On explicitly solvable Vekua equations and explicit solution of the stationary Schrödinger equation...

Complex Variables and Elliptic EquationsVol. 52, No. 5, May 2007, 353–366

On explicitly solvable Vekua equations and

explicit solution of the stationary Schrodingerequation and of the equation divðrJuÞV0

VLADISLAV V. KRAVCHENKO*y and HECTOR OVIEDOz

yDepartment of Mathematics, CINVESTAV, Queretaro, Libramiento NorponienteNo. 2000, Fracc. Real de Juriquilla, Queretaro, Qro., C.P. 76230, MexicozSeccion de Estudios de Posgrado e Investigacion, Escuela Superior de

Ingenierıa Mecanica y Electrica, Instituto Politecnico Nacional,C.P. 07738 Mexico D.F., Mexico

Communicated by W. Tutschke

(Received in final form 17 July 2006)

We consider the known classes of explicitly solvable Vekua equations and construct new ones.Using the relationship between the Vekua equations and second order equations we obtainexplicitly solvable stationary Schrodinger equations as well as explicit solutions for the equationdivð�ruÞ ¼ 0, when � is an even positive or negative power of an arbitrary harmonic function.

Keywords: Vekua equation; Pseudoanalytic function; Schrodinger equation; Conductivityequation

AMS 2000 Mathematics Subject Classification: 30G20

1. Introduction

In the recent work [8], a new simple relation was found between the two-dimensionalstationary Schrodinger equation and a special class of Vekua equations describingthe generalized analytic [12] or pseudoanalytic functions [3, 4]. Given a particular solu-tion of the Schrodinger equation with a real valued potential a Vekua equation is con-structed such that the real part of any of its solution is a solution of originalSchrodinger equation and the imaginary part is a solution of an associatedSchrodinger equation. The relation between the solutions of the two Schrodingerequations is a direct generalization of the relation between conjugate harmonicfunctions in complex analysis. Moreover, these results are easily generalized to the

*Corresponding author. Email: [email protected]

Complex Variables and Elliptic Equations

ISSN 1747-6933 print/ISSN 1747-6941 online � 2007 Taylor & Francis

http://www.tandf.co.uk/journals

DOI: 10.1080/17476930601140343

case of Schrodinger equations with complex valued potentials by considering bicomplexVekua equations [6].

The Vekua equation with this important property has a special form which resultedto be closely related to the form of explicitly solvable Vekua equations analysedby Bauer in [2] as well as of those studied by Canak in [5]. We use this relation forobtaining a class of explicitly solvable Schrodinger equations. Moreover, we establishthe relation of the same special class of Vekua equations with the equationdivð�ruÞ ¼ 0 which is extremely important in different applications, for example inelectrical impedance tomography (see, e.g., [1, 10, 11]). We obtain the general solutionof this equation in terms of analytic functions in the case when � is an even positive ornegative power of an arbitrary harmonic function.

As we show in this work, the class of Vekua equations which can be explicitly solvedwith the aid of Bauer’s procedure (we show that Canak’s result is a special case of themore general Bauer’s scheme) is closely related to the Laplace equation. We prove thata slightly modified procedure works in the case of an arbitrary Schrodinger equationand not necessarily for the Laplace equation. More precisely, we obtain that the generalsolution of a Vekua equation related to a Schrodinger equation in the sense explainedsubsequently can be expressed in terms of solutions of another Vekua equation thecoefficient of which is obtained from any particular solution of the Schrodingerequation. As a special case this general result gives us the Bauer and Canak scheme.

2. Notation

Denote @ z ¼ ð@=@xÞ þ ið@=@yÞ and @z ¼ ð@=@xÞ � ið@=@yÞ (usually these operators areintroduced with the factor 1=2; nevertheless here it is somewhat more convenientto consider them without it). The subindex z or z means the application of @ z or @z,respectively. For the sake of simplicity, throughout the whole article we suppose that� is a simply connected domain in C.

