Transcendental Trace Formulas For Finite-Gap Potentials

0.17 MB·English

by  Yu. V. Brezhnev

2
1 Theor.Math.Phys. 164(1)(2010), 920–928
0
2
TRANSCENDENTAL TRACE FORMULAS
n
a FOR FINITE-GAP POTENTIALS
J

3
1 YURIIV.BREZHNEV

]
I Abstract. Weshowthatformulasdifferingfromclassicalanaloguesofrationaltrace
S formulasforalgebraic-geometricpotentialsoccurinthetheoryoffinite-gapintegration
. ofspectralequations. Thenewformulascontaintranscendentalmodularfunctionsand
n
hypergeometricseries. Theyresultintranscendental relationsforthetafunctions.
i
l
n
[

1
Contents
v
6
1. Intorduction 1
2
2. The rational trace formula 2
8
22.1. The algebraic curve 2
.2.2. The rational trace formula 3
1
03. The transcendental trace formula 5
24. Transcendental identities for Θ-functions 7
1References 9
:
v
i
X
r 1. Intorduction
a
Trace identities and formulas are known in the integration theory of spectral problems.
This terminology originates from papers by Krein, Gel’fand, and Dickey on the spectral
theory of the ordinary differential equation of the form ψ u(x)ψ = λψ dating back
xx
−
to the 1950s. In 1975, stimulated by the appearance of finite-gap integration methods
(inthespectralformulation),V.Matveevestablishedanewandimportantinterpretation
of these formulas. These relations, together with their t-isospectral modifications, are
differential identities for exactly solvable potentials u(x). In particular, using one of the
series of these formulas, namely, the famous formula
u(x)=2 γ (x)+ +γ (x) +const, (1)
1 ··· g
(cid:8) (cid:9)
allows reconstructing the potentials from the quantities γ (x), which satisfy integrable
k
ordinary differential equations that are the Dubrovin equations. All the remaining trace
identities follow from this “master” formula. Here, we interpret these relations or, to be
precise, their analogues for higher-order spectral equations as formulas that express an
exactly solvable potential in terms of the quantities γ (x).
k
Key words and phrases. Spectral problem, finite-gap potential, trace formula, modular functions,
theta-functions, ellipticfunctions.
This work was supported in part by the Russian Federal Agency for Science and Innovation
(GrantNo.02.740.11.0238)andtheProgramforSupportingLeadingScientificSchools(GrantNo.NSh-
3400.2010.2).
2 YU.BREZHNEV
It is known that a potential u(x) determines a Liouville-integrable finite-dimensional
Hamiltoniansystemin whichx plays the role oftime. We pass to the canonicalvariables
(p,q) by combining the potential and its derivatives: p = p(u,u ,...), q = q(u,u ,...)
x x
[8]. The subsequenttransitionto separationvariablesis the transitiontothe variablesγ
k
andµ subject to the algebraicrelationW(γ ,µ )=0,whichis calledthe spectralcurve
k k k
equation. Formulas of the inverse transition p = p(γ ,µ ), q = q(γ ,µ ) and hence the
k k k k
formula of the form u=u(p,q) are well known [2] in Hamiltonian system theory, but we
can clearly identify formula (1) with a differential analogue of an operator trace only in
the Schr¨odinger equation case. A “trace” interpretation of all other algebraic-geometric
(finite-gap)operatorsislacking. Thisisbecauseboththespectralcurvecoordinatesγ and
µ must participate in a general theory of variable separation because Abelian functions,
generallyspeaking,areexpressedintermsofboththesevariables. Second-orderequations
are the only particular case where u=u(γ). Only isolated examples are known for more
complexspectralproblems[4],whileageneralschemeforconstructingtheprincipaltrace
formula
u=R(γ ,µ ) (2)
k k
with a rational function R is still lacking, to the best of our knowledge.
