Bulletin of the American Mathematical Society

MathSciNet review: 1567492
Full text of review: PDF   This review is available free of charge.
Book Information:

Authors: M. S. P. Eastham and H. Kalf
Title: Schrödinger-type operators with continuous spectra
Additional book information: Research Notes in Mathematics, Vol. 65, Pitman Advanced Publishing Program, Boston, 1982, 281 pp., $24.95. ISBN 0-2730-8526-3.

Shmuel Agmon, Lower bounds for solutions of Schrödinger equations, J. Analyse Math. 23 (1970), 1–25. MR 276624, DOI 10.1007/BF02795485

N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equations, Amer. J. Math. 79 (1957), 597–610. MR 88623, DOI 10.2307/2372564

J. E. Avron and I. W. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Comm. Math. Phys. 52 (1977), no. 3, 239–254. MR 468862

Joseph Avron and Barry Simon, Almost periodic Schrödinger operators. I. Limit periodic potentials, Comm. Math. Phys. 82 (1981/82), no. 1, 101–120. MR 638515

D. M. Èĭdus, The principle of limiting amplitude, Uspehi Mat. Nauk 24 (1969), no. 3(147), 91–156 (Russian). MR 0601072

Richard Froese and Ira Herbst, Exponential bounds and absence of positive eigenvalues for $N$-body Schrödinger operators, Comm. Math. Phys. 87 (1982/83), no. 3, 429–447. MR 682117

Richard Froese, Ira Herbst, Maria Hoffmann-Ostenhof, and Thomas Hoffmann-Ostenhof, On the absence of positive eigenvalues for one-body Schrödinger operators, J. Analyse Math. 41 (1982), 272–284. MR 687957, DOI 10.1007/BF02803406

8.

I. Ya. Gol'dshtein [sic], S. A. Molchanov and L. A. Pastur, A pure point spectrum of the stochastic one-dimensional Schrödinger operator, Functional Anal. Appl. 11 (1977), 1-8 (Translation of Funktsional. Anal. i Prilozhen. 11 (1977), 1-10.)

Ira W. Herbst, Dilation analyticity in constant electric field. I. The two body problem, Comm. Math. Phys. 64 (1979), no. 3, 279–298. MR 520094

Tosio Kato, Growth properties of solutions of the reduced wave equation with a variable coefficient, Comm. Pure Appl. Math. 12 (1959), 403–425. MR 108633, DOI 10.1002/cpa.3160120302

Jürgen Moser, An example of a Schroedinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv. 56 (1981), no. 2, 198–224. MR 630951, DOI 10.1007/BF02566210

Johann von Neumann, Mathematische Grundlagen der Quantenmechanik, Die Grundlehren der mathematischen Wissenschaften, Band 38, Springer-Verlag, Berlin-New York, 1968 (German). Unveränderter Nachdruck der ersten Auflage von 1932. MR 0223138

13.

J. R. Oppenheimer, Zur Quantentheorie kontinuierlicher Spektren, Z. Phys. 41 (1927), 268-293.

14.

J. R. Oppenheimer, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31 (1928), 66-81. Compare p. 74 with the original in Weyl, op. cit., p. 268.

D. B. Pearson, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), no. 1, 13–36. MR 484145

Michael Reed and Barry Simon, Methods of modern mathematical physics. I, 2nd ed., Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York, 1980. Functional analysis. MR 751959

17.

E. Schrödinger, Quantisierung als Eigenwertproblem. Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien, Ann. Phys. (Leipzig) 80 (1926), 437-490.

Barry Simon, On positive eigenvalues of one-body Schrödinger operators, Comm. Pure Appl. Math. 22 (1969), 531–538. MR 247300, DOI 10.1002/cpa.3160220405

Hermann Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), no. 2, 220–269 (German). MR 1511560, DOI 10.1007/BF01474161

Aurel Wintner, On the location of continuous spectra, Amer. J. Math. 70 (1948), 22–30. MR 23995, DOI 10.2307/2371929

Aurel Wintner, The adiabatic linear oscillator, Amer. J. Math. 68 (1946), 385–397. MR 16811, DOI 10.2307/2371822

1.

