Post on 05-May-2023
Complex Variables and Elliptic Equations, 2013
Vol. 58, No. 3, 415–430, http://dx.doi.org/10.1080/17476933.2011.593097
Schrodinger type equations with real-time variable and
complex spatial variables
Ciprian G. Gala*, Sorin G. Galb and Jerome A. Goldsteinc
aDepartment of Mathematical Sciences, The University of Missouri, Columbia,MO 65211, USA; bDepartment of Mathematics and Computer Science,University of Oradea, Str. Universitatii Nr. 1, 410087 Oradea, Romania;
cDepartment of Mathematical Sciences, The University of Memphis,Memphis, TN 38152, USA
Communicated by V. Radulescu
(Received 23 February 2011; final version received 27 May 2011)
In recent work, heat and Laplace equations, (un)damped wave equations,the Burgers and the Black–Merton–Scholes equations with real-timevariable and complex spatial variable were studied. The purpose of thisarticle is to make a similar study for the Schrodinger equation withreal-time variable and complex spatial variable. The complexification of thespatial variable in the case of the Schrodinger equation is made by twodifferent methods which produce different equations: first, one complexifiesthe spatial variable in the corresponding convolution formula by replacingthe usual sum of variables (translation) by an exponential product(rotation) and second, one complexifies the spatial variable in thecorresponding evolution equation and then one searches for non-analyticand for analytic solutions. By both methods, new kinds of evolutionequations (or systems of equations) in two-dimensional spatial variables aregenerated and their solutions are constructed. It is of interest to note that inthe case of the first method, solutions can be studied by employing thepowerful theory of groups of linear operators. Then, we show that thesesolutions preserve some geometric properties of the boundary function, likethe univalence, starlikeness, convexity and spirallikeness. In the case of thehigher order Schrodinger equation, the complexification of the spatialvariable is made in the corresponding convolution formula.
Keywords: Schrodinger equation; higher order Schrodinger equation;complex convolution integrals; complex spatial variables; groups of linearoperators on locally convex spaces; univalence; starlikeness; convexity;spirallikeness
AMS Subject Classifications: 47D03; 47D06; 47D60; 30C45
*Corresponding author. Email: galc@missouri.edu
� 2013 Taylor & Francis
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1. Introduction
Let us consider the Cauchy problem for the case of the one-dimensional Schrodingerequation
i@u
@tðt, xÞ þ
@2u
@x2ðt, xÞ ¼ 0, ðt, xÞ 2R�R,
uð0, xÞ ¼ f ðxÞ, x2R,
8<: ð1:1Þ
where f and u are complex-valued functions. It is well-known that under certainconditions, such as f2L1(R),
uðt, xÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
Z þ1�1
f ðx� uÞeiu2=ð4tÞdu ð1:2Þ
is a solution of the above initial value problem (see, e.g. [1]). In other spaces, likeL2(R), it is also the unique solution. In particular, the integral operator in (1.2)defines a group of isometries in the space L2(R).
It is natural to ask what happens if in the Schrodinger equation we complexifythe spatial variable and keep the time variable real? It is the goal of this article toanswer this question. To the best of our knowledge, it is the first time when such astudy is made. This will be done for the Schrodinger equation in Section 2 and for itshigher order version in Section 3. This topic was already developed in detail forcomplex heat and Laplace equations in [2,3], and complex wave and telegraphequations in [4]. Finally in [5], the Burgers and Black–Merton–Scholes equationswith real-time variable and complex spatial variable were also analyzed.
The complexification of the spatial variable in the Schrodinger equation is madeby two different methods which produce different equations: first, one complexifiesthe spatial variable in the corresponding formula (1.2) by replacing the usual sum ofvariables (translation) by an exponential product(rotation). We will then establishthat the resulting operator defines a (C0)-group of linear operators which is locallyequicontinuous on some locally convex space A(DR), for a discDR of radius R41.This method yields solutions that satisfy proper differential equations, and whichpreserve some geometric properties of the boundary function, such as, univalence,starlikeness, convexity and spirallikeness. Second, one directly complexifies thespatial variable in the corresponding evolution equation of (1.1), and then onesearches for analytic and non-analytic solutions for the resulting equation.
The plan of this article is as follows. Section 2 treats the case of the complexifiedSchrodinger equation, while Section 3 deals with the complexification of its higherorder version. For future study, we note that it would be plausible to seekcomparable results for regions other than the open unit discD and for otheroperators similar to the Laplacian D.
2. Schrodinger equation with complex spatial variable
For R41, let us define the open discDR¼ {z2C; jzj5R}. Let us consider the locallyconvex space
AðDRÞ ¼ f f:DR ! C; f is analytic on DRg,
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endowed with the countable family of seminorms generated by a sequence Rn%R,
Rn� 1, for all n2N, given by
k f kn ¼ supfj f ðzÞj; z2DRng:
It is well-known that, when the space A(DR) is endowed with the metric
dð f, gÞ ¼X1n¼0
1
2nk f� gkn
1þ k f� gkn,
we have that A(DR) is a Frechet space.For f2A(DR), we will complexify the integral solution (1.2), by replacing the
usual translations x� u2R in (1.2) with the rotations ze�iu2C. To this end, we
define
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
Z þ1�1
f ðze�iuÞeiu2=ð4tÞdu, z2DR, t2Rnf0g: ð2:1Þ
We aim to study this complex convolution integral (2.1) by showing that, for analytic
functions f2A(DR), the function u, defined by (2.1), satisfies a proper differential
equation in DR.