Note that the operator @ z applied to a real-valued function � can be regarded as itsgradient, and if we know that � z ¼ � in a whole complex plane or in a convex domain,where � ¼ �1 þ i�2 is a given complex-valued function such that its real part �1 andimaginary part �2 satisfy the equation

@y�1 � @x�2 ¼ 0, ð1Þ

then we can reconstruct � up to an arbitrary real constant c in the following way:

�ðx, yÞ ¼

Z x

x0

�1ð�, yÞd�þ

Z y

y0

�2ðx0, �Þd� þ c

where ðx0, y0Þ is an arbitrary fixed point in the domain of interest. This formula can beeasily extended to any simply connected domain by considering the integral along anarbitrary rectifiable curve � leading from ðx0, y0Þ to (x, y)

�ðx, yÞ ¼

Z�

�1dxþ�2dyþ c:

354 V. V. Kravchenko and H. Oviedo

By A we denote this integral operator:

A½��ðx, yÞ ¼

Z�

�1dxþ�2dy:

Thus, if � satisfies (1), there exists a family of real-valued functions � such that � z ¼ �,given by the formula � ¼ A½��.

3. Explicitly solvable Vekua equations

One of the main results of the monograph [2] reads as follows.

THEOREM 1 [2] Let � be an analytic function in �, ð�þ �Þ�z 6¼ 0 in � and m 2 N.Then for every solution W of the equation

Wz ¼m�z

ð�þ �ÞW in � ð2Þ

there exists an analytic function ’ in �, such that

W ¼Xmk¼0

ð�1Þm�kð2m� 1� kÞ!

k!ðm� kÞ!ð�þ �Þm�km

1

�z@z

� �k

’� ðm� kÞ1

�z@z

� �k

’

" #, ð3Þ

and for every analytic function ’ in �, (3) represents a solution of (2).

Remark 2 For m ¼ 1, expression (3) takes the form

W ¼’z�z

�ð’� ’Þ

ð�þ �Þ: ð4Þ

In the recent work [5] Canak obtained the following result claiming that this is thefirst example of an explicitly solved Vekua equation.

THEOREM 3 [5] Let � be an analytic function in �, �þ � 6¼ 0 in �. Then the generalsolution of the Vekua equation

! z ¼ ��z

�þ �! ð5Þ

has the form

! ¼ z � ð þ Þ�z

�þ �, ð6Þ

where is an arbitrary analytic function in �.

Explicitly solvable Vekua equations 355

Let us show that this result of Canak is a special case of a somewhat more generalresult of Bauer formulated as Theorem 1. Note that if ! is a solution of (5) then thefunction

W ¼i!

�zð7Þ

is a solution of (2) for m¼ 1.Moreover, taking ! defined by (6) we obtain

W ¼i z

�z�

i � i

�þ �

� �:

Denoting ’ ¼ i we arrive at (4). Thus, equation (5) by a simple transformation(7) turns into (2) with m¼ 1 and the form of the solution (6) coincides with thesolution (3) for m ¼ 1:

4. Special class of Vekua equations and its relation to second-order elliptic equations

In [6–9] the following class of Vekua equations was studied. Let f be a real-valued twicecontinuously differentiable nonvanishing function defined on �. Consider the equation

Wz ¼f zfW in �: ð8Þ

Denote

�1 ¼�f

f: ð9Þ

The following theorem establishes a simple relation between solutions of the Vekuaequation (8) and solutions of two stationary Schrodinger equations, one of themwith the potential �1 and the other with the potential

�1 ¼ 2f z�� f

� �2

� �1: ð10Þ

THEOREM 4 [8] If W ¼ W1 þ iW2 is a solution of (8) then W1 ¼ ReW is a solutionof the equation

��W1 þ �1W1 ¼ 0 in � ð11Þ

and W2 ¼ ImW is a solution of the associated Schrodinger equation

��W2 þ �1W2 ¼ 0 in � ð12Þ

where �1 and �1 are defined by (9) and (10), respectively.


Moreover, in [8] a simple formula was obtained which allows us for any givensolution W1 of (11) to construct such a solution W2 of (12) that W ¼ W1 þ iW2 willbe a solution of (8).

