Ames W.F., Harrel E.M., Herod J.V. (editors). Differential Equations with Applications to Mathematical Physics
Подождите немного. Документ загружается.
130
F.
Gesztesy
&
R.
Weikard
It should perhaps be pointed out that with the exceptions
of
[4]-
[6],
[31], [32],
[GO],
the corresponding complex-valued analog received
much less attention in the literature. In particular, the Jacobi inver-
sion problem on the noncompact Riemann surface
Km
associated
with
Vo
in the complex-valued periodic
or
almost-periodic infinite-
gap case (a crucial step
in
the corresponding generalization
of
the
Its-Matveev formula) appears to be open.
b)
Harmonic Oscillators etc:
The double commutation approa.ch in connection with (4.55) and
(4.56) can be used to produce families
of
isospectral unbounded po-
tentials with purely discrete spectra.
In
order
to
see the connection
with spectral deforma.tions in Section 4 consider the harmonic oscil-
lator example
l’&)
=
x2
-
1
(6.2)
and the (suitably
scaled)
Mathieu potential
I<(.)
=
2c2[1
-
COS(€X)]
-
1,
€
>
0.
(6.3)
As
is well known [57], all periodic and anti-periodic eigenvalues
of
i-
V,
restricted
to
[xo, xo
+
(27r/~)],
E
>
0
are simple and hence
d2
dx2
--
d2
(1x2
11,
=
--
i-
V,
on
H2(IR),
6
>
0
has infinitely ma.ny spectral
g
a(K)
=
u
jEN
As
€
1
0,
and, since
,ps
for all
E
>
0
K(X)
d
vqx)
=
22-
1
10
Spectral Deformations and Soliton Equations
131
(see, e.g.,
[33], [63]).
In this scaling limit
6
1
0,
the noncompact
Riemann surface
Kw(c)
associated with
V,,
6
>
0
degenerates into
a
highly singular curve consisting of infinitely many double points
{2(j
-
l)}jEw.
A
careful study of this limit on the level
of
degen-
erating hyperelliptic curves and their &functions, to the best of
our
knowledge, has not been undertaken yet. Isospectral families
of
the
limit potential
Vo(x)
=
x2
-
1
have been constructed in
[45]
and
[56]
but apart from the harmonic oscillator case we are not aware of any
other detailed study of isospectral families for unbounded potentials
with purely discrete spectra.
Finally, we mention another possible generalization in
a
bit more
detail:
c)
N-Soliton Solutioiis
as
N
+
00:
Here we choose
d2
dx2
110
=
--
on
H2(IR),
V0(x)
=
0
and choose double commutation to insert
N
eigenvalues
{Xj
=
-K~}K~,
tcj
>
0,
15
j
5
N,
~j
#
K~I
for
j
#
j'
(6.10)
into the spectral gap
po
=
(-w,O)
of
Ho.
The result is the N-soliton
potential
[22], [38]
d2
dx2
VN(X)
=
-2-
In det[lN
+
CN(Z)],
(6.11)
(6.12)
where
(6.13)
132
F.
Gesztesy
&
R.
Weikard
are (norming) constants (related
to
TI,+
in (4.47) by
cf
=
71,+,
1
5
I
5
N,
i.e.,
VN(X)
=
Vc:
,...,
c~,+
(AI,.
.
.
,AN,
x)).
Introducing
d2
dx2
HN
=
--
+
VN
on
H’(IR),
(6.14)
one verifies that
O(I1,)
=
u
[O,
CQ)
(6.15)
with purely absolutely continuous essential spectrum
of
multiplicity
two
oess
(I-TN)
=
oac
(HN)
=
[O,
001,
(6.16)
up
(fh)
n
[o,
CQ)
=
oSc
(HN)
=
0
(6.17)
and simple discrete eigenvalues
{
-.;}K1.
(Here
oess(.),
oat(.),
osc(
.),
and
up(
.)
denote the essential, absolutely continuous, singularly con-
tinuous, and point spectrum (the set
of
eigenvalues) respectively.)
The unitary scattering ma.trix
SN(~)
ilia2
associated with the pair
(HN,
Ho)
is reflectionless a.nd reads
(6.18)
(A
=
k2
the spectral parameter
of
Ho).
As
briefly mentioned in
Section 4, the singular curve associated with
HN
is
of
the type
which can be obtained from the nonsingular curve
Spectral
Deformations and Soliton Equations
133
by degenerating the compact spectral bands
[Ez(j-1),
E2j-11
into the
eigenvalues
-
63
[~2(j-11,
~2j-11-
-62
3'
1
5
j
5
N.
(6.21)
At this point it seems natural to ask what happens if
N
+
00.