THEOREM 2.1 Let R41 and f2A(DR), that is, f ðzÞ ¼P1
k¼0 akzk, for all z2DR.
(i) For all t2R\{0}, u(t, z), defined by (2.1) is analytic in DR, and we have on DR:
uðt, zÞ ¼X1k¼0
ake�ik2tzk:
Also, we have u(0, z)¼ f(z), for all z2DR.(ii) For all z2Dr with 1� r5R and t2R, the following estimate holds:
juðt, zÞ � f ðzÞj �t2
2
X1k¼0
jakjk4rk þ jtj
X1k¼0
jakjk2rk,
whereP1
k¼0 jakjk4rk 51, from f(4)2A(DR).
(iii) For all z2Dr with 1� r5R and t, s2R, we have:
juðt, zÞ � uðs, zÞj � 2X1k¼0
jakjk2rkjt� sj:
(iv) Denoting T(t)( f )(z)¼ u(t, z), where u(t, z) is given by (2.1), then {T(t)}t2R is a
(C0)-group of linear operators, locally equicontinuous (that is equicontinuous
for t2 [�a, a], with some a40) on the locally convex space A(DR), and the
unique solution of the Cauchy problem
i@u
@tðt, zÞ þ
@2u
@’2ðt, zÞ ¼ 0, ðt, zÞ 2R�DR, z ¼ rei’, 05 r5R, z 6¼ 0, ð2:2Þ
uð0, zÞ ¼ f ðzÞ, z2DR, f2AðDRÞ: ð2:3Þ
Moreover, u(t, z) belongs to A(DR), for each fixed t.
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Proof (i) For fixed z2DR, we get f ðze�iuÞ ¼P1
k¼0 ake�ikuzk: Since jake
�ikuj ¼ jakj,
for all u2R, and sinceP1
k¼0 akzk is absolutely convergent, it follows thatP1
k¼0 ake�ikuzk is uniformly convergent with respect to u2R. This immediately
implies that the series can be integrated term by term, that is,
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzk
Z 1�1
e�ikueiu2=ð4tÞdu
� �
¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzk
Z 1�1
cos ku�u2
4t
� �duþ i
Z 1�1
sinu2
4t� ku
� �du
� �:
Suppose now that t40. Writing ku� u2
4t ¼ k2t��u�2kt2ffiffitp�2
and making the followingchange of variable v ¼ u�2kt
2ffiffitp , we easily deduceZ 1
�1
cos ku�u2
4t
� �du
¼ cosðk2tÞ
Z 1�1
cosu� 2kt
2ffiffitp
� �2" #
duþ sinðk2tÞ
Z 1�1
sinu� 2kt
2ffiffitp
� �2" #
du
¼ 2ffiffitp
cosðk2tÞ
Z 1�1
cosðv2Þdvþ 2ffiffitp
sinðk2tÞ
Z 1�1
sinðv2Þdv
¼ffiffiffiffiffiffiffi2�tp
cosðk2tÞ þ sinðk2tÞ
:
Here we have used the well-known values of the Fresnel integrals (see, e.g.[6, Chapter 7]), Z 1
�1
cosðv2Þdv ¼
Z 1�1
sinðv2Þdv ¼
ffiffiffiffiffiffi2�p
2:
Analogously, we haveZ 1�1
sinu2
4t� ku
� �du
¼ cosðk2tÞ
Z 1�1
sinu� 2kt
2ffiffitp
� �2" #
du� sinðk2tÞ
Z 1�1
cosu� 2kt
2ffiffitp
� �2" #
du
¼ffiffiffiffiffiffiffi2�tp
cosðk2tÞ � sinðk2tÞ
:
On account of the last two identities, we get
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzkffiffiffiffiffiffiffi2�tp
½e�ik2t þ ie�ik
2t�
¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzkð1þ iÞ
ffiffiffiffiffiffiffi2�tp
e�ik2t
¼X1k¼0
akzke�ik
2t,
where we have used the fact thatffiffiip¼ ei�=4 ¼ cosð�=4Þ þ i sinð�=4Þ ¼
ffiffi2p
2 ð1þ iÞ. Thisproves (i), for positive times.