We devote this paper not to this problem but to the observation that in addition to
representing potentials in the form of Abelian functions as in (2), we have other rep-
resentations. They are provided by transcendental, not rational but still single-valued,
functions of symmetric combinations of the parameters (γ ,µ ). The example we use to
k k
demonstrate this is interesting not only in itself but also because it leads to new tran-
scendentalidentitiesforΘ-functions. WerecallthatpolynomialidentitiesforΘ-functions
are well known and are in fact uniformizing representations for Abelian varieties [3] and
Θ-functional representations for Abelian functions (2) respectively.
2. The rational trace formula
We consider the third-order spectral problem
1
Ψ′′′+u(x)Ψ′ u′(x)Ψ=λΨ. (3)
− 2
It is known that if we associate this problem with a nonlinear integrable equation, then
its isospectral deformations are described by the Kaup–Kupershmidt equation [11]
25
ut =uxxxxx +5uuxxx + 2 uxuxx +5u2ux. (4)
In what follows, it suffices to consider only stationary solutions or, equivalently, the
operator pencil
A(λ;[u])Ψ′′+B(λ;[u])Ψ′+C(λ;[u])Ψ=µΨ, (5)
which depends linearly on λ and commutes with (3).
2.1. The algebraic curve. The compatibility condition of Eq.(5) with the spectral
problem, i.e., with problem (3), customarily yields expressions for A, B, and C:
1 1 2
A= 9λ, B =u2+ u′′+c, C = A′′+ Au B′,
− 2 −3 3 −
The constant is related to the stationary dynamics u=u(x ct), and we have
−
1 1
9λΨ′′+ u2+ u′′+c Ψ′ 6uλ+2uu′′+ u′′′ Ψ=µΨ.
− (cid:16) 2 (cid:17) −(cid:16) 2 (cid:17)
TRANSCENDENTAL TRACE FORMULAS 3
The quantities λ and µ are algebraically dependent, i.e., they belong to the algebraic
curve
µ3+Q(λ;[u])µ+T(λ;[u])=0. (6)
We obtain the equation of this curve by eliminating Ψ from (3) and (5), with the result
9 27 1
µ3+ 27cλ2 Kλ+E µ+729λ5+E λ3+ uKλ2+E λ+ (u′′+2u2+2c)2K =0.
2 1 3
(cid:16) −2 (cid:17) 2 8
We present expressions for E below and here note that not only integrals of stationary
k
equation (4) but also the equation itself enter the spectral curve equation:
25
K =cu′+u(V)+5uu′′′+ u′u′′+5u2u′.
2
Byvirtueof(4)theconstantKmustvanishidentically. Buttheexpressionu′′+2u2+2c=
const also does not contradict it. If the constant in the right-hand side of this equation
is nonzero, then the compatibility with the equation K = 0 results in the trivial answer
u=const. Otherwise, we have the degenerate curve
3
µ3+27cλ2µ+729λ5 27 u′2+u3+3cu λ3 =0.
− (cid:18)4 (cid:19)
This curve has the genus g = 1, the corresponding solution u = 3℘ x+ 3g t can be
− 4 2
easily calculated, and we are not interested in it. In the nondegenera(cid:0)te case, c(cid:1)urve (6)
becomes
µ3+(27cλ2+E )µ+729λ5+81E λ3+E λ=0, (7)
2 1 3
it has the genus g = 4, and the problem of determining the corresponding potential is
nontrivial.
2.2. The rational trace formula. We firstobtain the standardtrace formula,i.e., the
representation of the potential in form (2). Let us determine the quantities γ . In the
k
finite-gap potential theory, these objects are the poles of Ψ(x), which are independent of
λ. Eliminating the Ψ-function and its first derivative from Eqs.(3) and (5), we obtain
the expression
Ψ′ 3µ2+Q 1
=27λ (u′′+2u2+2c). (8)
Ψ Π − 18λ
Here Q(λ):=27cλ2+E ([u]), and the differential polynomial Π=Π(λ;[u]) is
1
1 1
3−7Π(λ;[u])=λ4 u′λ3+ u(IV)+5uu′′+4u′2+2u3+2cu λ2+
− 6 18
(cid:0) (cid:1) (9)
1 1
+ (u′′′+4uu′)(u′′+2u2+2c)λ+ (u′′+2u2+2c)3.
