method, solutions can be studied by employing the powerful theory of groups of linear operators. Then, we show that these solutions preserve some geometric properties of the boundary function, like the univalence, starlikeness, convexity and spirallikeness. In the case of the higher order Schrödinger equation, the complexification of the spatial variable is made in the corresponding convolution formula. © 2013 Copyright Taylor and Francis Group, LLC.
Complex Variables and Elliptic Equations
Gal, C., Gal, S., & Goldstein, J. (2013). Schrödinger type equations with real-time variable and complex spatial variables. Complex Variables and Elliptic Equations, 58 (3), 415-430. https://doi.org/10.1080/17476933.2011.593097