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Communications in Mathematical Physics

, Volume 64, Issue 1, pp 1–34 | Cite as

Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems IV

  • P. Deift
  • W. Hunziker
  • B. Simon
  • E. Vock
Article

Abstract

We describe several new techniques for obtaining detailed information on the exponential falloff of discrete eigenfunctions ofN-body Schrödinger operators. An example of a new result is the bound (conjectured by Morgan)\(\left| {\psi (x_1 \ldots x_N )} \right| \leqq C\exp ( - \sum\limits_1^N {\alpha _n r_n )}\) for an eigenfunction ω of
$$H_N = - \sum\limits_{i = 1}^N {(\Delta _i - } \left. {\frac{Z}{{\left| {x_i } \right|}}} \right) + \sum\limits_{i< j} {\left| {x_i - x_j } \right|^{ - 1} }$$
with energyE N . In this boundr1r2...r N are the radii |x i | in increasing order and the α's are restricted by α n <(En−1E n )1/2, whereE n , forn=0, 1,...,N−1, is the lowest energy of the system described byH n . Our methods include subharmonic comparison theorems and “geometric spectral analysis”.

Keywords

Neural Network Statistical Physic Lower Energy Complex System Spectral Analysis 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Aguilar, J., Combes, J.M.: A class of analytic perturbations for one-body Schrödinger Hamiltonians. Commun. math. Phys.22, 269–279 (1971)Google Scholar
  2. 2.
    Alrichs, R.: Asymptotic behavior of atomic bound state wave functions. J. Math. Phys.14, 1860–1863 (1973)Google Scholar
  3. 3.
    Alrichs, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: Bounds for the long range behaviour of electronic wave functions. J. Chem. Phys.68, 1402–1410 (1978)Google Scholar
  4. 4.
    Alrichs, R., Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: “Schrödinger inequalities” and asymptotic behaviour of many electron densities. Phys. Rev. (to appear)Google Scholar
  5. 5.
    Balslev, E., Combes, J.M.: Spectral properties of many-body Schrödinger operators with dilation analytic interactions. Commun. math. Phys.22, 280–294 (1971)Google Scholar
  6. 6.
    Berthier, A., Gaveau, B.: Critère de convergence des fonctionelles de Kac et application en mécanique quantique et en géometrie. J. Func. Anal. (to appear)Google Scholar
  7. 7.
    Brown, A., Pearcy, C.: Spectra of tensor products of operators. Proc. Am. Math. Soc.17, 162–166 (1966)Google Scholar
  8. 8.
    Carmona, A.: Marseille-Luminy preprints (in preparation)Google Scholar
  9. 9.
    Combes, J.M., Thomas, L.: Asymptotic behavior of eigenfunctions for multiparticle Schrödinger operators. Commun. math. Phys.34, 251–270 (1973)Google Scholar
  10. 10.
    Davies, E.B.: Properties of Green's functions of some Schrödinger operators. J. Lond. Math. Soc.7, 473–491 (1973)Google Scholar
  11. 11.
    Deift, P., Simon, B.: A time dependent approach to the completeness of multiparticle quantum systems. Comm. Pure Appl. Math.30, 573–583 (1977)Google Scholar
  12. 12.
    Enss, V.: A note on Hunzikers theorem. Commun. math. Phys.52, 233–238 (1977)Google Scholar
  13. 13.
    Guerra, F., Rosen, L., Simon, B.: The vacuum energy forP(φ)2. Infinite volume limit and coupling constant dependence. Commun. math. Phys.29, 233–247 (1973)Google Scholar
  14. 14.
    Herbst, I., Sloan, A.: Perturbation of translation invariant positivity preserving semigroups onL 2(R N). Trans. Am. Math. Soc. (to appear)Google Scholar
  15. 15.
    Hoffmann-Ostenhof, M., Hoffmann-Ostenhof, T.: “Schrödinger inequalities” and asymptotic behavior of the electron density of atoms and molecules. Phys. Rev.16A, 1782–1785 (1977)Google Scholar
  16. 16.
    Hunziker, W.: Space-time behavior of Schrödinger wave functions. J. Math. Phys.7, 300–304 (1966)Google Scholar
  17. 17.
    Jörgens, K., Weidmann, J.: Spectral properties of Hamiltonian operators. Lecture notes in mathematics, Vol. 319. Berlin, Heidelberg, New York: Springer 1973Google Scholar
  18. 18.
    Kato, T.: Perturbation theory for linear operators. Berlin, Heidelberg, New York: Springer 1966Google Scholar
  19. 19.
    Kato, T.: Schrödinger operators with singular potentials. Is. J. Math.13, 135–148 (1973)Google Scholar
  20. 20.
    Lavine, R.: Private communicationGoogle Scholar
  21. 21.
    Mercuriev, S.P.: On the asymptotic form of three-particle wave functions of the discrete spectrum. Soviet J. Nucl. Phys.19, 222–229 (1974)Google Scholar
  22. 22.
    Morgan III, J.: The exponential decay of sub-continuum wave functions of two-electron atoms. J. Phys. A10, L91-L93 (1977)Google Scholar
  23. 23.
    O'Connor, T.: Exponential decay of bound state wave functions. Commun. math. Phys.32, 319–340 (1973)Google Scholar
  24. 24.
    Reed, M., Simon, B.: Methods of modern mathematical physics. II. Fourier analysis, self-adjointness. New York: Academic Press 1975Google Scholar
  25. 25.
    Reed, M., Simon, B.: Methods of modern mathematical physics. IV. Analysis of operators. New York: Academic Press 1978Google Scholar
  26. 26.
    Simon, B.: Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems. I. Proc. Am. Math. Soc.42, 395–401 (1974)Google Scholar
  27. 27.
    Simon, B.: Pointwise bounds on eigenfunctions and wave packets inN-body quantum systems. III. Trans. Am. Math. Soc.208, 317–329 (1975)Google Scholar
  28. 28.
    Simon, B.: Geometric methods in multiparticle quantum systems. Commun. math. Phys.55, 259–274 (1977)Google Scholar
  29. 29.
    Simon, B.: Functional integration and quantum physics. New York: Academic Press (expected 1979)Google Scholar
  30. 30.
    Slaggie, E.L., Wichmann, E.H.: Asymptotic properties of the wave function for a bound nonrelativistic system. J. Math. Phys.3, 946–968 (1962)Google Scholar
  31. 31.
    Vock, E.: ETH thesis (in preparation)Google Scholar
  32. 32.
    Zhislin, G.: Discussion of the spectrum of the Schrödinger operator for systems of many particles. Tr. Mosk. Mat. Obs.9, 81–128 (1960)Google Scholar

Copyright information

© Springer-Verlag 1978

Authors and Affiliations

  • P. Deift
    • 1
  • W. Hunziker
    • 2
  • B. Simon
    • 3
  • E. Vock
    • 2
  1. 1.Courant InstituteNYUNew YorkUSA
  2. 2.Institut für Theoretische PhysikETH ZïchZürichSwitzerland
  3. 3.Departments of Mathematics and PhysicsPrinceton UniversityPrincetonUSA

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