No data available.
Please log in to see this content.
You have no subscription access to this content.
No metrics data to plot.
The attempt to load metrics for this article has failed.
The attempt to plot a graph for these metrics has failed.
Proof that the H− ion has only one bound state. Details and extension to finite nuclear mass
1.Variational (Rayleigh—Ritz) upper bounds of sufficient accuracy to demonstrate that has at least one bound state were obtained only a few years after the discovery of the Schrödinger equation. H. Bethe, Z. Phys. 57, 815–21 (1929) obtained an upper bound of
1.and E. A. Hylleraas, Z. Phys. 63, 291–2 (1930) obtained an upper bound of Both bounds lie below the hydrogen ground state energy of
1.A large scale variational calculation by C. L. Pekeris, Phys. Rev. 126, 1470–6 (1962) used a 444 term trial function to obtain an upper bound of to the nonrelativistic energy.
1.The ground state has been discussed recently by C. D. Lin, Phys. Rev. A 12, 493–7 (1975).
2.R. N. Hill, Phys. Rev. Lett. 38, 643–6 (1977).
3.Tosio Kato, Trans. Am. Math. Soc. 70, 195–211 (1951).
4.The first results of this kind were obtained by G. M. Zhislin, Dokl. Akad. Nauk SSSR 128, 231–4 (1959),
4.G. M. Zhislin, Trudy Moskov. Mat. Obsc. 9, 81–120 (1960),
4.G. M. Zhislin, Dokl. Akad. Nauk SSSR 175, 521–524 (1967)
4.[G. M. Zhislin, Soviet Math. Dokl. 8, 878–82 (1967)],
4.G. M. Zhislin, Izv. Akad. Nauk SSSR, Ser. Mat. 33, 590–649 (1969)
4.[G. M. Zhislin, Math. USSR, Izvestija 3, 559–616 (1969/70)],
4.and by G. M. Zhislin and A. G. Sigalov, Izv. Akad. Nauk SSSR, Ser. Mat. 29, 835–60
4.[G. M. Zhislin and A. G. Sigalov, AMS Transl. Ser. II, 91, 263–95 (1970)],
4.G. M. Zhislin and A. G. Sigalov, Izv. Akad. Nauk SSSR, Ser. Mat. 29, 1261–72 (1965)
4.[G. M. Zhislin and A. G. Sigalov, AMS Transl. Ser. II, 91, 297–310 (1970)].
4.Extensions of the results and alternate methods of proof may be found in W. Hunziker, Helv. Phys. Acta 39, 451–62 (1966),
4.B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton U.P., Princeton, New Jersey, 1971), Chap. VII;
4.K. Jörgens and J. Weidmann, Spectral Properties of Hamiltonian Operators (Lecture Notes in Mathematics, Vol. 313, Springer‐Verlag, 1973),
4.J. Uchiyama, Publ. Res. Inst. Math. Sci. (Kyoto) A 2, 117–32 (1966);
4.V. Enss, Commun. Math. Phys. 52, 233–8 (1977).
5.Very general results of this kind have been obtained by A. O’Connor, Commun. Math. Phys. 32, 319–40 (1973).
5.See also B. Simon, Proc. Am. Math. Soc. 42, 395–401 (1974).
6.The earliest result of this kind for multiparticle atomic systems is due to J. Weidmann, Commun. Pure Appl. Math. 19, 107–10 (1966).
6.See also J. Weidmann, Bull. Am. Math. Soc. 73, 452–6 (1967);
6.S. Albeverio, Ann. Phys. (N.Y.) 71, 167–276 (1972);
6.B. Simon, Math. Ann. 207, 133–8 (1974).
7.A. G. Sigalov, Uspekhi Mat. Nauk 22, No. 2 (134), 3–20 (1967)
7.[A. G. Sigalov, Russ. Math. Surveys 22, No. 2, 1–18 (1967)].
8.T. Kato, Suppl. Prog. Th. Phys. 40, 3–19 (1967).
9.B. Simon, Quantum Mechanics for Hamiltonians Defined as Quadratic Forms (Princeton U.P., Princeton, New Jersey, 1971);
9.B. Simon, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann (Princeton U.P., Princeton, New Jersey, 1976), pp. 305–26.