324 368
We now present the Novikov integrals E :
k
15 5
E =u(IV)+5uu′′+ u′2+ u3+cu, (10)
1
4 3
1 1 1
E = u′′′(u′′′+8uu′) uu′′2+ u′′(E +3u′2 20u3 12cu)+ ,
2 1
−4 − 4 24 − − ···
3 1 3
E = u′′′u′(u′′+2u2+2c)+ u′′3+ (u2+c)u′′2 3u′2u′′+ ,
3
−4 4 4 − ···
where we truncate the expressions for E , E because we do not need them in what
2 3
follows.
4 YU.BREZHNEV
Theseparationvariablesγ determinethepolesoflogarithmicderivative(8),i.e.,they
k
factor the polynomial Π. We therefore define
Π=37(λ γ )(λ γ )(λ γ )(λ γ ). (11)
− 1 − 2 − 3 − 4
If we define the second coordinate µ =µ (x) by the formula
k k
µ3 +Q γ (x) µ +T γ (x) =0,
k k k k
(cid:0) (cid:1) (cid:0) (cid:1)
then we obtain ananalogueofthe Dubrovinequations from(8) in the case under consid-
eration. Indeed, passing to the limit λ γ in (8), we obtain
→ k
dγ γ 3µ2 +Q(γ )
k = k k k . (12)
dx −81 (γ γ )
k− j
jQ6=k
Additional explanations canbe found in [4], where the generalmethod canalso be found
for obtaining such equations and the second coordinate of the zero of the polynomial Π:
(u′′+2u2+2c)2
µ =3uγ + . (13)
k k 36γ
k
Comparing (9) and (11), for instance, we obtain the identities
4
u′ =6 γ , u(IV)+5uu′′+4u′2+2u3+2cu=18 γ γ . (14)
k k j
k
X1 Xk>j
Composing symmetric combinations of Eqs.(12) and using formula (13) for the second
coordinate, we can eliminate all the derivatives of the potential and thus obtain a linear
equation for u. This is the desired trace formula. Omitting the details, we eventually
obtain
1 4 γ2µ
u= k k . (15)
3 k (γ γ )
X1 k− j
jQ6=k
The last simplification of Eqs.(12) is to reduce them to quadratures. Again composing
symmetric combinations of the right-hand sides of (12), we obtain the equations
4 4
dγ γ dγ
k =0, k k =0,
k 3µ2+Q(γ ) k 3µ2 +Q(γ )
X1 k k X1 k k
4 µ dγ 4 γ2dγ 1
k k =0, k k = dx,
k 3µ2+Q(γ ) k 3µ2 +Q(γ ) −81
X1 k k X1 k k
which, as expected, result in a Jacobi problem, i.e., in the problem of inverting four
holomorphic integrals on curve (7):
4 (γk,µk) 4 (γk,µk)
dλ λdλ
=a , =a ,
k Z 3µ2+Q(λ) 1 k Z 3µ2+Q(λ) 2
X1 X1
4 (γk,µk) µdλ 4 (γk,µk) λ2dλ 1
=a , =a x.
k Z 3µ2+Q(λ) 3 k Z 3µ2+Q(λ) 4− 81
X1 X1
In what follows, we consider the dependence γ = γ (x) to be defined from these equa-
k k
tions.
TRANSCENDENTAL TRACE FORMULAS 5
We note thatthis scheme for obtaining the Dubrovinequationandtrace formulafrom
the definition of the differential polynomial Π and Ψ is intrinsic for general finite-gap
operators. The definitions of the quantities γ , the Dubrovin equations, and various
k
relations for the potential and its derivatives (different versions of the trace formula) are
“simultaneously entangled” in the general set of determining polynomial relations.