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Suppose now that t50. In this case, taking into account that cos(�) (and sin(�),
respectively) is an even (odd, respectively) function, by making a simple change of
variable v ¼ u�2kt2ffiffiffiffi�tp , we can getZ 1
�1
cos ku�u2
4t
� �du
¼ cosðk2tÞ
Z 1�1
cosu� 2kt
2ffiffiffiffiffiffi�tp
� �2" #
du� sinðk2tÞ
Z 1�1
sinu� 2kt
2ffiffiffiffiffiffi�tp
� �2" #
du
¼ 2ffiffiffiffiffiffi�tp
cosðk2tÞ
Z 1�1
cosðv2Þdv� 2ffiffiffiffiffiffi�tp
sinðk2tÞ
Z 1�1
sinðv2Þdv
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ð�tÞ
pcosðk2tÞ � sinðk2tÞ
:
Analogously, we haveZ 1�1
sinu2
4t� ku
� �du
¼ � cosðk2tÞ
Z 1�1
sinu� 2kt
2ffiffiffiffiffiffi�tp
� �2" #
du� sinðk2tÞ
Z 1�1
cosu� 2kt
2ffiffiffiffiffiffi�tp
� �2" #
du
¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ð�tÞ
p� cosðk2tÞ � sinðk2tÞ
:
Thus, we get
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzkffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ð�tÞ
p½e�ik
2t � ie�ik2t�
¼1ffiffiffiffiffiffiffiffiffi4�itp
X1k¼0
akzkð1� iÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ð�tÞ
pe�ik
2t
¼ð1� iÞ
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2�ð�tÞ
p2ffiffiffi�p ffiffiffiffiffiffiffiffiffi
ð�iÞp ffiffiffiffiffiffiffiffiffi
ð�tÞp ¼
X1k¼0
akzke�ik
2t,
where we have used the formulaffiffiffiffiffiffi�ip¼ e
12ði argð�iÞÞ ¼ e�i�=4 ¼ cosð�=4Þ�
i sinð�=4Þ ¼ffiffi2p
2 ð1� iÞ. This proves (i), for negative times.(ii) Let jzj � r. We obtain
juðt, zÞ � f ðzÞj ¼X1k¼0
akzk½e�ik
2t � 1�
����������
¼X1k¼0
akzk½�2 sin2ðk2t=2Þ þ i sinðk2t=2Þ � cosðk2t=2Þ�
����������
�X1k¼0
jakjrkj2 sin2ðk2t=2Þj þ
X1k¼0
jakjrkj2 sinðk2t=2Þj
�t2
2
X1k¼0
jakjk4rk þ jtj
X1k¼0
jakjk2rk,
since jsin(x)j � jxj, for all x2R.
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(iii) We have
juðt, zÞ � uðt, sÞj ¼X1k¼0
akzk½e�ik
2t � e�ik2s�
����������
¼X1k¼0
½cosðk2tÞ � cosðk2sÞ � iðsinðk2tÞ � sinðk2sÞÞ�
¼X1k¼0
akzk �2 sin k2ðtþ sÞ=2
� �sin k2ðt� sÞ=2� �������
�2i sin k2ðt� sÞ=2� �
cos k2ðtþ sÞ=2� � ��
� 4X1k¼0
jakjrk sin
k2ðt� sÞ
2
� ��������� � 2
X1k¼0
jakjk2rkjt� sj:
(iv) From (i), it is immediate that T(tþ s)¼T(t)(T(s)), for all t, s2R; using (ii),we easily get that T(0)¼ I. The strong continuity of T(�) follows from (iii). Inparticular, according to, e.g. [7, p. 232], (T(t))t2R is a C0-group of linear operators onA(DR). Since A(DR) is a Frechet space, it follows that (T(t))t2R is also locallyequicontinuous (see, e.g. [8,9]). For f2A(DR) such that f ðzÞ ¼
P1k¼0 akz
k, z2DR,z¼ rei’, z 6¼ 0, r5R, on account of (i), we can now write
TðtÞð f ÞðzÞ ¼X1k¼0
ake�ik2trkeki’:
Since this series is uniformly convergent in any compact disc included in DR, it can bedifferentiated term by term, with respect to t and ’. We then easily obtain thatT(t)( f )(z) satisfies (2.2) for all z2DR. Note that the generator of this group T(t),t2R, can be computed explicitly,
Gð f ÞðzÞ ¼d½TðtÞð f ÞðzÞ�
dtjt¼0 ¼ �i
X1k¼0
k2akzk ¼ i
@2TðtÞð f ÞðzÞ
@’2:
Also, using the same series representation, it is easy to check that
Tð0Þð f ÞðzÞ ¼ f ðzÞ, z2DR:
Finally, reasoning now exactly as in [10, Theorem 1.2, p. 83], it follows thatu(t, z)¼T(t)( f )(z) is the unique solution of the Cauchy problem (2.2)–(2.3). We alsonote that, in (2.2) we must take z 6¼ 0 simply because z¼ 0 has no polarrepresentation, that is, z¼ 0 cannot be represented as function of ’. This completesthe proof of the theorem. g
The next result shows that the solution of problem (2.2)–(2.3) preserves someinteresting geometric properties on the unit disc, of the boundary function. In orderto state these properties more precisely, we need to consider some definitions for thefollowing well-known subclasses of univalent analytic functions:
S�ðD1Þ ¼
(f2AðD1Þ; f ð0Þ ¼ f 0ð0Þ � 1 ¼ 0,Re
zf 0ðzÞ
f ðzÞ
� �4 0, for all z2D1
),
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S��ðD1Þ ¼
(f2AðD1Þ; fð0Þ¼ f 0ð0Þ�1¼ 0,Re ei�
zf 0ðzÞ
fðzÞ
� �40, for all z2D1
), j�j5�=2,
KðD1Þ ¼
(f2AðD1Þ; f ð0Þ ¼ f 0ð0Þ � 1 ¼ 0,Re
zf 00ðzÞ
f 0ðzÞþ 1
� �4 0, for all z2D1
),
S2ðD1Þ ¼
(f2AðD1Þ; f ðzÞ ¼
X1k¼1
akzk, ja1j �
X1k¼2
jakj
)
and
S1ðD1Þ ¼
�f2AðD1Þ, f ðzÞ ¼ zþ
X1k¼2
akzk,X1k¼2
kjakj � 1
�:
We recall that S�(D1), S��ðD1Þ and K(D1) are called the classes of normalized starlike,
spirallike and convex functions on D1, respectively. According to, e.g. [11, p. 97,
Exercise 4.9.1], if f2S1(D1), then jzf 00(z)/f(z)� 1j51 8z2D1, and therefore f is
starlike (and univalent) on D1. Also, if f2S2(D1), then f is starlike in D1 [12].
