Doktorov E.V., Leble S.B. A Dressing Method in Mathematical Physics

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1.3 Results for diﬀerential operators 9

that a given operator A = D

+ ... + a

having the kernel K intertwines

operators L and L

. The pro of is based on the prop erty o f the operator A

(Proposition 1.1) and the statement about division of L

by A [191]. Then

the theorem about the important property of the potential having only regu l ar

singularities g eneralizes the statement of [137] for the matrix potentials.

The pap er [191] describ es the matrix-valued potentials U(x) ∈ Mat

(C)

of the Schr¨odinger operator

= −

∂

∂x

+ U(x) (1.30)

having trivial monodromy. The monodromy operator is expressed via the

quasideterminant (see the deﬁnition in Sect. 1.9) as

AΨ = |



W (Ψ

, ..., Ψ

,Ψ)|

n+1,n+1

. (1.31)

The d × d matrices Ψ

, i =1,...,n and Ψ are eigenmatrices of the operator

and the Vandermond supermatrix



W (Ψ

, ..., Ψ

,Ψ) is deﬁned by its ﬁrst

superrow. Hence,

U = U

− 2∂



−1

), (1.32)

where Y is obtained from



W (Ψ

, ..., Ψ

) by deletion of the (n −1)th superrow.

The special case of U

=0yields

U(z)=



i=1

(z − z

)

=degdet



where the singularities of the potential, owing to (1.32), are the roots of

det



W = 0 , and the matrices A

have rank 1 in the generic case of the simple

roots z

The intertwining relation for a dressing in multidimensions in the case of

the zero potential was studied from a general mathematical p o int of view for

x ∈ C

[85]. The intertwining relation was used for studyi n g maps between

solutions of the Laplace equations with the seed operator

= ∆ =



i=1

∂

∂x

and the “dressed” operator

= ∆ + u(x)

with a rational potential u,

u(x)=



α∈S

+1)(α

⊥

,α

⊥

)

[α(x)]

10 1 Mathematical preliminaries

for α

⊥

= {a

}, a

∈ C. T h e set S contains a ﬁnite number of hy perplanes

α(x)=(α

⊥

,x)=a

. Relation (1.1) is considered with A(x,

∂

∂x

) and is named

as the DT.

The paper [85] formulates the suﬃcient condition (the results from [47] are

used) for the existence of the intertwining operator and describes all rational

potentials linked to ∆ by the intertwining relation.

1.4 Hyperspherical coordinate systems

and ladder operators

In the review paper [150] the hyperspherical coordinates [153, 154] are used to

describe quantum dynamical evolution of atomic and molecular aggregates,

ranging from their compact states to fragments. Using such a type of co or di-

nates is directly related to a generalization of the ladder operators’ structure

to many degrees of freedom.

In this approach, in contrast to the traditional independent-particle theory

[304], a quantum-mechanical multiparticle problem is parameterized by the

single collective radial parameter. The hyperradius

R =





i=1



1/2

,M=



i=1

serves as the basic aggregate co o r dinate of a wave function. Subsequent sepa-

ration of variables results in the equation for hyperspherical harmonics which

is a pro duct of the Legendre functions of subaggregate angle variables and the

Jacobi function.

The invariance of the total kinetic energy under multidimensional rotations

hints at an analogy with the angular momentum theory and hence with ladder

operators, new eig envalues, and new quantum numbers. A new basis is formed

by hyperspherical harmonics constructed by a kind of the “laddering” or, as

we call it here, the purely algebraic dressing procedure [456], without solving

diﬀerential equations.

