Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Schrödinger operators with rapidly oscillating central potentials


Author: Denis A. W. White
Journal: Trans. Amer. Math. Soc. 275 (1983), 641-677
MSC: Primary 35P25; Secondary 34B25, 35P10, 81F05
DOI: https://doi.org/10.1090/S0002-9947-1983-0682723-7
MathSciNet review: 682723
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Spectral and scattering theory is discussed for the Schrödinger operators $ H = - \Delta + V$ and $ {H_0} = - \Delta $ when the potential $ V$ is central and may be rapidly oscillating and unbounded. A spectral representation for $ H$ is obtained along with the spectral properties of $ H$. The existence and completeness of the modified wave operators is also demonstrated. Then a condition on $ V$ is derived which is both necessary and sufficient for the Møller wave operators to exist and be complete. This last result disproves a recent conjecture of Mochizuki and Uchiyama.


References [Enhancements On Off] (What's this?)

  • [1] S. Agmon, Spectral properties of Schrödinger operators and scattering theory, Ann. Scuola Norm. Sup. Pisa 2 (1975), 151-218. MR 0397194 (53:1053)
  • [2] V. De Alfaro and T. Regge, Potential scattering, Wiley, New York, 1965. MR 0191316 (32:8724)
  • [3] W. O. Amrein and D. B. Pearson, The scattering matrix for rapidly oscillating potentials, J. Phys. A 13 (1980), 1259-1264. MR 565769 (83c:81100)
  • [4] M. Arai, Eigenfunction expansions associated with Schrödinger operators with long-range potentials, Publ. Res. Inst. Math. Sci. 16 (1980), 35-39. MR 574029 (81h:47015)
  • [5] F. V. Atkinson, The asymptotic solution of second-order differential equations, Ann. Mat. Pura Appl. 37 (1954), 347-378. MR 0067289 (16:701f)
  • [6] M. L. Baetemann and K. Chadan, Scattering theory with highly singular, oscillating potentials, Ann. Inst. H. Poincaré Sect. A 24 (1976), 1-16. MR 0400975 (53:4805)
  • [7] M. Ben-Artzi, Eigenfunction expansion for a class of differential operators, J. Math. Anal. Appl. 69 (1979), 304-314. MR 538219 (80h:34025)
  • [8] -, On the absolute continuity of Schrödinger operators with spherically symmetric long range potentials. I, J. Differential Equations 38 (1980), 41-50. MR 592867 (82g:35083)
  • [9] -, On the absolute continuity of Schrödinger operators with spherically symmetric long range potentials. II, J. Differential Equations 38 (1980), 51-60.
  • [10] -, Spectral properties of linear ordinary differential operators with slowly decreasing coefficients, J. Math. Anal. Appl. 68 (1979), 68-91. MR 531423 (80i:47066)
  • [11] M. Ben-Artzi and A. Devinatz, Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. 20 (1979), 594-607. MR 529723 (82a:35088a)
  • [12] B. Bourgeois, On scattering theory for oscillatory potentials of slow decay, Ann. Physics 121 (1979), 415-431. MR 548178 (80j:81065)
  • [13] E. A. Coddington and N. Levinson, Theory of ordinary differential equations, McGraw-Hill, New York, 1955, p. 92. MR 0069338 (16:1022b)
  • [14] M. Combescure, Spectral and scattering theory for a class of strongly oscillating potentials, Comm. Math. Phys. 73 (1980), 43-62. MR 573612 (84k:81030)
  • [15] M. Combescure and J. Ginibre, Spectral and scattering theory for the Schrödinger operator with strongly oscillating potentials, Ann. Inst. H. Poincaré Sect. A 24 (1976), 17-29. MR 0400976 (53:4806)
  • [16] A.Devinatz, The existence of wave operators for oscillating potentials, J. Math. Phys. 21 (1980), 2406-2411. MR 585593 (83c:35097)
  • [17] J. D. Dollard, Asymptotic convergence and the Coulomb interactions, J. Math. Phys. 5 (1964), 729-738. MR 0163620 (29:921)
  • [18] -, Quantum-mechanical scattering theory for short-range and Coulomb interactions. Rocky Mountain J. Math. 1 (1971), 5-88. MR 0270673 (42:5561)
  • [19] J. D. Dollard and C. N. Friedman. Existence of Moller wave operators for $ V(r) = \lambda \sin (\mu \,{r^\alpha })/{r^\beta }$, Ann. Physics 111 (1978), 251-266. MR 0489510 (58:8931)
  • [20] N. Dunford and J. T. Schwartz, Linear operators. Part II, Interscience, New York, 1963. MR 0188745 (32:6181)
  • [21] V. Enss, Asymptotic completeness for quantum mechanical potential scattering. I. Short range potentials, Comm. Math. Phys. 61 (1978), 285-291. MR 0523013 (58:25583)
  • [22] -, Asymptotic completeness for quantum mechanical potential scattering. II. Singular and long range potentials, Ann. Physics 119 (1979), 117-132. MR 535624 (80k:81144)
  • [23] J. Ginibre, La méthode 'dépendant du temps' dans le problème de la complétude asymptotique, preprint.
  • [24] T. A. Green and O. E. Lanford, III, Rigourous derivation of the phase shift formula for the Hilbert space scattering operator of a single particle, J. Math. Phys. 1 (1960), 139-148. MR 0128356 (23:B1399)
  • [25] W. A. Harris, Jr. and D. A. Lutz, Asymptotic integration of adiabatic oscillators, J. Math. Anal. Appl. 51 (1975), 76-93. MR 0369840 (51:6069)
