Abstract
The purpose of this paper is to order and briefly describe, via two diagrams (Diagram 1 and Diagram 5 below), various spectra and subspaces that are closely related to scattering spectra and scattering subspaces. This will be done from the operator-theoretic viewpoint and the orderings are obtained primarily via Fredholm theory and Fourier theory, respectively. Both self-adjoint and non-self-adjoint operators will be considered, inasmuch as non-self-adjoint operators and methods are now being used in scattering theory.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Weyl, Uber beschränkte quadratische Formen, deren Differenz vollstetig ist, Rend. Circ. Mat. Palermo 27 (1909), 373–392.
F.E. Browder, On the spectral theory of elliptic differential operators I, Math. Ann. 142 (1961), 22–130.
M. Schechter, On the essential spectrum of an arbitrary operator I, J. Math. Anal. Applic. 13 (1966), 205–215.
F. Wolf, On the essential spectrum of partial differential boundary problems, Comm. Pure Appl. Math. 12 (1959), 211–228.
K. Gustafson and J. Weidmann, On the essential spectrum, J. Math. Anal. Applic. 25 (1969), 121–127.
J.T. Schwartz, Some results on the spectra and spectral resolutions of a class of singular integral operators, Comm. Pure Appl. Math. 15 (1962), 75–90.
T. Kato, Perturbation theory for linear operators, Springer, Berlin, (1966).
S. Goldberg, Unbounded linear operators. McGraw-Hill, New York (1966).
N. Akhiezer and I.M. Glazman, Theory of linear operators in Hilbert spacer, Ungar, New York, (1961).
M.H. Stone, Linear transformations in Hilbert space and their applications to analysis, Amer. Math. Soc. Colloq. Publ. 15, Providence, (1932).
N. Dunford and J.T. Schwartz, Linear operators, Interscience, New York, (1958).
B. Sz. Nagy, Spectraldarstellung linear transformationen des Hilbertschen Raumes, Springer, Berlin, (1942).
D. Hilbert, Grundzüge einer allgemeinen Theorie der linear Integralgleichungen, Leipzig, (1912).
F. Riesz and B. Sz. Nagy, Functional analysis, Ungar, New York, (1955).
P. Halmos, Introduction to Hilbert space and the theory of spectral multiplicity, Chelsea, New York, (1951).
S.K. Berberian, The Weyl spectrum of an operator, Indiana Univ. Math J. 20 (1970), 529–544.
K. Gustafson, Weyl’s Theorems, Proc. Oberwolfach Conf. on Linear Operators and Approximation, Int. Series. Num. Math. 20, Birkäuser, Basel, (1972), 80–93.
G. Huige, The spectral theory of some non-self-adjoint differential operators, Comm. Pure Appl. Math. 21 (1968), 25–50.
J.C. Combes, N-body systems, Proc. Denver Scattering Conf. (1973).
I.C. Gohberg and M.G. Krein, The basic propositions in defect numbers, root numbers, and indices of linear operators, Amer. Math. Soc. Transl. (2), 13 (1960), 185–264.
M.A. Kaashoek and D.C. Lay, Ascent, descent, and commuting perturbations, Trans. Amer. Math. Soc. 169 (1972), 35–47.
K. Gustafson, Some examples for the ascent-descent spectral theory (to appear).
N. Salinas, A characterization of the Browder spectrum, Proc. Amer. Math. Soc. 38 (1973), 369–373.
M. Schechter, Principles of functional analysis, Academic Press, New York, (1971).
A. Weinstein and W. Stenger, Methods of intermediate problems for eigenvalues, Academic Press, New York, (1972).
W. Amrein and V. Georgescu, On the characterization of bound states and scattering states in quantum mechanics, (to appear).
W. Amrein, Some fundamental questions in the non-relativistic quantum theory of scattering, Proc. Denver Scattering Conf. (1973).
J.M. Jauch, Theory of the scattering operator, Helv. Phys. Acta 31 (1958), 127–158.
T. Kato, Wave operators and similarity for non-self-adjoint operators, Math. Ann. 162 (1966), 258–279.
A. Wintner, Asymptotic distributions and infinite convolutions, Edwards Bros. Inc., Ann Arbor, (1938).
Y. Katznelson, An introduction to harmonic analysis, Wiley, New York, (1968).
P. Lax and R.S. Phillips, Scattering theory, Academic Press, New York, (1967).
L. Horwitz, J. LaVita, J.P. Marchand, The inverse decay problem, J. Math. Phys. 12 (1971), 2537–2543.
K. Sinha, On the decay of an unstable particle, Helv. Phys. Acta. 45 (1972), 619–628.
C. Wilcox, Scattering states and wave operators in the abstract theory of scattering, J. Funct. Anal. 12 (1973), 257–274.
D. Reulle, A remark on bound states in potential scattering theory, Nuovo Cim. 61A (1969), 655–662.
R. Haag and J. Swieca, When does a quantum field theory describe particles, Comm. Math. Phys. 1 (1965). 308–320.
I. Schoenberg, Über total monotone Folgen mit stetiger Belegungsfunktion, Math. Zeit. 30 (1929), 761–767.
K. Gustafson and G. Johnson, On the absolutely continuous subspace of a self-adjoint operator, Helvetica Physica Acta (to appear)
P. Martin and B. Misra, Kato-Rosenbloom Lemma, Proc. Denver Scattering Conf. (1973).
R. Lavine, Commutators and scattering theory II. A class of one body problems, Indiana Univ. Math. J. 21 (1972), 643–656.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1974 D. Reidel Publishing Company, Dordrecht-Holland
About this paper
Cite this paper
Gustafson, K. (1974). Candidates for óac and Hac . In: Lavita, J.A., Marchand, JP. (eds) Scattering Theory in Mathematical Physics. NATO Advanced Study Institutes Series, vol 9. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-2147-0_7
Download citation
DOI: https://doi.org/10.1007/978-94-010-2147-0_7
Publisher Name: Springer, Dordrecht
Print ISBN: 978-94-010-2149-4
Online ISBN: 978-94-010-2147-0
eBook Packages: Springer Book Archive