THEOREM 2.2 Let u(t, z) be the unique solution of the Cauchy problem (2.2)–(2.3),
given by (2.1). Let f2A(DR) with R41.
(i) As function of z, u(t, z) has the following properties: if f2S1(D1), then
eitu(t, z)2S1(D1), for all t2R\{0} and, if f2S2(D1), then u(t, z)2S2(D1),
for all t2R.(ii) For all t2 [�tf, tf]\{0} with sufficiently small tf40 (depending on f ), the
solution u(t, z) preserves, as function of z, the starlikeness, convexity and
spirallikeness of the boundary function f in D1.
Proof (i) Let t2R\{0} and assume f2S1(D1) such that f ðzÞ ¼ zþP1
k¼2 akzk, withP1
k¼2 kjakj � 1. First, it follows from Theorem 2.1, (i), that eitu(t, 0)¼ 0 and
eit½uðt, zÞ� 0z¼0 ¼ 1. Moreover, we can write
eituðt, zÞ ¼X1k¼0
akeite�ik
2tzk,
and
X1k¼2
kjake�ik2teitj ¼
X1k¼2
kjakj � 1,
which proves that u(t, z)2S1(D1), as function of z.
Now, let f2S2(D1). It follows that a0¼ 0 andP1
k¼2 jakj � ja1j. By Theorem 2.1(i),
we get
uðt, zÞ ¼X1k¼0
ake�ik2tzk ¼
X1k¼1
ake�ik2tzk,
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which implies
X1k¼2
jakjje�ik2tj �
X1k¼2
jakj � ja1j ¼ ja1jje�itj,
and, therefore, u2S2(D1).(ii) From Theorem 2.1(i) and (ii), it is readily seen that u(t, z) is analytic in DR
and u(t, z) converges to f(z), as t! 0, uniformly in D1. This convergence together
with an application of a well-known Weierstrass’s result yield that [u(t, z)]0 ! f 0(z)
and [u(t, z)]00 ! f 00(z), uniformly in D1, as t! 0. In what follows, from now on,
Pt( f )(z) will denote u(t, z)/b1(t), where b1(t)¼ a1e�it is the coefficient of z, in the
Taylor series representation of the analytic function u(t, z). We will also denote by
P 0t ð f ÞðzÞ and P 00t ð f ÞðzÞ, the corresponding first- and second-order derivatives of
Pt( f )(z) with respect to z.
If f(0)¼ f 0(0)� 1¼ 0, it is not difficult to observe that b1(t) 6¼ 0, Pt( f )(0)¼ f(0)/
b1(t)¼ 0, P 0t ð f Þð0Þ ¼ uðt, 0Þ 0=b1ðtÞ ¼ 1 and b1(t) converges to f 0(0)¼ 1 as t! 0. This
obviously implies that, for t! 0, we have Pt( f )(z)! f(z), P 0t ð f ÞðzÞ ! f 0ðzÞ and
P 00t ð f ÞðzÞ ! f 00ðzÞ, uniformly in D1. The rest of the proof is identical to the proof of
[2, Theorem 2.2]. g
Remark 2.3 (a) Taking into account the fact that @@’ ¼
@@z �
@z@’ ¼
@@z � ðizÞ and
@2
@’2¼ �z @
@z� z2 @2
@z2, Equation (2.2) is equivalent to
i@u
@tðt, zÞ ¼ z
@u
@zðt, zÞ þ z2
@2u
@z2ðt, zÞ:
(b) Since the constant eit does not influence the starlikeness and the univalence of
u(t, z) in the above result, it follows that for f2S1(D1) and t2R\{0}, the solution of
(2.2)–(2.3) is starlike (and univalent) in D1, as function of z. Moreover, in view of the
Riemann mapping theorem (see, e.g. [13, Chapter I]), all the geometric properties
concerning the solution of problem (2.2)–(2.3), stated by Theorem 2.2, can be readily
translated into statements which hold in any disc of radius r with 1� r5R.