A corresponding theory follows from the angular momentum theory of

quantum mechanics, where the components L

, L

,andL

of one-particle

angular momentum combine to reproduce the ladder operators’ algebra as in

(1.15) and (1.16). In spherical coordinates, l

= L

= −i(∂/∂φ), l

= L

±iL

Generalizing to the case of arbitrary number of particles, we take

= −i



∂

∂y

− y

∂

∂x



= −i

∂

∂φ

. (1.33)

The celebrated example of the helium atom [154] involves three indepen-

dent rotations of two Cartesian sets in the center-of-mass system. As a

1.5 Laplace transformations 11

result, two one-particle op erators l

and l

are combined with the third

(“mixed”) one,

−i



∂

∂z

− z

∂

∂z



. (1.34)

The operators (1.33) and (1.34) commute; hence, they form the Cartan

subalgebra h

∈H[205] in the Lie algebra of angular-momentum-like op-

erators formed by x

(∂/∂y

)andx

(∂/∂x

). The ladder operators in d

dimensions

= −i



∂

∂y

− y

∂

∂x



= −i



∂

∂x

− x

∂

∂x



(1.35)

raise or lower eigenvalues (or “weights”, in the nomenclature of the semisimple

Lie al gebra representation theory) of the operators h

.Inthewholealgebra,

some commutator relations can be read as eigenvalue problems of the adjoint

representation of h

ad(h

=[h

]=α

,i=1,...,l=dimH. (1.36)

The ladder operators are then the eigenvectors (the Cartan–Weyl basis), and

the lowering and raising properties are the direct consequences of (1.36). The

relations (1.36) generalize (1.6). So, the hyperspherical harmonics form the

basis of a representation of the Lie algebra of rank l.

The transition from the angular vector R to homogeneous comp onents

of the Jacobi coordinates [404] is performed by means of recursive change

→ ξ

. The recursion looks as follows: a single vector



+ M

+ r

)

is formed for a pair of the Jacobi vectors ξ

and ξ

of two subaggregates of

particles centered at r

and r

with masses M

and M

. The construction is

called a timber transformation by association with the Jacobi tree. For the

comprehensive outlook, details, and examples, see again [150].

1.5 Laplace transformations

The g eneral (hyperbolic) Laplace equation

+ α(x, y)φ

+ β(x, y)φ

+ γ(x, y)φ =0

go es to

+ aψ

+ bψ = 0 (1.37)

12 1 Mathematical preliminaries

after the gauge transformation

φ = gψ, g =exp





[a(x

′

,y) −β(x

′

,y)]dx

′

−



α(x, y

′

)dy

′



if the Laplace inva riants of both equations have the same values [168], or

−α

= a

−β

= b −γ + αβ. The Laplace transformation (LT) for (1.37) has

the form

a → a

−1

= a − ∂

ln(b − a

),b→ b

−1

= b − a

,ψ→ ψ

−1

= ψ

+ aψ,

a → a

= a + ∂

ln b, b → b

= b + ∂

(a + ∂

ln b) ,ψ→ ψ

and can be taken as a starting point in the theory of soliton equations in 2+1

dimensions [34, 168]. The LT is also a kind of a dr essing procedure; it leads

to a “partial” factorization of the operator of (1.37) and in the case of zero

Laplace invariants at some step of the LT iterations allows us to bu il d expl icit

solutions.

Important progress in the development of the LT theory was achieved in

[431]. Consider the operator

L =



i=0

2−i

+ a

(x, y)D

+ a

(x, y)D

+ c(x, y)

with arbitrary p

= p

(x, y). Solutions of the characteristic equation m

−

+ n

= 0 deﬁne the ﬁrst-order characteristic operators X

(x, y)D

+ n

(x, y)D

, which are strictly hyperbolic if the roots are dif-

ferent. The equation Lu = 0 can b e rewritten in the characteristic form

+α

)u =0, (X

+β

+α

)u =0, (1.38)

where α

= α

(x, y)andβ

= β

(x, y), the coeﬃcients of the ﬁrst-order char-

acteristic operators X

, can be found [up to a rescaling X

→ γ

(x, y)X

Since the operators X

do not commute, we have to take into consideration

the commutation rule

]=X

− X

= P (x, y)X

+ Q(x, y)X

. (1.39)

