Different treatments of the important potential 1/r4 are correlated to understand their interrelations and to clear up the connection between the eigenvalues of Mathieu’s equation and the poles of the S matrix. We also derive a new solution of the modified Mathieu equation. Mathematical and physical implications are also discussed.

1.
For a review see the article by H. H. Aly, W. Güttinger, and H. J. W. Müller in Lectures on Particle and Field, edited by H. H. Aly (Gordon and Breach, New York, 1969).
For a list of references to earlier literature see also
F.
Calogero
,
Phys. Rev.
139
,
B602
(
1965
).
2.
W.
Güttinger
,
R.
Penzl
, and
E.
Pfaffelhuber
,
Ann. Phys. (N.Y.)
33
,
246
(
1965
);
R.
Rajaraman
,
Phys. Rev.
178
,
2211
,
2221
(
1969
).
3.
R. M.
Spector
,
J. Math. Phys.
5
,
1185
(
1964
);
H. H.
Aly
and
H. J. W.
Müller
,
J. Math. Phys.
7
,
1
(
1966
).
4.
J.
Challifour
and
R. J.
Eden
,
J. Math. Phys.
4
,
359
(
1963
);
N.
Dombey
and
R. H.
Jones
,
J. Math. Phys.
9
,
986
(
1968
);
D.
Yuan‐Ben
,
Sci. Sinica (Peking)
13
,
1319
(
1964
).
5.
L.
Bertocchi
,
S.
Fubini
, and
G.
Furlan
,
Nuovo Cimento
35
,
633
(
1965
);
H. H.
Aly
and
H. J. W.
Müller
,
J. Math. Phys.
8
,
367
(
1967
).
6.
H. H.
Aly
,
H. J. W.
Müller
and
N.
Vahedi‐Faridi
,
Lett. Nuovo Cimento
2
,
485
(
1969
).
7.
D.
Masson
,
Nuovo Cimento
35
,
125
(
1965
), Appendix.
8.
H. H.
Aly
and
P.
Narayanaswamy
,
Phys. Lett.
28B
,
603
(
1969
).
9.
J. Meixner and F. W. Schäfke, Mathieusche Funktionen und Sphäroidfunktionen (Springer, Berlin, 1953, 1954, 1956).
10.
The disadvantage of such trajectories for singular potentials is, of course, that they do not lead to narrow width Regge recurrences. From the point of view of application energy‐dependent Yukawa‐like potentials as discussed by (a)
G.
Tiktopoulos
,
Phys. Lett.
28B
,
185
(
1969
),
(b) U. Trivedi, Cal. I. Tech. preprint 68–199, 1969,
(c) J. W. Johnson, Case Western Reserve University preprint, 1969 are therefore more realistic.
11.
For a review see H. J. W. Müller, “Perturbation Approach for Regular Interactions,” in Lectures in High Energy Theoretical Physics, edited by H. H. Aly (Wiley‐Interscience, New York, 1968).
12.
R. B.
Dingle
and
H. J. W.
Müller
,
J. Reine Angew. Math.
211
,
11
(
1962
).
13.
H. J. W.
Müller
and
K.
Schilcher
,
J. Math. Phys.
9
,
255
(
1968
).
14.
H. J. W.
Müller
,
J. Math. Phys.
11
,
355
(
1969
).
15.
J. B.
Keller
,
S. I.
Rubinow
, and
M.
Goldstein
,
J. Math. Phys.
4
,
829
(
1963
).
16.
W.
Magnus
and
L.
Kotin
,
Numerische Math.
2
,
228
(
1960
).
17.
This rise seems consistent with the general behaviour of the trajectories for the repulsive potential i/r4 as discussed by
N.
Limic
,
Nuovo Cimento
26
,
581
(
1962
), who has shown that here all Regge poles (for physical k) are located in the sectors which, incidentally, also implies that there can be no narrow‐width Regge recurrences.
18.
T. F.
O’Malley
,
L.
Spruch
, and
L.
Rosenberg
,
J. Math. Phys.
2
,
491
(
1961
).
19.
F.
Calogero
,
Nuovo Cimento
37
,
756
(
1965
);
F.
Calogero
,
Phys. Rev.
139
,
B
602
(
1965
).
20.
F.
Calogero
,
Phys. Rev.
135
,
B693
(
1964
).
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