THEOREM 5 [8] Let W1 be a solution of (11). Then a solution W2 of (12) such thatW ¼ W1 þ iW2 is a solution of (8), is constructed according to the formula

W2 ¼ f �1A if 2@ zð f�1W1Þ

� �: ð13Þ

It is unique up to an additive term cf �1 where c is an arbitrary real constant. Conversely,given a solution W2 of (12), the corresponding solution W1 of (11) can be constructed asfollows:

W1 ¼ �fA if�2@ zð fW2Þ� �

: ð14Þ

It is unique up to an additive term cf:

When �1 � 0 and f � 1, equalities (13) and (14) turn into the well known in complexanalysis formulae for constructing conjugate harmonic functions. By analogy, we cansay that in general (13) and (14) give us a procedure for constructing conjugatemetaharmonic functions.

Let us notice that Theorems 4 and 5 are easily generalized onto the case of complexSchrodinger equations (11) and (12) by consideration of a bicomplex Vekua equationof the form (8) [6].

Remark 6 [8] Equation (8) can be written as follows

f @ z f �1W1

� �þ if �1@ zð fW2Þ ¼ 0,

where W1 ¼ ReW and W2 ¼ ImW:In the following theorem we show a simple relation between solutions of (8) and

solutions of another important second-order equation which arises in different branchesof physics in static models for inhomogeneous media, for example, in electro- ormagnetostatics.

THEOREM 7 Let W ¼ W1 þ iW2 be a solution of (8) where f ¼ �1=2: Then u ¼ ��1=2W1 isa solution of the equation

divð�ruÞ ¼ 0 in �, ð15Þ

and v ¼ �1=2W2 is a solution of the equation

divð� �1rvÞ ¼ 0 in �: ð16Þ

Proof According to Remark 6, equation (8) where f ¼ �1=2 can be written in thefollowing form

�1=2@ z ��1=2W1

� �þ i��1=2@ z �

1=2W2

� �¼ 0: ð17Þ


Multiplying by �1=2 and applying @z gives

@z�@ z ��1=2W1

� �þ i� �1=2W2

� �¼ 0

from where we have that Re @z�@ z ��1=2W1

� �� ¼ 0 which is equivalent to (15)

where u ¼ ��1=2W1.

Multiplying (17) by ��1=2 and applying @z gives

� ��1=2W1

� �þ i@z�

�1@ z �1=2W2

� �¼ 0

from where we have that Re @z��1@ z �

1=2W2

� �� ¼ 0 which is equivalent to (16)

where v ¼ �1=2W2. g

Remark 8 Equation (8) with f ¼ �1=2 is equivalent to the equation

wz ¼1� �

1þ �wz ð18Þ

where w ¼ uþ iv: The relation between (18) and (15), (16) was observed in [1]and resulted to be essential for solving the Calderon problem in the plane.

The following result is a direct corollary of the preceding two theorems.

THEOREM 9 Let u be a solution of (15). Then a solution v of (16) such that

W ¼ �1=2uþ i��1=2v

is a solution of (8) with f ¼ �1=2, is constructed according to the formula

v ¼ Aði�uzÞ:

It is unique up to an additive constant. Conversely, given a solution v of (16), thecorresponding solution u of (15) can be constructed as follows:

u ¼ �Aði� �1v zÞ:

It is unique up to an additive constant.

Proof Consists in substitution of W1 ¼ �1=2u and of W2 ¼ ��1=2vinto (13) and (14). g

With the aid of Theorems 4 and 5 in [8] a simple procedure based on Bers’ theory ofTaylor series in formal powers for pseudoanalytic functions [3, 4] was proposed for con-structing in explicit form a complete system of exact solutions for a wide class ofSchrodinger equations. For the procedure a particular solution of the Schrodingerequation is needed. Theorems 7 and 9 show us that the same procedure is obviouslyapplicable to equations (15) and (16). It works if the coefficient � fulfills the so-calledcondition S [8], this is automatically true when, for example, � depends on oneCartesian, spherical, parabolic, or elliptic variable but is not limited to these cases.


Moreover, we do not even need a given particular solution of (15) or (16). Meanwhilethis observation is to be developed in a subsequent publication, in the next sectionwe show how Bauer’s results can be applied to the considered second-order ellipticequations with the help of the relations established in this section.