3. The transcendental trace formula
Deriving the preceding formulas, we manipulated with polynomials, which is common in
polynomialidealtheory[7]. Thebasesofsuchidealsarenotunique,whichmeansthatwe
can presumably find other relations determining the potential. This is indeed the case.
Eliminatingthe higherderivativeu(IV) fromthe NovikovintegralE andfromthe second
1
identity in (10), we obtain the quadratic equation for u′:
3u′2+4u3+12cu+E =216 γ γ .
1 k j
Xk>j
But u′ can be expresseduniquely in terms of γ by virtue of the first identity in (14). We
then obtain the cubic polynomial in the variable u:
4
u3+3cu+3E +27 γ2 =0. (16)
1 k
k
X1
It can be interpreted as an algebraic (implicit) variant of the principal trace formula.
On the other hand, finite-gap potentials are single-valued functions of the variable x.
They satisfy autonomous ordinary differential equations, pass the Painlev´e test, and are
expressedintermsofthetafunctions. Itisalsoknown(butseldomused)thatrootsofany
3rd or 4th degree polynomial can be expressed analytically via its coefficients in terms
of elliptic functions [1]. For exhaustive information on this topic see [5]. In our case the
situationsimplifies since we maydirectly considerpolynomial(16)asa Weierstrasscubic
polynomial
4
4u3+12cu+12E +108 γ2(x)=4u3 au b=
1 k k − −
X1
=4(u e)(u e′)(u e′′),
− − −
where the points e, e′, and e′′ depend on the coefficients a and b, which in turn depend
on x. The roots of this polynomial are
u= e,e′,e′′ =℘ ω (x);a,b ,
{ } k
(cid:0) (cid:1)
where
a= 12c, b= 108 γ2(x) 12E ,
− − k − 1
are expressed in terms of the theta constants [1]P
1 π2
e= ϑ4(τ)+ϑ4(τ) ,
ω2 12 3 4
(cid:8) (cid:9)
1 π2
e′ = ϑ4(τ)+ϑ4(τ) , (17)
−ω2 12 2 3
(cid:8) (cid:9)
1 π2
e′′ = ϑ4(τ) ϑ4(τ) ,
ω2 12 2 − 4
(cid:8) (cid:9)
6 YU.BREZHNEV
and the quantities ω and τ = ω′/ω are expressed in terms of the coefficients a and b. It
suffices to take just one such expression and express the ϑ-constants in (17) using the
classic series [1]:
∞ ∞ ∞
ϑ (τ):= eπ4iτ e(k2+k)πiτ, ϑ (τ):= ek2πiτ, ϑ (τ):= ( 1)kek2πiτ.
2 3 4
k k k −
X−∞ X−∞ X−∞
(we let the symbol ez denote the exponential function to distinguish it from the Weier-
strass e-points).
We thus obtaina classicelliptic modular function inversionproblem, i.e., the problem
of expressing the periods (2ω,2ω′) of the elliptic curve w2 =4z3 az b in terms of its
− −
coefficients. Thesolutionschemeisknown. Wemustfindtherootτ ofthetranscendental
equation
a3
J(τ)= ,
a3 27b2
−
where J(τ) is the classic modular Klein function [1, 10]. After finding the root τ, we
calculate the half-periods (ω, ω′) by the formula
a g (τ)
ω2 = 3 , ω′ =τω, (18)
b g (τ)
2
where g (τ) are the known modular forms [10]. We have the Eisenstein series, the
2,3
Hurwitz–Lambert series, or the ϑ-constant representations for these forms:
π4
g (τ)= ϑ8(τ)+ϑ8(τ)+ϑ8(τ)
2 24 2 3 4
(cid:8) (cid:9) (19)
π6
g (τ)= ϑ4(τ)+ϑ4(τ) ϑ4(τ)+ϑ4(τ) ϑ4(τ) ϑ4(τ) .
