The abstract theory of scattering

Kato, Tosio; Kuroda, S. T.

doi:10.1216/rmj-1971-1-1-127

Cited by 121 publications

(73 citation statements)

References 2 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In our previous work (Ikebe [1]) we have obtained a spectral representation for the Schrodinger operator -A + V(x) with a purely long-range, real-valued potential F(x) acting in the Euclidean three-space U 3 .…”

mentioning

confidence: 99%

Spectral Representation for Schrödinger Operators with Long-Range Potentials, II — Perturbation by Short-Range Potentials

Ikebe

1975

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

mentioning

confidence: 99%

Spectral Representation for Schrödinger Operators with Long-Range Potentials, II — Perturbation by Short-Range Potentials

Ikebe

1975

Publ. Res. Inst. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…In two papers [362,363], Kato and Kuroda developed what they called an abstract theory of scattering. As Kuroda told me "it was too abstract to become popular" (blaming himself for this).…”

Section: Sgn(v (Y)) and ξ(X) = |V (X)| 1/2 ϕ(X)mentioning

confidence: 99%

“…KatoKuroda [362,363] have arguments to get this. In his original announcement, Agmon [3] didn't mention scattering.…”

Section: Sgn(v (Y)) and ξ(X) = |V (X)| 1/2 ϕ(X)mentioning

confidence: 99%

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

Simon¹

2018

Bull. Math. Sci.

View full text Add to dashboard Cite

show abstract

“…The (stationary) wave operators and scattering matrix can be expressed in terms of the resolvents R(z) = (77 -z)~x and R0(z) = (T~z)-x (see [14], [19]), e.g., (111.2) Z + = s-lim Z(e), Z<e) = -f R0(A -ie)R(X + ie) dX.…”

Section: I->±00mentioning

confidence: 99%

Mathematical theory of single channel systems. Analyticity of scattering matrix

Sigal¹

1982

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

Abstract. We show that the S-matrix of a quantum many-body, short-range, single-channel system has a meromorphic continuation whose poles occur at most at the dilation-analytic resonances [28], [24] and at the eigenvalues of the Hamiltonian. In passing, we prove the main spectral theorem (on location of the essential spectrum) and asymptotic completeness for the mentioned class of systems.Introduction. In this paper we study the analytic properties of the scattering matrix for many-body, short-range, single-channel systems. Our main theorem asserts that the scattering matrix has a meromorphic continuation in the energy parameter into a certain sector of the complex plane. The poles of this continuation occur only at eigenvalues of the dilation analytic family //(f) associated with the Hamiltonian H. By Balslev's and Combes' theorem the latter eigenvalues lead to the poles of the meromorphic continuation of the matrix elements , (7/-z)_1F> on the dilation analytic vectors across the continuous spectrum into the second Riemann sheet. Moreover, if the potentials are nice enough so that the negative axis belongs to the meromorphic domain, then the poles on this axis occur at most at the negative eigenvalues of H.To prepare the ground for the proof of our main theorem we prove most of the theorems of the spectral and scattering theory for single-channel systems. These results are not new, but some of the theorems (e.g. the asymptotic completeness) contain improvements over the previous results.Most of the work is done in an abstract setting and only at the end we check the assumptions of the abstract theorems. This allows us to treat the problems of different character separately. The abstract approach has forced us to introduce the notions of the abstract Schrödinger operator and abstract many-body system. We hope they will be useful in extending our methods to other systems.The paper is self-contained; all necessary definitions are given in the text. § §I-V contain the preparatory abstract results; the main theorem is formulated and proved in §VI, where all relevant definitions and preliminary results for A/-body systems are also given. § §VII and VIII deal with the estimates of certain operators built out of the potentials and the free resolvent. These estimates are needed in the theorem of §VI on the behavior of (77 -z)"1 near the continuous spectrum. In

show abstract

The abstract theory of scattering

Cited by 121 publications

References 2 publications

Spectral Representation for Schrödinger Operators with Long-Range Potentials, II — Perturbation by Short-Range Potentials

Spectral Representation for Schrödinger Operators with Long-Range Potentials, II — Perturbation by Short-Range Potentials

Tosio Kato’s work on non-relativistic quantum mechanics: part 2

Mathematical theory of single channel systems. Analyticity of scattering matrix

Contact Info

Product

Resources

About