10.K. Jörgens and J. Weidmann, Spectral Properties of Hamiltonian Operators (Lecture Notes in Mathematics, Vol. 313. Springer Verlag, 1973).
11.W. Hunziker, Acta Phys. Austr., to appear.
12.T. Kato, Trans. Am. Math. Soc. 70, 212–18 (1951), first showed that helium in the infinite nuclear mass approximation has an infinity of bound states.
12.For general atoms, it is a result of G. M. Zhislin, Trudy Moskov. Mat. Obsc. 9, 81–128 (1960).
12.See also E. Balslev, Ann. Phys. (N.Y.) 73, 49–107 (1972);
12.B. Simon, Helv. Phys. Acta 43, 607–30 (1970);
12.J. Uchiyama, Publ. Res. Inst. Math. Sci. (Kyoto) A 2, 117–132 (1966/67).
13.The finiteness of the number of bound states for negative ions has been discussed in a rigorous way by J. Uchiyama, Publ. Res. Inst. Math. Sci. (Kyoto) A 5, 51–63 (1969);
13.G. M. Zhislin, Teor. Mat. Fiz. 7, 332–341 (1971)
13.[G. M. Zhislin, Theor. Math. Phys. 7, 571–578 (1971)];
13.M. A. Antonets, G. M. Zhislin, and I. A. Shereshevskii, Teor. Mat. Fiz. 16, 235–46 (1973)
13.[M. A. Antonets, G. M. Zhislin, and I. A. Shereshevskii, Theor. Math. Phys. 16, 800–9 (1974)];
13.D. R. Yafeev, Funkcional. Anal. Prilozen. 6, 103–4 (1973)
13.[D. R. Yafeev, Funct. Anal. Appl. 6, 349–50 (1972)]. In particular, Yafeev has proven that has only finitely many bound states even when the nuclear mass is finite.
14.See H. S. W. Massey, Negative Ions (Cambridge U.P., Cambridge, 1976), 3rd ed., pp. 9–11 and pp. 25–30.
15.This belief is based on computer solutions to the Hartree‐Fock equations. To the best of the author’s knowledge, a rigorous proof has not been given.
16.This case has been reviewed by B. Simon, in Studies in Mathematical Physics, Essays in Honor of Valentine Bargmann (Princeton U.P., Princeton, New Jersey, 1976), pp. 305–26
16.and by A. Martin (CERN preprint, Ref. TH. 2294‐CERN 10 March 1977).
17.F. H. Gertler, H. B. Snodgrass, and L. Spruch, Phys. Rev. 172, 110–8 (1968);
17.I. Aronson, C. J. Kleinman, and L. Spruch, Phys. Rev. A 4, 841–6 (1971). Some of their results are not rigorous because they are based on approximate adiabatic potentials for which rigorous error bounds are not supplied.
18.I. Aronson, C. J. Kleinman, and L. Spruch, Phys. Rev. A 12, 349–52 (1975).
19.N. W. Bazley, Proc. Natl. Acad. Sci. U.S.A. 45, 850–3 (1959);
19.N. W. Bazley, Phys. Rev. 120, 144–9 (1960).
20.N. W. Bazley and D. W. Fox, Phys. Rev. 124, 483–92 (1961);
20.N. W. Bazley and D. W. Fox, J. Res. Nat. Bur. Standards Sect. B 65, 105–11 (1961).
20.Certain difficulties of principle in extending the Bazley—Fox method to lithium are resolved in D. W. Fox, SIAM J. Math. Anal. 3, 617–24 (1972);
20.D. W. Fox and V. G. Sigillito, Chem. Phys. Lett. 13, 85–7 (1972);
20.D. W. Fox and V. G. Sigillito, 14, 583–5 (1972); , Chem. Phys. Lett.
20.D. W. Fox and V. G. Sigillito, J. Appl. Math. Phys. 23, 392–411 (1972).
20.See also C. E. Reid, Int. J. Quantum Chem. (U.S.A.) 6, 793–5 (1972).
21.S. H. Gould, Variational Methods for Eigenvalue Problems: An Introduction to the Weinstein Method of Intermediate Problems (University of Toronto Press, Toronto, 1966), 2nd ed.
22.A. Weinstein and W. Stenger, Methods of Intermediate Problems for Eigenvalues (Academic, New York, 1972).
