## Abstract

In this chapter, we consider the inverse problem for multi-dimensional Schrödinger operators appearing in quantum scattering theory. After a brief explanation of the scattering phenomena and related spectral properties, we define wave operators and the S-matrix, and then we review the achievements obtained in the study of multi-dimensional Gel’fand–Levitan theory.

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## Notes

- 1.
However, in general, the differential cross section does not determine the scattering amplitude uniquely. See [153].

- 2.
A necessary and sufficient condition guaranteeing for a given function to be the scattering matrix of some potential

- 3.
Since \(f \in \mathcal H_{ac}(H)\), (

*e*^{−itH}*f*,*g*) → 0 as*t*→±*∞*for any \(g \in \mathcal H\). - 4.
In disordered systems the singular continuous spectrum appears.

- 5.
Given two Hilbert spaces \(\mathcal H_i\) and closed subspaces \(S_i \subset \mathcal H_i\),

*i*= 1, 2, a linear operator \(A : \mathcal H_1 \to \mathcal H_2\) is said to be a partial isometry with initial set*S*_{1}and final set*S*_{2}if*Ax*= 0 for \(x \in S_1^{\perp }\) and ∥*Ax*∥ = ∥*x*∥ for*x*∈*S*_{1}, moreover*AS*_{1}=*S*_{2}. - 6.
This proof is formal in the sense that precise conditions on the kernel

*K*(*x*,*y*) are lacking. We do not pursue them, however, since we are now interested in the algebraic aspect of Gel’fand–Levitan theory. - 7.
For the sake of simplicity, we consider here only a perturbation by bounded operators. The extension to relatively bounded perturbations is not difficult.

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Isozaki, H. (2020). Multi-Dimensional Gel’fand–Levitan Theory. In: Inverse Spectral and Scattering Theory. SpringerBriefs in Mathematical Physics, vol 38. Springer, Singapore. https://doi.org/10.1007/978-981-15-8199-1_3

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