In what follows, the Schrodinger equation is complexified, by replacing x2R
with z2C. More precisely, we aim to study the following initial value problem for a
Schrodinger-type equation
i@u
@tðt, zÞ þ
@2u
@z2ðt, zÞ ¼ 0,
uð0, zÞ ¼ f ðzÞ:
8<: ð2:4Þ
We will first consider the case when f is analytic and our goal is to search for the
analytic solutions u(t, z), as functions of z, for any t2R. In this case, by taking
u(t, z)¼ u1(t, x, y)þ iu2(t, x, y), for z¼ xþ iy, we have @u@z ¼
@u1@x þ i @u2@x and
@2u@z2¼ @2u1
@x2þ i @
2u2@x2
; hence, the initial value problem (2.4) reduces to solving
i@u1@tþ i
@u2@t
� �þ@2u1@x2þ i
@2u2@x2¼ 0,
uð0, zÞ ¼ f ðzÞ:
8<: ð2:5Þ
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Now, setting f(z)¼ f1(x, y)þ if2(x, y), it is immediate that (2.5) is equivalent to the
following system:
@u1@t¼ �
@2u2@x2
,@u2@t¼@2u1@x2
, u1ð0,x, yÞ ¼ f1ðx, yÞ, u2ð0, x, yÞ ¼ f2ðx, yÞ: ð2:6Þ
Since, by hypothesis the couple ( f1, f2) satisfies the Cauchy–Riemann equations with
respect to x and y, we can prove the following result.
THEOREM 2.4 For d40, denote by Sd the strip {z¼ xþ iy; x, y2R, jyj � d}, and
define as L2xðSdÞ, the class of all functions f : Sd!C, f-analytic on Sd, such that if
f(z)¼ f1(x, y)þ if2(x, y), z¼xþ iy2Sd, then f1(x, y), f2(x, y),@f1@x ðx, yÞ,
@f1@y ðx, yÞ,
@f2@x ðx, yÞ,
@f2@y ðx, yÞ are in L2(R), as functions of x, for any fixed jyj � d. Then, for
f2L2xðSdÞ,
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
Z þ1�1
f ðz� uÞeiu2=ð4tÞdu
is the unique (analytic, as function of z) solution of (2.4), for any t2R\{0}.
Proof For f(z)¼ f1(x, y)þ if2(x, y), we can write
uðt, zÞ ¼1ffiffiffiffiffiffiffiffiffi4�itp
Z þ1�1
f1ðx� u, yÞeiu2=ð4tÞduþ i
1ffiffiffiffiffiffiffiffiffi4�itp
Z þ1�1
f2ðx� u, yÞeiu2=ð4tÞdu
:¼ u1ðt,x, yÞ þ iu2ðt, x, yÞ,
for all z¼ xþ iy2Sd. Translating the initial condition of (2.5) into the initial
conditions of (2.6) for both u1 and u2, and employing a well-known result concerning
problem (1.1), we deduce that for each fixed, arbitrary y, with jyj � d, u1 and u2 are
the unique solutions of the following Cauchy problems
i@u1@tðt, x, yÞ þ
@2u1@x2ðt,x, yÞ ¼ 0, ðt, xÞ 2R� R,
u1ð0,x, yÞ ¼ f1ðx, yÞ, x2R,
8<:
and
i@u2@tðt, xÞ þ
@2u2@x2ðt,xÞ ¼ 0, ðt, xÞ 2R� R,
u2ð0, x, yÞ ¼ f2ðx, yÞ, x2R:
8<:
Thus, it is easy to check that u¼ u1þ iu2 satisfies (2.5), which is equivalent to saying
that u is the unique solution of (2.4).It remains to show that u(�, z) is analytic on Sd. This immediately follows from the
analyticity of f¼ f1þ if2 and the continuity on Sd, of the partial derivatives@f1@x ðx, yÞ,
@f1@y ðx, yÞ,
@f2@x ðx, yÞ,
@f2@y ðx, yÞ. The fact that f2L2
xðSdÞ assures, in addition, that u1, u2exist and that we can differentiate u1 and u2 under the sign of the improper
integrals (for such a criterion, see, e.g. [14, p. 293, Exercise 607]). By the above
considerations, it is easy to check that u1 and u2 satisfy the Cauchy–Riemann
conditions and that they have continuous partial derivatives of first order. This
proves the theorem. g
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Remark 2.5 Here, we shall use the opportunity to correct some mistakes concerning
some hypotheses of some results in [2]. A much closer look at the main formulae in
[2, Theorems 2.4 and 3.4], indicates that the boundary function f(z) (and
consequently the solution u(�, z)) need to be defined in a strip that contains the
x-axis and not only in the closed unit disc D as they were in [2, Theorems 2.4
and 3.4]. Moreover, we must also make some corrections in the statements of
[2, Theorems 2.7 and 3.6], where the non-analytic boundary function f(z) (and,
consequently, the corresponding non-analytic solution u(�, z)) has to be defined on
the whole C and not only in the closed unit disc. The statements of these results
remain true even for these minor modifications.