5. Explicitly solvable second-order equations

In section 3 we formulated the result of Bauer giving a general solution of theVekua equation of the form (2). In the following statement we show that equation (2)can be written in the form (8) where f ¼ �m0 and �0 is a harmonic function.

LEMMA 10 Let � be an analytic function in� with �0 6¼ 0, �0 ¼ Re � and �1 ¼ Im �. Then

m�z�þ �

¼@ zf

f

where f ¼ �m0

Proof Consider

m�z�þ �

¼m@ z�

2�0¼

m@zð2�0 � ð�0 þ i�1ÞÞ

2�0¼@ z�

m0

�m0:

As a result of this observation we can reformulate Theorem 1 as follows.

THEOREM 11 Let f ¼ �m0 , where �0 is a nonvanishing real valued harmonic function in� and m 2 N. Then the general solution of (8) has the form

W ¼Xmk¼0


k!ðm� kÞ!ð�þ �Þm�km

1

�z@z

� �k

’� ðm� kÞ1

�z@z

� �k

’

" #ð19Þ

where � is an analytic function in � such that Re � ¼ �0 and ’ is an arbitrary analyticfunction in �:

Proof From Lemma 10 we have that equation (8) can be written in the form (2) where� is an analytic function in � such that Re� ¼ �0. Then application of Theorem 1 givesus the result.

The relationship between pseudoanalytic functions and solutions of the Schrodingerequation (Theorems 4 and 5 ) allows us to obtain the general solution for a class ofSchrodinger equations in terms of analytic functions.

THEOREM 12 The general solution of the Schrodinger equation

��W1 þ �1W1 ¼ 0 in � ð20Þ

with

�1 ¼ mðm� 1Þr�0�0

� �2

, ð21Þ


where m 2 N and �0 is a nonvanishing real-valued harmonic function in �, is given by theformula

W1 ¼ ReXmk¼0


k!ðm� kÞ!ð2�0Þm�k

mð1

�z@zÞ

k’� ðm� kÞ1

�z@z

� �k

’

" #ð22Þ

where � is an analytic function in � such that Re � ¼ �0 and ’ is an arbitrary analyticfunction in �. Analogously, the general solution of the Schrodinger equation

��W2 þ �1W2 ¼ 0 in � ð23Þ

with

�1 ¼ mðmþ 1Þr�0�0

� �2

ð24Þ

is given by the formula

W2 ¼ ImXmk¼0



m1

�z@z

� �k

’� ðm� kÞ1

�z@z

� �k

’

" #ð25Þ

where � is an analytic function in � such that Re � ¼ �0 and ’ is an arbitrary analyticfunction in �.

Proof It is easy to see that if W is a solution of (8) where f ¼ �m0 with �0 being aharmonic function, then W1 ¼ ReW and W2 ¼ ImW are solutions of (20) and (23),respectively, and conversely due to Theorem 5 any solution of (20) is the real part ofa solution of (8) with the coefficient described previously and any solution of (23)is the imaginary part of a solution of (8). Consequently, separating the real and theimaginary parts in (19) we finish the proof. g

Thus, the general solution of Schrodinger equations with potentials having theform (21) or (24) reduces to analytic functions.

Remark 13 The representation (25) of solutions of (23) was obtained in [2, Chapter I]though without using a relation between the Schrodinger equation and the Vekuaequation.

With the aid of Theorems 7 and 9 we obtain a similar result for equations (15)and (16).

THEOREM 14 Let � ¼ �2m0 , where �0 is a nonvanishing real valued harmonic function in� and m 2 N. Then the general solution of the equation

divð�ruÞ ¼ 0 in � ð26Þ

has the form

u ¼ ��1=2ReXmk¼0



m1

�z@z

� �k

’� ðm� kÞ1

�z@z

� �k

’

" #ð27Þ


where � is an analytic function in � such that Re � ¼ �0 and ’ is an arbitrary analyticfunction in �. Analogously, the general solution of the equation

divð� �1rvÞ ¼ 0 in � ð28Þ

has the form

v ¼ �1=2ImXmk¼0



m1

�z@z

� �k

’� ðm� kÞ1

�z@z

� �k

’