3 432 2 3 3 4 4 − 2
(cid:8) (cid:9)(cid:8) (cid:9)(cid:8) (cid:9)
Itremainstowritetheexpressionfortherootτ. Strangelyenough,anexplicitanalytic
solution of this classic problem is lacking1, although it is common knowledge that this
solutioncanbewrittenintermsofratiosofhypergeometric F -series. Becausetheseries
2 1
F (J) converges only inside the unit circle, the formulas differ depending on whether
2 1
J > 1 or J < 1. Altogether, this results in cumbersome expressions also containing
| | | |
functional series in the logarithmic derivative of the Euler Γ-function. For example, the
corresponding expression (22)–(27) in Sec.14.6.2 of [10] is half a page long (even if we
forgive the incorrectness of formula (23)). But the problem admits a simple solution.
Weusethewell-knownfactthatthefunctionJ iscloselyrelatedtothehypergeometric
equation of the form
1 1
J(J 1)ψ′′+ (7J 4)ψ′+ ψ =0.
− 6 − 144
Solutionsofgeneralhypergeometricequationsarehypergeometricfunctions F (a,b;cz),
2 1
|
but these functions can be rewritten in terms of classic special functions under special
restrictions imposed on the parameters a, b, and c. We mean not reductions of the
function F butthecaseswherethe F -seriesadmitsaquadratictransformation,andthe
2 1 2 1
hypergeometricequationthenreducestoatwo-parameterequation,e.g.,totheLegendre
equation. This is true for the above equation. Its solution is a linear combination
ψ = √6J APµ(√1 J)+BQµ(√1 J)
ν − ν −
(cid:8) (cid:9)
1Unfortunately, solutions of this problem are given incorrectly both on page 789 of encyclopedia [6]
andintheendofSect.11oftheremarkablebook[1];thisisnotduetomisprints.
TRANSCENDENTAL TRACE FORMULAS 7
of the Legendre functions with the parameters (ν,µ) = 1, 1 (see [9] for detailed
−2 3
information about these functions). From the above, we con(cid:0)clude (cid:1)that the formula
aP(√1 J)+bQ(√1 J)
τ = − − (20)
cP(√1 J)+dQ(√1 J)
− −
holds at certain numericalvalues of a,b,c,d (we omit the indices ν, µ of the Legendre
{ }
functions for brevity). We present only the final answer; see [5] for derivation.
Proposition 1. For the elliptic curve w2 =4z3 az b, the period ratio τ =ω′/ω is
− −
P √1 J πi a3
τ = πi − 1 e3 , where J:= . (21)
(cid:26) Q(cid:0)√1 J(cid:1) − (cid:27) a3 27b2
− −
(cid:0) (cid:1)
Remark 1. If J is substituted for J(τ) in equality (21), then this equality becomes an
identity that holds for all τ in the upper half-plane H+. The left- and right-handsides of
this identity must be calculated up to the action of the total modular group PSL (Z)=:
2
Γ(1). Relation(21)isthereforeaΓ(1)-equivalentofthesolutionofthemodularinversion
problemin the Legendre representation. Namely, giventhe elliptic curvein the Legendre
form w2 = (1 z2)(1 k2z2), we can calculate the elliptic modulus τ by the celebrated
− −
Jacobi formula
K′(k)
τ =i mod Γ(2), (22)
K(k)
where K and K′ are the complete Legendre elliptic integrals of the first kind [1, 10, 13].
Returningtothepotential,weuseformulas(17)and(18)with(19)takenintoaccount
to obtain the desired transcendental “trace formula”:
3 4 ϑ8(τ)+ϑ8(τ)+ϑ8(τ)
u(x)= E +9 γ2(x) 2 3 4 , (23)
2c(cid:26) 1 k k (cid:27) ϑ4(τ)+ϑ4(τ) ϑ4(τ) ϑ4(τ)
X1 2 3 4 − 2
(cid:8) (cid:9)(cid:8) (cid:9)
where τ is the function
P √1 J πi πi 4c3
τ(x)=Γ πi − e3 e3 , J:= , (24)
(cid:18) Q(cid:0)√1 J(cid:1) − (cid:19) 4c3+9 E +9 γ2(x) 2
− 1 k
b (cid:0) (cid:1) (cid:8) P (cid:9)
and we let the symbol Γ denote the operation of setting a number to the fundamental
domain of the group. Formulas (23)–(24) do not contain the coordinates µ . Verifying
b k
thatfunction(23)indeedsatisfiesrelation(10)orastationaryversionofEq.(4)isagood
and rather nontrivial exercise.