Using the Laplace invariants of (1.38),

h = X

(α

)+α

− α

,k= X

(β

)+β

− α

we represent the original operator L in two partially factorized forms

L =(X

+ α

)(X

+ α

) −h =(X

+ β

)(X

+ β

) − k. (1.40)

1.5 Laplace transformations 13

Each of them allows us to introduce the LT and leads to a factorization if

either h or k equals zero. Note also that the equation Lu = 0 is equivalent to

any of the ﬁrst-order systems (for this idea see also [34, 35])



u = −α

u + v,

v = hu − α

⇔



u = −β

u + w,

w = ku −β

(1.41)

Making use of the matrix nomenclature, we read a pair of u and w as columns,

introducing the coeﬃcient matrices α

for the right-hand side of (1.41).

The generalization of the classical LT originates from the central idea of swap-

ping the operators X

and X

: after using the commutator (1.39), the coeﬃ-

cients of the ﬁrst-order operators are changed, which produces the transformed

operator L[1]. The explicit procedure is described in [431] and is summarized

in the following steps:

1. If we have to solve an equation Lu = 0, transform it into the characteristic

form (1.41).

2. If the matrix [α

(x, y)] of the characteristic system is upper- or lower-

triangular, solve the equations consecutively.

3. If the matrix is block-triangular, the system factors into several lower-order

systems; try for each subsystem step 2.

4. In the general case of a nontriangular matrix [α

(x, y)], p erform several

(consecutive) generalized LTs , using diﬀerent choices of the pivot element

= 0. The goal is to obtain a block-triangular matrix for one of the

transformed systems.

In [405] the general hyperbo li c quasilinear equation u

= F (x, y, u, u

)

is treated from the Laplace theory point of view. The Laplace invariants H

are introduced via the recurrence

[log H

]=H

i+1

+ H

i−1

− 2H

,i∈ Z,

where D

x,y

are total derivatives and the ﬁrst terms of the r ecurrence are [359]

= D

) − F

− F

= D

) − F

− F

The recurrence obviously simpliﬁes in the cas e of (1.37). The following theorem

is the result of joint eﬀorts of the authors of [22] and [405].

Theorem 1.5. A break oﬀ of the recurrent sequence at both sides, i.e., ∃n, m,

such as H

= H

=0me ans the Darboux integrability, i.e., there exists a

pair of functions P and Q on prolonged space such that P

=0and Q

=0.

The famous example of such a (nonlinear) equation is the Liouville equation

=exp(u). The other one is concerned with the linear equation (1.37).

Recent important results are reported in [431], where a matrix version of

the classical LT is given. Let us repr oduce the main proposition of [431]:

14 1 Mathematical preliminaries

Proposition 1.6. Suppose the linear partial diﬀerential equation of order

n ≥ 2,

Lu =



i+j≤n

i,j

(x, y)D

u = 0 (1.42)

is strictly hyperbolic, i.e., the characteristic equation



i+j=n

i,j

=0has n

simple real roots λ

(x, y). Any strictly hyperbolic linear partial diﬀerential

equation (1.42) is e quivalent to an n × n ﬁrst-order system in characteristic

form



(x, y)u

. (1.43)

This statement opens the way for a generalization of the Laplace theory by

means of the procedure described above by steps 1–3.

The recurr ences introduced in this section are a kind of dressing in the

sense of a procedure that algebraically connects equations of the same form

with diﬀerent coeﬃcients. It resembles the Schlesinger transformation (degen-

erate elementary DT) that we will deal with in Chap. 3.

1.6 Matrix factorization

In this section we establish the link between the matrix factorization and the

intertwining relations and recall basic facts of matrix factorization in terms

of dressing procedures.