This can be answered
as
follows.
Theorem
6.1
[as],
[29] Assume
{~j
>
O}jE~
E
P(IN),
Kj
#
~jl
for
j
#
j'
and choose {cj
>
O}jEl~
such that {cj/~j}~~~
E
ll(IN). Then
VN
converges pointwise to some
V,
E
C"(IR)
n
LW(IR)
as
N
-+
00
and
(i)
liin
V,(x)
=
0
and
Xd+W
lim
sup
IV$m)(x)
-
Vp)(x)l
=
0,
rn
E
IN
U
(0)
(6.22)
XE
I<
n--.co
for
any compact
IC
c
IR.
(ii) Denoting
(6.23)
we have
Oess(fI~)
=
{-K~}SE~N
u
[o,
m),
(6.24)
aac(KQ)
=
[OF),
(6.25)
(6.26)
{-lC;}j~iN
OP(EIco)
{-Kjz}j~N-
(6.27)
The spectral multiplicity
of
EI,
on
(0,~)
equals two while
ap(H,)
is simple. In addition, if
{Kj}jEN
is
a
discrete subset of
(0,00)
(i.e.,
0
is its only limit point) then
[opW,)
u
o~~(IL,)]
n
(0,
00)
=
0,
%C(H,)
=
0,
(6.23)
o(H~)
n
(-00,o)
=
Q(H,)
=
{-~jz}~~N.
(6.29)
134
F.
Gesztesy
&
R.
Weikard
More generally, if
{~j}'.
Here
A'
denotes the derived set of
A
c
IR
(i.e., the set of accumu-
lation points of
A)
and
ad(.)
denotes the discrete spectrum (cf. also
the paragraph following
(6.17)).
We refer to
[29]
for
a
complete proof of this result. Here we only
mention that the condition
{C,~/K~}~~N
E
1'(IN)
implies convergence
in trace norm topology of the
N
x
N
matrix
CN(Z)
(see
(6.12))
embedded into
Z2(IN)
to the trace class operator
C,(z)
in
12(IN)
given
c,(x)
=
~
e-(nl+"l')z
I.
(6.30)
is
countable then
(6.28)
holds.
I€N
by
[
Kl
clcl'
+
KlI
l,lI€lN
Moreover, one has in analogy to
(G.ll),
(6.31)
Vm(x)
=
-2-
In det
1[1
t
C,(x)],
where detl(.) denotes the Fredholm determinant associated with
We emphasize that Theorem
6.1
solves the following inverse spec-
tral problem:
Given any bounded and countable subset
{-K;}~€~N
of
(-oo,O),
construct a (smooth and real-valued) potential
V
such that
+
V
has a purely absolutely continuous spectrum equal
d2
dx2
to
[O,oo)
and the set
of
eigenvalues
of
H
includes the prescribed set
{-K;}~~~N.
(In paaticular,
{-K?}~€N
can be dense in
a
bounded
subset of
(-00,
O).)
d2
dX2
P(
IN).
H
=
--
Under the stronger hypothesis
{K~}~€IN
E
Z'(IN)
one obtains
Theorem
6.2
[28],
[29]
Assume
{~j
>
O}jEl~
E
ll(IN),
~j
#
r;j~
for
j
#
j' and choose {cj
>
O}jEl~
such that
{c?/~j}~€~
E
I1(IN).
Then
in
addition to the conclusions
of
Theorem
6.1
we have
Spectral Deformations
and
Soliton Equations
135
(ii)
aess
(Hm)
=
aac(Hoo)
=
[0700)7
up
(KO)
n
(07
..)
=
asc(HC0)
=
0,
(6.33)
(6.34)
ad
(Hco)
=
{-";)jclN*
(6.35)
The unitary scattering matrix
S,(k)
ins'
associated with the
pair
(H,,
Ho)
is reflectionless and given by
(6.36)
Note that Theorem
6.2
constructs
a
new class
of
reflectionless
potentials involving
an
infinite negative point spectrum
of
H,
accu-
mulating
at
zero.
For
a
detailed proof of Theorem
6.2
see
[29].
We remark that
the condition
{~j)~~~v
E
I'(1N)
implies that
V,
E
L'(IR)
(but
V,
6
L'(1R;
(1+
1x1)
dz))
and that the product
TN(~)
converges absolutely
to
T,(k)
as
N
-+
00.
We conclude with the observation that the simple substitution
(6.37)
63
t
in
(6.30)
and
(6.31),
denoting the result in
(6.31)
by
V,(z,t),
pro-
duces solutions
of
the KdV1 equation (see
(2.8))
cj
+
cje
J
,
j
E
IN
(6.38)
3
1
4
03*xxx
2
In particular, substitutions
of
the type
(6.37)
together with Theorem
6.2
provide new soliton solutions
of
the I<dV hierarchy
[28],[29].