  • [26] L. Hörmander, The existence of wave operators in scattering theory, Math. Z. 146 (1976), 69-71. MR 0393884 (52:14691)
  • [27] T. Ikebe and H. Isozaki, Completeness of modified wave operators for long-range potentials, Publ. Res. Inst. Math. Sci. 15 (1979), 679-718. MR 566076 (81i:35131)
  • [28] H. Isozaki, Eikonal equations and spectral representations for long range Schrödinger Hamiltonians, J. Math. Kyoto Univ. 20 (1980), 243-261. MR 582166 (81i:35042)
  • [29] A. R. Its and V. B. Matveev, Co-ordinate asymptotics for the Schrödinger equation with a rapidly oscillating potential, J. Soviet Math. 11 (1979), 442-444.
  • [30] K. Jorgens and J. Weidmann, Spectral properties of Hamiltonian operators, Springer-Verlag, Berlin, 1973. MR 0492941 (58:11990)
  • [31] T. Kato, Perturbation theory for linear operators, 2nd ed., Springer-Verlag, Berlin, 1976. MR 0407617 (53:11389)
  • [32] H. Kitada, Scattering theory for Schrödinger operators with long-range potentials. I. Abstract theory, J. Math. Soc. Japan 29 (1977), 665-691. MR 0634802 (58:30372a)
  • [33] -, Scattering theory for Schrödinger operators with long-range potentials. II. Spectral and scattering theory, J. Math. Soc. Japan 30 (1978), 603-632. MR 0634803 (58:30372b)
  • [34] H. Kitada and K. Yajima, A scattering theory for time-dependent long-range potentials, preprint. MR 659945 (83i:35137)
  • [35] V. B. Matveev, Wave operators and positive eigenvalues for a Schrödinger equation with oscillating potential, Theoret. and Math. Phys. 15 (1973), 574-583.
  • [36] V. B. Matveev and M. M. Skriganov, Wave operators for the Schrödinger equation with rapidly oscillating potential, Dokl. Akad. Nauk SSSR 13 (1972), 185-188. MR 0300135 (45:9183)
  • [37] K. Mochizuki and J. Uchiyama, Radiation conditions and spectral theory for $ 2$-body Schrödinger operators with 'oscillating' long-range potentials. I, J. Math. Kyoto Univ. 18 (1978), 377-408. MR 0492943 (58:11992)
  • [38] -, Radiation conditions and spectral theory for $ 2$-body Schrödingerr operators with 'oscillating' long-range potentials. II, J. Math. Kyoto Univ. 19 (1979), 47-70. MR 527395 (80g:35100)
  • [39] -, Time dependent representations of the stationary wave operators for 'oscillating' long-range potentials, preprint.
  • [40] J. von Neumann and E. Wigner, Über merkwürdige diskrete Eigenwerte, Z. Phys. 30 (1929), 465-467.
  • [41] D. B. Pearson, Scattering theory for a class of oscillating potentials, Helv. Phys. Acta 52 (1979), 541-554. MR 566255 (81f:34030)
  • [42] P. A. Perry, Propagation of states in dilation analytic potentials and asymptotic completeness, Comm. Math. Phys. 81 (1981), 243-259. MR 632760 (84f:81097)
  • [43] M. Reed and B. Simon, Methods of modern mathematical physics, vol. II, Academic Press, New York, 1975. MR 0493420 (58:12429b)
  • [44] -, Methods of modern mathematical physics, vol. III, Academic Press, New York, 1979.
  • [45] Y. Saito, Spectral representations for Schrödinger operators with long range potentials, Springer-Verlag, Berlin, 1979. MR 540891 (81a:35083)
  • [46] M. Schechter, Spectra of partial differential operators, North-Holland, Amsterdam, 1971. MR 869254 (88h:35085)
  • [47] -, Scattering theory for elliptic operators of arbitrary order, Comment. Math. Helv. 49 (1974), 84-113. MR 0367484 (51:3726)
  • [48] B. Simon, Phase space analysis of simple scattering systems: Extensions of some work of Enss, Duke Math. J. 46 (1979), 119-168. MR 523604 (80j:35081)
  • [49] M. M. Skriganov, On the spectrum of the Schrödinger operator with rapidly oscillating potential, Trudy Mat. Inst. Steklov 125 (1973), 177-185. MR 0344705 (49:9444)
  • [50] -, The eigenvalues of the Schrödinger operator situated on the continuous spectrum, J. Soviet Math. 8 (1977), 464-467.
  • [51] H. Tamuro, The principle of limiting absorption for uniformly propagative systems with perturbations of long-range class, Nagoya Math. J. 82 (1981), 141-174. MR 618813 (82h:35084)
  • [52] M. Wolfe, Asymptotic behavior of solutions of the radial Schroedinger equation and applications to long-range potential scattering, Dissertation, University of Texas at Austin, 1978.
  • [53] D. Yafaev, On the proof of Enss of asymptotic completeness in potential scattering, preprint, Steklov Institute, Leningrad. MR 1084881 (92b:35118)
  • [54] B. Bourgeois, Quantum mechanical scattering for potentials of the form $ \sin \,r/{r^\beta },\frac{1} {3} < \beta \leqslant \frac{1} {2}$, J. Math. Phys. 23 (1982), 790-797.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 35P25, 34B25, 35P10, 81F05

Retrieve articles in all journals with MSC: 35P25, 34B25, 35P10, 81F05


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1983-0682723-7
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society