S. Agmon, Lower bounds for solutions of Schrödinger's equation, J. Analyse Math. 23 (1970), 1-25. MR 0276624

2.

N. Aronszajn, On a problem of Weyl in the theory of singular Sturm-Liouville equation, Amer. J.Math. 79 (1957), 597-610. MR 88623

3.

J. E. Avron and I. Herbst, Spectral and scattering theory of Schrödinger operators related to the Stark effect, Comm. Math. Phys. 52(1977), 239-254. MR 468862

4.

J. Avron and B. Simon, Almost periodic Schrödinger operators. I: Limit periodic potentials, Comm. Math. Phys. 82 (1981), 101-120. MR 638515

5.

D. M. Eidus, The principle of limit amplitude, Russian Math. Surveys 24 (1969), no. 3, 97-167. (Translation of Uspekhi Mat. Nauk 24 (1969), no. 3, 91-156.) MR 601072

6.

R. Froese and I. Herbst, Exponential bounds and absence of positive eigenvalues for N-body Schrödinger operators, Comm. Math. Phys. 87 (1982), 429-447. MR 682117

7.

R. Froese, I. Herbst, M. Hoffmann-Ostenhof and T. Hoffmann-Ostenhof, On the absence of positive eigenvalues for one-body Schrödinger operators, J. Analyse Math. 41 (1982), 272-284. MR 687957

8.

I. Ya. Gol'dshtein [sic], S. A. Molchanov and L. A. Pastur, A pure point spectrum of the stochastic one-dimensional Schrödinger operator, Functional Anal. Appl. 11 (1977), 1-8 (Translation of Funktsional. Anal. i Prilozhen. 11 (1977), 1-10.)

9.

I. Herbst, Dilation analyticity in constant electric field. I: The two body problem, Comm. Math. Phys. 64 (1979), 279-298. MR 520094

10.

T. Kato, Growth properties of solutions of the reduced wave equation with variable coefficients, Comm. Pure Appl. Math. 12 (1959), 403-425. MR 108633

11.

J. Moser, An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum, Comment. Math. Helv. 56 (1981), 198-224. MR 630951

12.

J. von Neumann, The mathematical foundations of quantum mechanics, Princeton Univ. Press, Princeton, 1955 (Translation of Die mathematische Grundlagen der Quantenmechanik, Springer, Berlin, 1932). MR 223138

13.

J. R. Oppenheimer, Zur Quantentheorie kontinuierlicher Spektren, Z. Phys. 41 (1927), 268-293.

14.

J. R. Oppenheimer, Three notes on the quantum theory of aperiodic effects, Phys. Rev. 31 (1928), 66-81. Compare p. 74 with the original in Weyl, op. cit., p. 268.

15.

D. B. Pearson, Singular continuous measures in scattering theory, Comm. Math. Phys. 60 (1978), 13-36. MR 484145

16.

M. Reed and B. Simon, Methods of modern mathematical physics, Vol. IV: Analysis of operators, Academic Press, New York, 1978. MR 751959

17.

E. Schrödinger, Quantisierung als Eigenwertproblem. Dritte Mitteilung: Störungstheorie, mit Anwendung auf den Starkeffekt der Balmerlinien, Ann. Phys. (Leipzig) 80 (1926), 437-490.

18.

B. Simon, On the positive eigenvalues of one-body Schrödinger operators, Comm. Pure Appl. Math. 22 (1969), 531-538. MR 247300

19.

H. Weyl, Über gewöhnliche Differentialgleichungen mit Singularitäten und die zugehörigen Entwicklungen willkürlicher Funktionen, Math. Ann. 68 (1910), 220-269. MR 1511560

20.

A. Wintner, On the location of continuous spectra, Amer. J. Math. 70 (1948), 22-30. MR 23995

21.

A. Wintner, The adiabatic linear oscillator, Amer. J. Math. 68 (1946), 385-397; Asymptotic integrations of the adiabatic oscillator, Amer. J. Math. 69 (1947), 251-272. MR 16811