-
-
v,
V,,x
=
0.
KdV'(V,)
=
I/,,t
+
-
v
Acknowledgements
We would like to thank W. Karwowski, I<. Unterkofler, and
Z.
Zhm
for numerous discussions on this subject. It is
a
great pleasure to
thank W.
F.
Ames, E.
M.
Harrell, and
J.
V. Herod
for
their kind
invitation
to
a
highly stimulating conference.
136
F.
Gesztesy
&
R.
Weikard
Bibliography
[l]
P.
B. Abraham and
€1.
E.
Moses, Phys. Rev.
A22
(1980),
1333-
1340.
[2] M. Abramowitz and
I.
A. Stegun,
Handbook
of
Mathematical
Functions,
Dover, New York, 1972.
[3]
S.
I.
Al’ber, Commun. Pure Appl. Math.
34
(1981), 259-272.
[4] B. Birnir, Commun. Pure Appl. Math.
39
(1986), 1-49.
[5]
B. Birnir, Commun. Pure Appl. Math.
39
(1986), 283-305.
[GI
B. Birnir, SIAM
J.
Appl. Math.
47
(1987), 710-725.
[7]
J.
L. Burchnall and
T.
W.
Chaundy, Proc. London Math. SOC.
Ser. 2,
21
(1923), 420-440.
[8]
J.
L. Burchnall and
T.
W.
Chaundy, Proc, Roy.
SOC.
London
A118
(1928), 557-583.
[9] M. Buys and A. Finkel,
J.
Diff.
Eqs.
55
(1984), 257-275.
[lo]
W.
Craig, Commun. Math. Phys
126
(1989), 379-407.
[ll]
M. M. Crum, Quart.
J.
Math. Oxford (2),
6
(1955), 121-127.
[12]
G.
Darboux, C. R. Acad. Sci. (Paris)
94
(1882), 1456-1459.
[13]
P.
A. Deift, Duke Math.
J.
45
(1978), 267-310.
[14]
P.
Deift and
E.
Trubowitz, Commun. Pure Appl. Math.
32
(1979), 121-251.
[15]
B.
A. Dubrovin, Russ. Math. Surv.
36:2
(1981), 11-92.
[lG]
B. A. Dubrovin,
I.
M. Krichever, and
S.
P.
Novikov, in
Dynam-
ical
Systems IV,
V.
I.
Arnold,
S.
P.
Novikov (eds.), Springer,
Berlin, 1990,
pp.
173-280.
[17] B. A. Dubrovin,
V.
I3.
Matveev, and
S.
P.
Novikov, Russ. Math.
Surv.
31:l
(1976), 59-146.
Spectral Deformations and Soliton Equations
137
[18]
F.
Ehlers and 1-1. Knorrer, Comment. Math. Helv.
57
(1982),
1-10.
[19]
N.
M. Ercolani and 11. Flaschka, Phil. Trans. Roy. SOC. London
A315
(1985), 405-422.
[20]
N.
Ercolani and
H.
P.
McKean, Invent. math.
99
(1990), 483-
544.
[21]
A. Finkel,
E.
Isaacson, and
E.
Trubowitz, SIAM
J.
Math. Anal.
18
(1987), 46-53.
[22]
C.
S.
Gardner,
J.
M. Greene,
M.
D.
Kruskal, and
R.
M.
Miura,
Commun. Pure Appl. Math.
27
(1974), 97-133.
[23]
I. M. Gel’fand and
B.
M.
Levitan, Amer. Math. SOC. Transl.
Ser.
2,
1
(1955), 253-304.
[24]
F.
Gesztesy, in
Ideas and Methods in Mathematical Analysis,
Stochastics, and Applications,
Vol.
1,
S.
Albeverio,
J.
E.
Fen-
stad, H. Elolden, and
T.
Lindstroin (eds.), Cambridge Univ.
Press,
1992,
pp.
428-471.
[25]
F.
Gesztesy, A complete spectral characterization
of
the double
commutation method, preprint,
1992.
[26]
F.
Gesztesy and
R.
Svirsky,
(m)
KdV -solitons on
the
back-
ground
of
quasi-periodic finite-gap solutions,
preprint,
1991.
[27]
F.
Gesztesy and
R.
Weiltard, in preparation.
[28]
F.
Gesztesy,
W.
I<arwowsl<i, and Z. Zhao, Bull. Amer. Math.
SOC.,
to
appear.
[29]
F.
Gesztesy,
W.
Karwowski, and
Z.