" #ð29Þ

Proof According to Theorem 7 for any solution u of (26) there exists a solution W of(8) such that its real part W1 is equal to �

1=2u. Note that for � ¼ �2m0 we have that in (8):f ¼ �m0 . Then applying Lemma 10 and Theorem 11 we obtain (27). Formula (29)is obtained in a similar way considering the imaginary part of a correspondingpseudoanalytic function. g

Remark 15 The class of second-order equations which can be solved explicitly onthe basis of Bauer’s Theorem 1 is not limited by Theorems 12 and 14. The explicitlysolvable Schrodinger equation (23) may serve for construction of other solvableSchrodinger equations in the following way. Consider equation (23) with the potential(24). Its solution is given by (25). Denote �2 ¼ �1 and take any solution f of (23).Consider equation (8). According to Theorem 4, the real part of any solution of (8)is now a solution of the equation

��W1 þ �2W1 ¼ 0 in � ð30Þ

and the imaginary part is a solution of the equation

��W2 þ �2W2 ¼ 0 in � ð31Þ

where �2 ¼ 2ð f z�� =f Þ2 � �2. Solutions of (31) are obtained from solutions of (30) using

formula (13). In this way the general solution of (31) reduces to analytic functions also.Obviously this procedure can be continued denoting �3 ¼ �2 and repeating the logicalsteps described previously.

Remark 16 It is interesting to notice that meanwhile for the coefficient � in (26) beingan even positive or negative power of an arbitrary harmonic function, the general solu-tion of (26) is given by Theorem 14, how to solve (26) with � being an odd power ofa harmonic function (including the case of � being a harmonic function itself ) is stillan open question.

6. An extension of the class of explicitly solvable Vekua equations.

The results of Bauer and of Canak give us a class of Vekua equations the solutionof which reduces to the Cauchy–Riemann system.


In this section we show how solution of a Vekua equation reduces to solutionof another Vekua equation. The result of Bauer for m¼ 1 and the result of Canakare special cases of this more general scheme.

Assume that f is a real-valued nonvanishing solution of the equation

ð��þ �Þf ¼ 0 in �,

where � is a real-valued function. Take an arbitrary solution of the equation

z ¼f zf in �, ð32Þ

and consider the function

! ¼ z þ p þ q : ð33Þ

Note that due to Theorem 4

@ z z ¼�

2ð þ Þ þ

�

2ð � Þ,

where � ¼ 2ðjf zj=f Þ2� �:

Then application of @ z to (33) gives us the expression

@z! ¼�

2ð þ Þ þ

�

2ð � Þ þ pz þ p

fzf þ qz þ q z

¼ q z þ qz þ pfzfþ��

2

� � þ pz þ

�þ �

2

� � :

On the other hand we have

q! ¼ q z þ pq þ qq :

We observe that ! is a solution of the equation

! z ¼ q! ð34Þ

iff the following system of equations is satisfied

qz þ pf zfþ��

2¼ pq, ð35Þ

pz þ�þ �

2¼ qq: ð36Þ


Noting that ð�þ �Þ=2 ¼ ðj f zj=f Þ2 and looking for a function p in the form p ¼ qþ u

where u is a complex-valued function, we rewrite the system as follows

qz � qqþ qf zf� quþ u

f zfþ��

2¼ 0, ð37Þ

qz � qqþ uz þf z�� f

� �2

¼ 0: ð38Þ

A general analysis of the nonlinear system (35), (36) or of its equivalent form (37), (38)is pretty much desirable because any of its solutions gives us an explicitly solvableVekua equation (34) in terms of solutions of equation (32). Here we study the casewhen q has the following special form

q ¼ �gzg, ð39Þ

where g is a real-valued, nonvanishing function. Then we have

qz � qq ¼ ��g

g: ð40Þ

Substitution of (39) and (40) gives us the following system

��g

g�gzg

f zfþgzguþ u

f zfþ��

2¼ 0, ð41Þ

��g

gþ uz þ

fz�� f

� �2

¼ 0: ð42Þ

The imaginary part of the last equation reads as follows

@yRe u� @xIm u ¼ 0:

That is, there exists such a real-valued function � that

u ¼ @ z�:

From (41) it is seen that it is convenient to choose � ¼ ln f. That is

u ¼f zf:

Then (41) takes the form

��g

gþ

f z�� f

� �2

þ��

2¼ 0: ð43Þ


Note that

uz ¼��

2:

Thus, system (41), (42) reduces to equation (43). Note that

��

2¼ �

f z�� f

� �2

þ�,

and equation (43) turns into the Schrodinger equation for g:

��gþ �g ¼ 0:

In this way we arrive at the following statement.

