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Multi-Dimensional Gel’fand–Levitan Theory

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Inverse Spectral and Scattering Theory

Part of the book series: SpringerBriefs in Mathematical Physics ((BRIEFSMAPHY,volume 38))

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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. 1.

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

  2. 2.

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

  3. 3.

    Since \(f \in \mathcal H_{ac}(H)\), (e itHf, g) → 0 as t →± for any \(g \in \mathcal H\).

  4. 4.

    In disordered systems the singular continuous spectrum appears.

  5. 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. 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. 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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