Zhao, Duke Math.
J.,
to
appear.
[30]
P.
G.
Grinevich and
I.
M. Krichever, in
Soliton theory:
a
survey
of
results,
A.
P.
Fordy (ed.), Manchester Univ. Press, Manch-
ester,
1990,
pp.
354-400.
138
F.
Gesztesy
&
R.
Weikard
[31]
V.
Guillemin and A. Uribe, Trans. Amer. Math. SOC.
279
(1983),
759-771.
[32]
V.
Guillemin and A. Uribe, Commun. Part. Diff.
Eqs.
8
(1983),
1455-1474.
[33]
E.
M. Harrell, Ann. Phys.
119
(1979), 351-369.
[34]
A.
R. Its and
V.
B. Matveev, Theoret. Math. Phys.
23
(1975),
343-355.
[35]
K. Iwasaki, Ann. Mat. Pure Appl.
Ser.4,
149
(1987), 185-206.
[36]
C.
G.
J.
Jacobi,
J
Reine angew. Math.
17
(1837), 68-82.
[37]
T.
Kappeler, Ann. Inst. Fourier, Grenoble
41
(1991), 539-575.
[38]
I.
Kay and H.
E.
Moses,
J.
Appl. Phys.
27
(1956), 1503-1508.
[39]
S.
Kotani and M. Krishna,
J.
Funct. Anal.
78
(1988), 390-405.
[40]
B.
M.
Levitan, Math. USSR Izvestija
18
(1982), 249-273.
[41]
B. M. Levitan, Math USSR Izvestija
20
(1983), 55-87.
[42]
B. M. Levitan, Trans. Moscow Math. SOC.
45:l
(1984), 1-34.
[43]
B. M. Levitan, Math. USSR Sbornik
51
(1985), 67-89.
[44]
B. M. Levitan,
Inverse Sturnz-Liouville Problems,
VNU
Science
Press, Utrecht,
1987.
[45]
B. M. Levitan, Math USSR Sbornik
60
(1988), 77-106.
[46]
V.
A. Marchenko,
Sturnz-Liouville Operators and Applications,
Birkhauser, Basel,
1986.
[47]
V.
A. Marchenko,
I.
V.
Ostrovskii, Math. USSR Sbornik
26
(1975), 493-554.
[48]
V.
B. Matveev,
Abelian Functions
and
Solitons,
1976,
preprint.
Spectral Deformations and Soliton Equations
139
[49]
H.
P.
McKean, in
Partial Differential Equations and Geometry,
C.
I.
Byrnes (ed.), Marcel Dekker, New York,
1979,
pp.
237-254.
[50]
H.
P.
McKean, in
G60k~l
Analysis,
M. Grmela and
J.
E.
Marsden
(eds.), Lecture Notes in Math., Vol.
755,
1979,
pp.
83-200.
(511
H.
P.
McKean, Commun. Pure Appl. Math.
38
(1985), 669-678.
[52]
H.
P.
McKean, Rev. Mat. Iberoamericana
2
(1986), 235-261.
[53]
H.
P.
McKean,
J.
Stat. Phys.
46
(1987), 1115-1143.
[54]
H.
P.
McKean and
E.
Trubowitz, Commun. Pure Appl. Math.
29
(1976), 143-226.
[55]
H.
P.
h/IcI<ean and
E.
Trubowitz, Bull. Amer. Math. SOC.
84
(1978), 1042-1085.
[56]
H.
P.
McKean and
E.
Trubowitz, Commun. Math. Phys.
82
(1982), 471-495.
[57]
N.
W.
McLachlan,
Theory and Application
of
Muthieu Func-
tions,
Clarendon Press, Oxford,
1947.
[58]
S.
Novikov,
S.
V. Manaliov, L.
P.
Pitaevskii, and
V.
E.
Zakharov,
Theory
of
Solitons,
Consultants Bureau, New York,
1984.
[59]
D.
L. Pursey, Phys. Rev.
D33
(198G), 1048-1055.
[GO]
J.
Ralston a.nd
E.
Trubowitz, Ergod. Th. Dyn. Syst.
8
(1988),
301-358.
[61]
P.
Sarnak, Commun. Math. Phys.
84
(1982), 377-401.
[G2]
U.-W. Schmincke, Proc.
Roy.
SOC. Edinburgh
80A
(1978), 67-
84.
[G3]
T.
Sunada, Duke Math.
J.
47
(19SO), 529-546.
[G4]
M. I. Weinstein a.nd
J.
B. Keller, SIAM
J.
Appl. Math.
45
(1985), 200-214.
[G5]
J.
Zagrodzihki,
J.
Phys.
A17
(1984), 3315-3320.