THEOREM 17 Let g be a real valued nonvanishing solution of the equation

ð��þ �Þg ¼ 0 in �, ð44Þ

where � is a real valued function. The general solution of the equation

! z þgzg! ¼ 0 in � ð45Þ

can be represented in the form

! ¼ z þ@zð f=gÞ

ð f=gÞ �

gzg ,

where f is a real-valued nonvanishing solution of (44) and is an arbitrary solution of theequation

z ¼f zf :

Remark 18 Let in Theorem 17 � � 0 in �, f � 1 and g be any harmonic function. Thenwe obtain Theorem 3 as a corollary of Theorem 17.

Remark 19 In [8] it was observed that the Vekua equations

! z ¼ �gzg! and Wz ¼

gz

gW in �,

where g is a real-valued function, are closely connected. Namely, if W is a solution ofthe second equation, then

! ¼ Wz �gzgW ð46Þ


is a solution of the first and conversely, if ! is a solution of the first equation, then

WðzÞ ¼1

2gðzÞRe

Z z

z0

!ð�Þd�

gð�Þþ

i

gðzÞIm

Z z

z0

gð�Þ!ð�Þd�

� �ð47Þ

is a solution of the second equation. Here z0 is an arbitrary point in � and the integral ispath-independent.

Expression (46) is the ðg, i=gÞ-derivative ofW in the sense of Bers [3, 4], and (47) is theðg, i=gÞ-antiderivative of !.

Due to this observation we obtain the following corollary of the preceding theorem.

THEOREM 20 Let g be a real-valued nonvanishing solution of equation (44) where � is areal-valued function. The general solution of the equation

Wz ¼gz

gW in �

can be represented in the form

WðzÞ ¼gðzÞ

2Re

Z z

z0

1

gð�Þ@� ð�Þ þ

@�ð fð�Þ=gð�ÞÞ

fð�Þ=gð�Þ ð�Þ �

@�gð�Þ

gð�Þ ð�Þ

� �d�

þi

2gðzÞIm

Zzz0

gð�Þ @� ð�Þ þ@�ð fð�Þ=gð�ÞÞ

fð�Þ=gð�Þ ð�Þ �

@�gð�Þ

gð�Þ ð�Þ

� �d�,

where f is a real-valued nonvanishing solution of (44) and is an arbitrary solution ofthe equation

z ¼f zf :

Example 21 Consider the equation

z ¼ c , ð48Þ

where c is an arbitrary complex constant. It is not more difficult than theCauchy–Riemann system (corresponding to c¼ 0). It is easy to see that being a solutionof (48), the function is necessarily a solution of the Helmholtz equation

�� þ cj j2 ¼ 0: ð49Þ

If a real-valued function 1 is a solution of (49) then it is easy to construct such areal-valued function 2 that ¼ 1 þ i 2 will be a solution of (48). For this purposelet us observe that (48) can be written in the form (8) where f ¼ eaxþby, a ¼ Re c


and b ¼ Im c (such f is obviously a nonvanishing solution of (49)). Then according toTheorem 5,

2 ¼ e�ðaxþbyÞA ie2ðaxþbyÞ@z e�ðaxþbyÞ 1

� �� and

1 ¼ �eaxþbyA ie�2ðaxþbyÞ@z eaxþby 2

� �� :

Thus, equation (48) is related to the Helmholtz equation (49) in the same sense as theCauchy–Riemann system is related to the Laplace equation, and Theorems 17 and20 give us the general solution of the equations

! z ¼ �gzg! and Wz ¼

gz

gW in �,

where g is a nonvanishing solution of (49), in terms of the solutions of equation (48).

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