4. Transcendental identities for Θ-functions
The results in the preceding section demonstrate that we can analogously obtain many
more examples of Abelian functions that are radicalsor roots of algebraicequations. For
example, the last summand in polynomial (9) provides a single-valued representation of
the Abelian function radical of the form
18√3γ γ γ γ =u′′+2u2+2c,
1 2 3 4
andthegeneralmechanismforconstructing“transcendentaltraces”isbasedoneliminat-
ingderivativesofthepotentialfromthetotalsetofrelationsdeterminingtheintegralsE
k
andthevariablesγ . Solvingtheobtainedequationsinsingle-valuedfunctions(whichare
k
8 YU.BREZHNEV
always theta functions), we obtain the answer. These equations (and even possibly their
orders) are not uniquely defined and may depend on the genus, but transitions between
them are just different representations of the same Abelian function. They differ even in
the (γ,µ)-representation because the coordinates γ and µ are algebraically related.
Now equating the transcendental and rational representations, we obtain an unusual
identity containing theta functions. Indeed, comparing Eqs.(15) and (23), for example,
(23)
1 4 γ2µ 3 4 ϑ8(τ)+ϑ8(τ)+ϑ8(τ)
k k = E +9 γ2 2 3 4 , (25)
3 k (γ γ ) 2c(cid:26) 1 k k(cid:27) ϑ4(τ)+ϑ4(τ) ϑ4(τ) ϑ4(τ)
X1 k− j X1 2 3 4 − 2
jQ6=k (cid:8) (cid:9)(cid:8) (cid:9)
we can in principle represent the Abelian functions rational in (γ,µ) in this formula in
the formofΘ-functionrelations[3]. The obtainedrelationforΘ-functionsofgenusg =4
is a transcendental identity containing the theta constants ϑ τ(x) , although curve (7),
generally speaking, is not related to any elliptic curve. In ad(cid:0)dition(cid:1)to the curve moduli,
this identity also contains an arbitrary parameter x. We can also move all symmetric
functions that are rational in (γ,µ) to the left-hand side of the equality. We then obtain
an Abelian function for the Jacobian of trigonal curve (7) in the left-hand side and an
“exotic”representationofthiscurveintermsoftheellipticthetaconstantsoftheLegendre
functions in the right-hand side:
ϑ4(τ)+ϑ4(τ) ϑ4(τ) ϑ4(τ)
u2 = 2c 2 3 4 − 2 3c. (26)
− (cid:8) ϑ8(τ)+ϑ(cid:9)8((cid:8)τ)+ϑ8(τ) (cid:9) −
2 3 4
In turn, arguments of the Legendre functions (hypergeometric functions) in this repre-
sentation are expressions containing another Abelian function via the quantity √1 J,
−
i.e., γ2(x), by virtue of formula (24). This function is represented by its own rational
k
fractiPon of Θ-functions of the type Θ(xU +D). The nature of transcendental relations
is therefore rather involved and still needs to be understood. The transcendency is pre-
servedeven when the Jacobianof curve (7) splits completely into elliptic curves,and the
multidimensional Θ-functions become the elliptic θ-functions. Polynomials of type (16)
then remain nondegenerate, and it is natural to expect the appearance of transcendental
θ-identities.
The type of transcendency obtained above is related to the elliptic modular inversion
becausethisinversionisgeneratedbyasolutionofanalgebraicequationwithapplication
of the elliptic modular functions. The general theta functions and constants appear in
more-general methods (see, e.g., Umemura’s contribution to [12]), and it is therefore
natural to expect the appearance of other types of transcendental identities.