1.6.1 Example

It was shown that a factorization (1.5) yields the intertwining relation (1.1)

automaticall y. Taking the simplest example of 2 × 2 matrices, let us consider

a Hermitian matrix L as a product of mutually conjugate matrices A and A

L = A

A =



∗







|a|

+ |c|

∗

b + c

∗

a + d

∗

c |b|

+ |d|



Introducing the column vectors ψ =





and φ =





, we obtain the

following relations for these vectors:

|ψ|

= L

, |φ|

= L

, (ψ, φ)=L

It is easy to check that the n × n matrix can be factorized in the similar

way. For a diagonal matrix L, the scalar product (ψ, φ) is zero and one has

an orthogonal set of vectors. The matrix A made of the orthogonal vectors

1.6 Matrix factorization 15

intertwines the diagonal matrix L = A

A and the matrix AA

built from

norms of the row vectors and their scalar products:





∗





|a|

+ |b|

∗

+ bd

∗

+ db

∗

|c|

+ |d|



Generally, for an n×n matrix it is a set of n columns ψ

that forms factorizing

matrices.

1.6.2 QR algorithm

Another example of the factorization of an (invertible) n ×n matrix M into a

product of an orthogonal matrix Q and an upper-triangular

matrix R,

M = QR, (1.44)

is well known becau se it produces an algorithm for computing eigenvalues of

the matrix M [403]. A proof is provided by the Gramm–Schmidt orthogonal-

ization procedure [453].

The algorithm is the following. We start from (1.44) and factorize the

result of the transp osition

= RQ = Q

−1

MQ = Q

as M

= Q

, which allows us to construct M

= Q

. The repetition

of the factorization produces the chain

k+1

= Q

. (1.45)

This iterative process (which is also named as “dressing” ) is applied in the

theory of integrable Toda systems.

1.6.3 Factorization of the λ matrix

A λ matrix is determined as the polynomial

L(λ)=



k=0

(1.46)

with matrix coeﬃcients. Among the problems connected with operators of

the form (1.46), there are the eigenvalue problem for the equation L

ψ +

λL

ψ =0whenn = 1 and “bundle” problems [338] with arbitrary n. A sp ecial

factorization

L(λ)=M(λ)D

(1.47)

16 1 Mathematical preliminaries

introduces a right divisor of the matrix L with respect to the ﬁrst-order poly-

nomial D

= Aλ + B,where

M(λ)=

n−1



k=0

The substitution of D

and M(λ) into (1.47),

n−1



k=0

Aλ

k+1

n−1



k=0

Bλ



k=0

yields the recurrence

k−1

A + M

B = L

,k=1,...,n− 1,M

B = L

n−1

A = L

(1.48)

To calculate recurrently M

by means of (1.48), the existence of A

−1

and B

−1

is necessary.

1.7 Elementary factorization of matrix

For future use of the theory, the special (idempotent) case of the degenerate

matrix D

necessitates special attention. Namely, let two orthogonal idemp o-

tents p and q be given by p

= p, q

= q,andp + q = e,wheree is the identity

element. Hence, the space A of the matrices allows a natural splitting into

the column subspaces A

and A

. The splitting is realized by the following

identity for an arbitrary matrix M: M(p+q)=Mp+Mq.Consideraλ matrix

= pλ − σ that intertwines the polynomials of the ﬁrst order:

λ + B

)(pλ − σ)=(pλ − σ)(Aλ + B).

It leads to the conditions

p = pA, B

p − A

σ = −σA + pB, B

σ = σB.

The equivalent set corresponding to the decomposition of the unit matrix by

the projectors p and q looks as follows:

pAp = pA

p, qA

p =0,pAq=0, (1.49)

p − pA

(p + q)σp = −pσ(p + q)Ap + pBp, (1.50)

p − qA

(p + q)σp = −qσ(p + q)Ap, (1.51)

−pA

(p + q)σq = −pσqAq + pBq, (1.52)

−qA

qσq = −qσqAq , (1.53)

(p + q)σp = pσ(p + q)Bp, qB

(p + q)σp = qσ(p + q)Bp , (1.54)

(pB

p + pB

q)σq = pσ(pBq + qBq), (qB

p + qB

q)σq = qσ(pBq + qBq).