Another natural way of complexification of the Schrodinger equation would be
to replace @2
@x2by @2
@z2þ @2
@z2(since @
@z ¼ 0, for analytic functions). In this case, by
definition we have
@u
@z¼
1
2
@u
@xþ
1
2i
@u
@y,
@u
@z¼
1
2
@u
@x�
1
2i
@u
@y,
and these identities imply
@u
@x¼@u
@zþ@u
@z,
@u
@y¼ i
@u
@z�@u
@z
� �:
We may, formally, write @x ¼ @z þ @z, @y ¼ i @z � @zð Þ. Therefore, we have
@2u
@x2¼@
@x
@u
@x
� �¼ ð@z þ @zÞ
2
and
@2u
@y2¼@
@y
@u
@y
� �¼ ½ið@z � @zÞ�
2,
which immediately implies that
@2u
@z2þ@2u
@z2
� �¼
1
2
@2u
@x2�@2u
@y2
� �:
Consequently, in this case we obtain the following differential equation:
i@u
@tþ1
2
@2u
@x2�@2u
@y2
� �¼ 0:
Replacing now, u(t, z)¼ u1(t, x, y)þ iu2(t, x, y) and f(z)¼ f1(x, y)þ if2(x, y) (assuming
that f1, f22C2), we may obtain the following coupled system of partial differential
equations:
@u1@t¼
1
2
@2u2@y2�@2u2@x2
� �,
@u2@t¼
1
2
@2u1@x2�@2u1@y2
� �, ð2:7Þ
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subject to the initial conditions:
u1ð0, x, yÞ ¼ f1ðx, yÞ, u2ð0,x, yÞ ¼ f2ðx, yÞ, ð2:8Þ
where x2þ y2� 1.Concerning the solutions of (2.7)–(2.8), we present the following result.
THEOREM 2.6 Assuming that f1, f22C1 have all their partial derivatives bounded by
the same constant, a solution U ¼ u1u2
� �of the system (2.7)–(2.8) is given by
U(t)¼ etA(U0), where U0 ¼f1f2
� �, and A is the matrix A¼ (ai, j), i, j¼ 1, 2, with
a1,1 ¼ a2,2 ¼ 0, a1,2 ¼ DH :¼1
2
@2
@y2�@2
@x2
� �, a2,1 ¼ �DH
and
A2n ¼ ðai, jÞ, i, j ¼ 1, 2, a1,2 ¼ a2,1 ¼ 0, a1,1 ¼ ð�1ÞnD2n
H , a2,2 ¼ a1,1, n � 1,
A2n�1 ¼ ðai, jÞ, i, j ¼ 1, 2, a1,1 ¼ a2,2 ¼ 0, a1,2 ¼ ð�1Þn�1D2n�1
H , n � 1,
a2,1 ¼ �a1,2:
Proof The system (2.7)–(2.8) can be rewritten as
dU
dt¼ AU, Uð0Þ ¼
f1f2
� �:¼ U0:
The solution has the form U(t)¼ etA(U0). Finally, the powers An, n2N can be
obtained by simple calculations. g
Now, since @u@z ¼ 0 for any analytic function u, the analytic case can also be
generalized, by replacing @2
@x2with @2
@z2� @2
@z2. In this case, following a similar calculation
as above, we easily deduce
@2
@z2�@2
@z2¼
1
i
@2
@x@y: ð2:9Þ
This immediately leads to the following uncoupled system of evolution equations:
@u1@t¼@2u1@x@y
,@u2@t¼@2u2@x@y
, ð2:10Þ
u1ð0, x, yÞ ¼ f1ðx, yÞ, u2ð0,x, yÞ ¼ f2ðx, yÞ, ð2:11Þ
where x2þ y2� 1.In this case, we have the following result.
THEOREM 2.7 If f1 and f2 have partial derivatives of any order, bounded by the
same constant, then a solution of the problem (2.10)–(2.11) is given by
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uj(t,x, y)¼ etL( fj)(x, y), j¼ 1, 2, where LðF Þðx, yÞ :¼ @2F@x@y ðx, yÞ, that is,
ujðt, x, yÞ ¼X1k¼0
tk
k!Lkð fjÞðx, yÞ,
for j¼ 1, 2, all t2R and (x, y)2R2, with x2þ y2� 1.
