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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.

MSC:
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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