(1.55)

1.7 Elementary factorization of matrix 17

Let us consider the particular case of A = A

= ap + bq, a, b ∈ C and

commuting elements of σ, B,andB

. The conditions (1.49) and (1.53) hold

automatically, while (1.50) gives

p = pBp. (1.56)

Then (1.51) and (1.52) connect elements of B, B

,andσ:

p =(b − a)qσp, (1.57)

pσq =(b − a)

−1

pBq. (1.58)

Equations (1.56) and (1.57) can be read as a part of the transformation

B → B

for which we are searching.

Excluding pB

p from (1.54) yields

pBp σp + pB

qσp = pσpBp + pσqBp. (1.59)

Plugging the transform (1.57) and the link (1.58) into the second relation of

(1.55) produces

(b − a)qσp σp + qB

qσp = qσpBp + qσqBp. (1.60)

In the case of Abelian elements, the condition (1.59) is simpliﬁed:

qσp = pσqBp. (1.61)

If (1.55) and (1.61) can be solved with respect to elements of B

[we assume

∃(qσp)

−1

], then (1.61) accomplishes a construction of the transformation

B → B

, together with (1.56) and (1.57).

An important case of intert wining operators with degenerate coeﬃcients

was mentioned in the previous section. Indeed, let there exist matrices φ

∈ A

such that ∃pφ

−1

. Let us ﬁx the kernel of D

by means of the elements

of φ

(pλ

− σ)φ

=0. (1.62)

It can be read as

pσ(p + q)φ

= λ

pφ

,qσ(p + q)φ

=0.

These equations connect the matrix elements of σ with elements of φ

pσp= λ

pφ

− (b − a)

−1

pBqφ

,qσp= −qφ

(pφ

)

−1

, (1.63)

where (1.58) for pσq was used.

18 1 Mathematical preliminaries

Plugging the second equation in (1.63) into (1.57) gives the expressions

for qB

p =(a − b)qφ

(pφ

)

−1

,pB

q = −(b − a)

−1

qBpφ

(qφ

)

−1

. (1.64)

The equation (1.60), after substitutions from (1.64), determines qB

q,thelast

element of B. For the Abelian elements, we have

−qφ

(pφ

)

−1

[λ

(b − a)pφ

− pBqφ

+ qB

q − pBp ]=qσqB

Finally,

q = −λ

(b − a)pφ

+ pBqφ

+ pBp − (pφ

)(qφ

)

−1

qσqBp. (1.65)

The matrix elements of B

are expressed through elements of B and elements

of φ

, up to the “free parameter” qσq.

The general setting of the elementary DT theory is as follows:

1. One begins with

(Aλ

+ B

)φ

= 0 (1.66)

and note that owing to (1.62) the intertwining relation is valid identically.

2. For any λ and ψ(λ), if

(Aλ + B)ψ(λ)=0, (1.67)

the intertwining relation means that

(Aλ + B

ψ(λ)=0, (1.68)

where B

is determined by a solution of (1.66) and the parameter qσq

is given via (1.56), (1.64), and (1.65). This procedure links the known

problem (1.66) with another o ne (1.68) in a “covariant” way.

Remark 1.7. All matrices in this section could be considered as elements of a

ring as in Chaps. 2 and 3.

Remark 1.8. The case of higher polynomials in λ is studied similarly.

1.8 Matrix factorizations and integrable systems

The title of this section co incides with the title of the paper [111]. We give here

a concise overview of this paper and two preceding ones [109, 110] devoted

to the Toda lattice [152] and its applications in algorithms for computing

matrix eigenvalues [189]. The main idea of the QR algorithm is based on the

factorization of a matrix. Namely, if the chain of factorizations (1.45) yields

k+1

= Q

, (1.69)