Proof First, note that the two equations in (2.10) are not coupled. If f1 and f2 have
partial derivatives of any order, bounded by the same constant, then the solutions of
(2.10)–(2.11) are given by uj(t, x, y)¼ etL( fj)(x, y), j¼ 1, 2. More precisely, we have
ujðt, x, yÞ ¼X1k¼0
tk
k!Lkð fjÞðx, yÞ,
for j¼ 1, 2, all t2R and (x, y)2R2, where x2þ y2� 1. Note that by hypothesis, it
follows that the series representing the solutions uj(t,x, y) are absolutely and
uniformly convergent with respect to x2þ y2� 1 and t2 [�c, c], c40 arbitrary. g
A third kind of modification of the spatial variable in the non-analytic case
would be to replace @2
@x2in the Schrodinger equation of real spatial variable, with 4 @2
@z@z.
By means of calculations, we can easily deduce the relation
@2u
@z@z¼
1
4
@2u
@x2þ@2u
@y2
� �: ð2:12Þ
The decomposition uðt, z, zÞ ¼ u1ðt,x, yÞ þ iu2ðt, x, yÞ, z¼ xþ iy, leads to the follow-
ing coupled systems:
@u1@t¼ �Du2, u1ð0, x, yÞ ¼ f1ðx, yÞ, ð2:13Þ
@u2@t¼ Du1, u2ð0, x, yÞ ¼ f2ðx, yÞ: ð2:14Þ
Reasoning exactly as in the proof of Theorem 2.6, the following is also
immediate.
THEOREM 2.8 Assuming that f1, f22C1 have all their partial derivatives bounded by
the same constant, a solution U ¼�u1u2
�of (2.13)–(2.14) is given by U(t)¼ etA(U0),
where U0 ¼�f1f2
�, A is the matrix A¼ (ai, j), i, j¼ 1, 2, with
a1,1 ¼ a2,2 ¼ 0, a1,2 ¼ �D ¼ �@2
@x2þ@2
@y2
� �, a2,1 ¼ D
and
A2n ¼ ðai, jÞ, i, j ¼ 1, 2, a1,2 ¼ a2,1 ¼ 0, a1,1 ¼ ð�1Þn½�D�2n, a2,2 ¼ a1,1, n � 1,
A2n�1 ¼ ðai, jÞ, i, j ¼ 1, 2, a1,1 ¼ a2,2 ¼ 0, a1,2 ¼ ð�1Þn�1½�D�2n�1, n � 1,
a2,1 ¼ �a1,2:
426 C. G. Gal et al.
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Proof The system (2.13)–(2.14) can be rewritten as
dU
dt¼ AU, Uð0Þ ¼
f1f2
� �:¼ U0:
The solution has the form U(t)¼ etA(U0). Finally, the powers An, n2N, can be
obtained from simple calculations. g
Remark 2.9 The formula U(t)¼ etA(U0), in the statements of Theorem 2.7 (for
L¼DH) and Theorem 2.8 (for L¼�D) can be used to deduce more explicit
representations of the solutions U ¼�u1u2
�. In particular, we have
u1ðt, x, yÞ
u2ðt, x, yÞ
� �¼X1k¼0
tk
k!Ak f1ðx, yÞ
f2ðx, yÞ
� �,
where
Ak f1ðx, yÞ
f2ðx, yÞ
� �¼ð�1Þk=2Lkð f1Þðx, yÞ
ð�1Þk=2Lkð f2Þðx, yÞ
!,
for k-even number and
Ak f1ðx, yÞ
f2ðx, yÞ
� �¼ð�1Þðk�1Þ=2Lkð f2Þðx, yÞ
ð�1Þðk�1Þ=2Lkð f1Þðx, yÞ
!,
for k-odd number. Simple calculations show that
u1ðt, x, yÞ ¼X1
k¼0;k-even
tk
k!ð�1Þk=2Lkð f1Þðx, yÞ þ
X1k¼0;k-odd
tk
k!ð�1Þðk�1Þ=2Lkð f2Þðx, yÞ
and
u2ðt, x, yÞ ¼X1
k¼0;k-even
tk
k!ð�1Þk=2Lkð f2Þðx, yÞ þ
X1k¼0;k-odd
tk
k!ð�1Þðk�1Þ=2Lkð f1Þðx, yÞ:
3. Higher order Schrodinger type equations with complex spatial variables
The representation (2.1) of the solution in Theorem 2.1, (i), suggests attaching to
f2A(DR) and p2N, the more general transformation (well-defined for all jzj5R)
T ½ p�ðtÞð f ÞðzÞ ¼X1k¼0
ake�ikptzk ¼
X1k¼0
ake�ikptrkei’k, z ¼ rei’, r5R, t2R: ð3:1Þ
If p¼ 2, then clearly T [2](t)( f )(z) is the same as the complex convolution integral
given by (2.1). Following a similar strategy that was applied to higher order heat
equations in [3], a simple calculation shows that if p¼ 1, then T [p](t)( f )(z) satisfies
Complex Variables and Elliptic Equations 427
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the differential equation
i@T ½ p�ðtÞð f ÞðzÞ
@t�@T ½ p�ðtÞð f ÞðzÞ
@’¼ 0,
while for p� 2, T [p](t)( f )(z) satisfies
i@T ½ p�ðtÞð f ÞðzÞ
@t�@pT ½ p�ðtÞð f ÞðzÞ
@’p¼ 0, if p ¼ 2ð2j Þ, j ¼ 1, 2, . . . ,
and we can ask
i@T ½ p�ðtÞð f ÞðzÞ
@tþ@pT ½ p�ðtÞð f ÞðzÞ
@’p¼ 0, if p ¼ 2ð2jþ 1Þ, j ¼ 0, 1, . . . , :
Denoting p¼ 2q, it follows that u2q(t, z)¼T [2q](t)( f )(z) satisfies
i@u2qðt, zÞ
@tþ ð�1Þqþ1
@2qu2qðt, zÞ
@’2q¼ 0, ðt, zÞ 2 0, þ1ð Þ �DR, z ¼ rei’, z 6¼ 0 ð3:2Þ
subject to the initial condition
u2qð0, zÞ ¼ f ðzÞ, z2DR, f2AðDRÞ: ð3:3Þ
Our goal now is to study (3.2)–(3.3).