In conclusion, we note that the simultaneous and transcendental unification of theta
functions and hypergeometric series is in fact not unexpected, because a natural feature
of F series are their monodromy groups, which are in turn automorphisms of analytic
2 1
automorphic functions. All such functions that are currently known are ratios of theta
constants. Informally speaking, the “theta” geometry and “hypergeometry” are mutu-
ally inverse (they realize inversions of fractions of type (20)), and our transcendental
identities reflect both this fact and the properties of functions of the type Θ(xU +D).
Namely, Abelian manifolds are parameterized as algebraic manifolds by polynomials in
theta functions while linear sections of Jacobians appear from integrable equations. A
TRANSCENDENTAL TRACE FORMULAS 9
partial illustration is already provided by Jacobi equation (22) rewritten in the form
K′ ϑ22(τ)
τ i (cid:16)ϑ23(τ)(cid:17) mod Γ(2) τ H+.
≡ K ϑ22(τ) ∀ ∈
ϑ2(τ)
(cid:16) 3 (cid:17)
Considering this equation in a sufficiently small vicinity of a point τ , we can drop out
0
a “not completely analytic” operation of bringing a point into the fundamental domain
of the group, and this equation then becomes an exact analytic transcendental identity.
We can proceed in the same way with the theta versions of our relations (25) and (26)
interpreting them as functions of x.
References
[1] Akhiezer,N.I. Elements of the Theory of Elliptic Functions. Transl. Math. Monogr. 79. Amer.
Math.Soc.,Providence, R.I.(1990).
[2] Bl aszak,M. Multi-Hamiltonian Theory of Dynamical Systems. Texts and Monograph in Physics.
Springer-Verlag: Berlin(1998).
[3] Buchstaber,V.M.,Enolskii, V.Z.&Leykin,D.V.Uniformization ofJacobi’s varietiesof trig-
onal curvesand nonlinear equations.Funct.Anal.Appl.34(2)(2000), 79–85.
[4] Brezhnev,Yu.V.Dubrovinequationsforfinite-gapoperators.Russ.Math.Surveys57(2002),415–
417.
[5] Brezhnev,Yu.V.Non-canonicalextensionofθ-functionsandmodularintegrabilityofϑ-constants.
http://arXiv.org/abs/1011.1643.
[6] Виноградов,И.М.(ред.)Математическая энциклопедия.III.Москва(1982);
Hazewinkel,M. (Ed.) Encyclopaedia of Mathematics. Kluwer Academic Publishers; on-line:
http://eom.springer.de
[7] Cox,D., Little,J. & O’Shea,D. Ideals, Varieties, and Algorithms: An Introduction to Compu-
tational Algebraic Geometry and Commutative Algebra.(3rded.)Springer: NewYork(2007).
[8] Dickey,L.A. Solitons Equations and Hamiltonian Systems. 2nd edn. Advance Series in Math.
Physics26.WorldScientific: Singapore,2003.
[9] Erd´elyi,A.,Magnus,W.,Oberhettinger,F.&Tricomi,F.G.HigherTranscendentalFunctions
I. The Hypergeometric Function, Legendre Functions.McGraw–Hill: NewYork(1953).
[10] Erd´elyi,A.,Magnus,W.,Oberhettinger,F.&Tricomi,F.G.HigherTranscendentalFunctions
III. Elliptic and automorphic functions.McGraw–Hill: NewYork(1955).
[11] Kaup,D.J. On the inverse scattering problem for cubic eigenvalue problems of the class ψxxx +
6Qψx+6Rψ=λψ.Stud.Appl.Math.62(1980), 189–216.
[12] Mumford,D.TataLecturesonTheta.I–II.ProgressinMath.28,43.Birkh¨auser: Boston(1983)–
(1984).
[13] Whittaker,E.T.&Watson,G.N.ACourseofModernAnalysis: AnIntroductiontotheGeneral
TheoryofInfiniteProcessesandofAnalyticFunctions,withanAccountofthePrincipalTranscen-
dental Functions.CambridgeUniversityPress: Cambridge(1996).