23.H. F. Weinberger, Variational Methods for Eigenvalue Approximation (SIAM Regional Conference Series in Applied Mathematics, 1974).
24.D. S. Hughes and Carl Eckart, Phys. Rev. 36, 694–8 (1930).
25.Essentially this theorem appears as Theorem 1 on p. 21 of Weinstein and Stenger, Ref. 22, as Theorem 9 on p. 62 of Weinberger, Ref. 23, and is implicit in the discussion of intermediate problems of the second type in Gould, Ref. 21.
26.There are actually two distinct minimax principles, one going back to H. Weyl and R. Courant, the other due to Poincaré. See Weinstein and Stenger, Ref. 22, Chap. 3, for a careful discussion of the distinction and of the relation between the two principles. See also Weinberger, Ref. 23, pp. 55–57.
27.A formula such as (3.12) for the projection onto the span of the ranges of two projection operators is apparently not in the literature. A formula equivalent to (3.12) has been obtained independently by Percy Deift (private communication).
28.The decomposition can be verified by carrying out the integration indicated in (3.21). The existence of such a decomposition is suggested by the fact that is an inverse of while is an inverse of The decomposition is essentially the formal relation corrected by the subtractions needed to obtain a convergent integral in (3.21).
29.This method of counting bound states is originally due independently to J. Schwinger, Proc. Natl. Acad. Sci. U.S.A. 47, 122–9 (1961)
29.and M. S. Birman, Mat. Sb. 55, 124–74 (1961)
29.[M. S. Birman, AMS Transl. Ser. II 53, 23–80 (1966)]. See B. Simon, Ref. 16, for a review.
30.The notation for the modified Bessel functions is that used in W. Magnus, F. Oberhettinger, and R. P. Soni, Formulas and Theorems for the Special Functions of Mathematical Physics (Springer, New York, 1966), 3rd ed., p. 66.
31.Ref. 30, p. 37.
32.Ref. 30, p. 49.
33.See, for example, S. G. Mikhilin, Integral Equations and Their Applications to Certain Problems in Mechanics, Mathematical Physics, and Technology (Pergamon, New York, 1957), 2nd revised ed., Chap. II
33.or F. G. Tricomi, Integral Equations (Interscience, New York, 1957), Chap. III.
34.A useful general reference is L. M. Delves and J. Walsh, Numerical Solution of Integral Equations (Oxford U.P., New York, 1974).
34.Rigorous error bounds to the eigenvalues of integral equations which are solved by numerical methods are discussed by I. P. Mysovskih, Mat. Sb. (N.S.) 48 (90), 137–148 (1959)
34.[I. P. Mysovskih, AMS Transl. Ser. II, 35, 237–50 (1964)].
35.This trick, which is outlined below, is discussed in S. G. Mikhilin, Ref. 33, pp. 88–94
35.and in F. G. Tricomi, Ref. 33, pp. 119–24.
36.The method has some features in common with the partitioning technique of P. O. Lowdin, J. Math. Phys. 3, 969 (1962);
36.P. O. Lowdin, Phys. Rev. 139, A357 (1965);
36.P. O. Lowdin, J. Chem. Phys. 43, 5175 (1965).
36.See also T. M. Wilson, J. Chem. Phys. 47, 3912, 4706 (1967).
37.This trick for constructing lower bounds is known as truncation. It was introduced in H. F. Weinberger, “A Theory of Lower Bounds for Eigenvalues,” Tech. Note BN‐183, IFDAM, Univ. of Maryland, College Park, Maryland (1959). It was developed for quantum mechanical problems by N. W. Bazley and D. W. Fox, Ref. 20. See also Refs. 21–23.
38.See S. G. Mikhilin, Ref. 33, pp. 19–22
38.or F. G. Tricomi, Ref. 33, pp. 55–64.
39.See Ref. 30, p. 344, or
39.Milton Abramowitz and Irene A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), p. 230. is tabulated on pp. 238–51 of Abramowitz and Stegun.
40.H. Shull and P.‐O. Löwdin, J. Chem. Phys. 25, 1035–40 (1956).
40.See also H. S. W. Massey, Ref. 14, pp. 9–10.
41.Ref. 30, p. 68.
Article metrics loading...
Full text loading...