Continuity theorems for a class of convolution operators and applications to superoscillations
In this paper, a very large, and so far not yet considered, class of potentials \(V(t, x)\) for the Schrödinger equation \[ i\frac{\partial\psi(t, x)}{\partial t}=\left(-\frac12\frac{\partial^2}{\partial x^2}+V(t, x)\right)\psi(t, x) \] is considered. A new theorem on the continuity of convolution operators with variable coefficients is proved. It is used to deduce that the limit of a superoscillating sequence maintains the superoscillatory behaviour for all values of time, when evolved according to Schrödinger equations with time-dependent potentials. For a potential \(V=V(t, x)\), the definitions of the Green function \(G_V\) of type (I) and of type (II) are introduced. For example, if \(G_V\) is the Green function of type (I), then the solution of Cauchy problem \[ i\frac{\partial\varphi(t, x)}{\partial t}=\left(-\frac12\frac{\partial^2}{\partial x^2}+V(t, x)\right)\psi(t, x), \quad\varphi(0, x)=e^{iax}\,, \] can be written as \(\psi_n(t, x)=\mathcal{U}(t, x,\partial_x )F_n(x, a)\), where the operator \(\mathcal{U}(t, x,\partial_x )\) is defined, \[ F_n(x, a):=\left(\cos\left(\frac nx\right)+ ia\sin\left(\frac xn\right)\right)^n \] and \[ \lim\limits_{n\to\infty}\psi_n(t, x)=\varphi_a(t, x)=\int_{\mathbb{R}}G_V(t,x,y)e^{iay}\,dy\,. \]
convolution operators with variable coefficients
scientific article; zbMATH DE number 7001291