Remark 3.1 The initial value problem (3.2)–(3.3) suggests that one may also
consider the Cauchy problem for the following higher order Schrodinger equation
with real spatial variable x, as follows:
i@u2qðt, xÞ
@tþ ð�1Þqþ1
@2qu2qðt, xÞ
@x2q¼ 0, ðt, xÞ 2 0,þ1ð Þ � R, ð3:4Þ
with initial condition
u2qð0, xÞ ¼ f ðxÞ, x2R, ð3:5Þ
where f and u¼ u2q, q� 2, are complex valued. Fourth-order Schrodinger equations
(q¼ 2) have been introduced by Karpman [15] to study the role of fourth-order
dispersion terms in the propagation of intense laser beams in a bulk medium. Sharp
dispersive estimates for the biharmonic (i.e. q¼ 2) Schrodinger operator in (3.4) have
recently been obtained in [16]. The study of (3.4)–(3.5) for arbitrary q� 2 remains an
open question.
In the remaining part of this article, we will only deal with (3.2)–(3.3). Following
our work in [3], and arguing as in the proof of Theorem 2.1, the main result of this
section is as follows.
428 C. G. Gal et al.
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THEOREM 3.2 Let q2N, q� 2, R41 and f2A(DR). Let u2q(t, z)¼T [2q](t)( f )(z) begiven by (3.1).
(I) For all z2Dr with 1� r5R and t2R, the following estimate holds:
ju2qðt, zÞ � f ðzÞj �t2
2
X1k¼0
jakjk4qrk þ jtj
X1k¼0
jakjk2qrk,
whereP1
k¼0 jakjk4qrk 51, from f (4q)2A(DR).
(II) For all z2Dr with 1� r5R and t, s2R, we have:
ju2qðt, zÞ � u2qðs, zÞj � 2X1k¼0
jakjk2qrkjt� sj:
(III) (T [2q](t), t2R) is a (C0)-group of linear operators, locally equicontinuous onthe locally convex space A(DR) and the unique solution (that belongs toA(DR), for each fixed t) of the Cauchy problem (3.2)–(3.3).
Proof (I) Let jzj � r. We have
ju2qðt, zÞ � f ðzÞj ¼X1k¼0
akzk½e�ik
2qt � 1�
����������
¼X1k¼0
akzk½�2 sin2ðk2qt=2Þ þ i sinðk2qt=2Þ � cosðk2qt=2Þ�
����������
�X1k¼0
jakjrkj2 sin2ðk2qt=2Þj þ
X1k¼0
jakjrkj2 sinðk2qt=2Þj
�t2
2
X1k¼0
jakjk4qrk þ jtj
X1k¼0
jakjk2qrk,
where we have used the well-known inequality jsin(x)j � jxj for all x2R.(II) We have
ju2qðt, zÞ � u2qðt, sÞj ¼X1k¼0
akzk½e�ik
2qt � e�ik2qs�
����������
¼X1k¼0
½cosðk2qtÞ � cosðk2qsÞ � iðsinðk2qtÞ � sinðk2qsÞÞ�
����������
¼X1k¼0
akzk �2 sin
k2qðtþ sÞ
2
� �sin
k2qðt� sÞ
2
� ��������2i sin
k2qðt� sÞ
2
� �cos
k2qðtþ sÞ
2
� ������� 4
X1k¼0
jakjrk sin
k2qðt� sÞ
2
� ��������� � 2
X1k¼0
jakjk2qrkjt� sj:
(III) It is immediate that T [2q](tþ s)¼T [2q](t)(T [2q](s)), for all t, s2R. From (ii),we have T [2q](0)¼ I; the strong continuity of T [2q](�) follows from (iii). Therefore,according to, e.g. [7, p. 232], (T [2q](t))t2R is a C0-group of linear operators. Moreover,
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since A(DR) is a Frechet space, (T [2q](t))t2R is locally equicontinuous on A(DR) [8,9].
Finally, reasoning exactly as in the proof of Theorem 2.1, (iv), it follows that
u(t, z)¼T [2q](t)( f )(z) is the unique solution of (3.2)–(3.3). g
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