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Solving the dynamical inverse problem for the Schrödinger equation by the boundary control method. (English) Zbl 1001.35121
For an unknown potential $q(x)$, the Schrödinger equation $$i\ \frac{\partial u(x,t)}{\partial t}+\frac{\partial^2u(x,t)}{\partial x^2}-q(x)u(x,t) =0$$ with boundary conditions $$\frac{\partial u(0,t)}{\partial x} = f(t),\ u(\ell,t)=0,\ t\in (0, T)$$ and initial condition $$u(x,0) =0,$$ gives rise to a response operator (or, in control terms, input-output map) $$(R^Tf)(t)=u^f(0,T),$$ for each control $f\in L^2(0,T)$. The problem considered here is to determine $q$ from $R^T$. If $u^f(0,t)$ denotes the solution, for a given control $f$, at time $T$ and $x = 0$, it is shown that the `connection’ operator $$C^T=(U^T)^*U^T,$$ where $U^Tf = u^f(\cdot ,T)$, satisfies $C^T=i[R^T-(R^T)^*].$ The spectral data of the operator $\cal L$ given by $$({\cal L}\phi)(x) = -\phi''(x) + q(x)\phi(x)$$ can be found from $C^T$ (and hence from $R^T$) by the usual variational principle. Finally, the unknown potential $q(x)$ is determined by using the boundary controllability of the associated wave equation. This is based on the fact that the above spectral data completely determine the connecting operator for the associated wave equation.

MSC:
 35R30 Inverse problems for PDE 35Q40 PDEs in connection with quantum mechanics 35J10 Schrödinger operator
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