Regularity for a Schrödinger equation with singular potentials and application to bilinear optimal control. (English) Zbl 1109.35094

The authors study the Schrödinger equation \[ i\partial_tu+\Delta u+| x-a(t)| ^{-1}u+V_1u=0 \] on \(\mathbb{R}^3\times (0,T)\), where \(a\in W^{2,1}(0,T;\mathbb{R}^3)\) and \(V_1\) is a real valued electric potential, possibly unbounded, that may depend on space and time variables. They assume that the initial data \(u_0\in H^2(\mathbb{R}^3)\) is such that \[ \int_{\mathbb{R}^3}(1+| x| ^2)^2| u_0(x)| ^2\,dx<\infty. \] Assuming a sufficiently high regularity of \(V_1\) and that \(V_1\) is at most quadratic at infinity, the authors establish the well-posedness of the problem and show that the regularity of the solution is the same as that of the initial data.


35Q40 PDEs in connection with quantum mechanics
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
35B65 Smoothness and regularity of solutions to PDEs
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