Group of Weakly Continuous Operators Associated to a Generalized Schrödinger Type Homogeneous Model ()
1. Introduction
First, we begin by commenting that [1] has proven the existence of a solution of the Schrodinger type equation in the Hilbert space
. Also in [1] a family of bounded operators is introduced in the Hilbert space
and it is proved that forms a unitary group. For the justification of the model, we suggest reviewing the references cited in [1] . Thus motivated by these ideas we will solve the problem (
) in the topological dual of P:
, which is not a Banach space.
In this article, we will prove the existence and uniqueness of the solution of (
) in
. Furthermore, we will demonstrate that the solution depends continuously with respect to the initial data in
, considering the weak convergence in
. And we will prove that the introduced family of operators forms a group of weakly continuous linear operators. Thus, with this family we will rewrite our result in a fine version.
We also want to highlight the wealth of information from Terence [2] , Kato [3] , Linares and Ponce [4] . We can also cite works of existence solution using Semigroup theory by Liu-Zheng [5] , Muñoz [6] , Pazy [7] , Santiago [8] [9] and Raposo [10] .
Our article is organized as follows. In Section 2, we indicate the methodology used and cite the references used. In Section 3, we put the results obtained from our study. This section is divided into three subsections. Thus, in Subsection 3.1 we prove that the problem (
) has a unique solution and also demonstrate that the solution depends continuously with respect to the initial data. In Subsection 3.2, we introduce families of weakly continuous linear operators in
that manage to form a group. In Subsection 3.3 we improve Theorem 3.1.
Finally, in Section 4 we give the conclusions of this study.
2. Methodology
As theoretical framework in this article we use the references [1] [11] [12] [13] and [14] for Fourier Theory in periodic distributional space, periodic Sobolev spaces, topological vector spaces, weakly continuous operators, group of operators and existence of solution of a distributional differential equation.
We will use this theory in the analysis of the existence and continuous dependence of the solution of (
), carrying out a series of calculations and approximations in the process.
Thus, below we will briefly give some definitions necessary for the development (understanding) of this work. It is suggested, for an in-depth study, to refer to the cited references.
Let be
that is, the space of the functions
infinitely differentiable and periodic with period 2π. It’s known that this space is a complete metric space.
Also,
That is,
is the topological dual of P.
is known as the space of periodic distributions.
We want to summarize the properties of
with the following diagram:
where the inclusions are continuous with dense image,
is the space of Rapidly Decreasing sequences (R.D.), defined by
and
is the space of Slowly Growing sequences (S.G.), defined by
3. Main Results
The presentation of the results obtained has been organized in subsections and is as follows.
3.1. Solution of the Schrödinger Equation (
)
In this subsection we will study the existence of a solution to the problem (
) and the continuous dependence of the solution with respect to the initial data in
.
Theorem 3.1 Let
,
, m even not multiple of four and the distributional problem
then (
) has a unique solution
. Furthermore, the solution depends continuously on the initial data. That is, given
such that
implies
,
, where
is solution of (
) with initial data
and u is solution of (
) with initial data f.
Proof.- We have organized the proof as follows.
1) Suppose there exists
satisfying (
); this will allow us to obtain the explicit form of u. Then taking the Fourier transform to the equation
we get
which for each
is an ODE with initial data
.
Thus, we propose an uncoupled system of homogeneous first-order ordinary differential equations
and we get
from where we obtain the explicit expression of u, candidate for solution:
(1)
(2)
Since
then
. Thus, we affirm that
(3)
Indeed, let
, since
then satisfies:
,
such that
,
, using this we get
Then,
If we define
(4)
we have that
,
, since we apply the inverse Fourier transform to
.
2) We will prove that u defined in (4) is solution of (
) and
.
Evaluating (2) at
, we obtain
Also, the following statements are verified.
a)
in
,
. That is, we will prove that the following equality
is satisfied, for all
.
Indeed, let
,
and
, we denote
Thus, we get
(5)
Let
, we have
(6)
Taking norm to equality (6) we obtain
(7)
That is, from (7) we get
(8)
Note that (8) is valid for
.
Using the inequality (8) and that
we obtain
since
.
Using the Weierstrass M-Test, the series
is absolute and uniformly convergent. Then we can take limit and get
(9)
Using (9) and that
for
,
, we have
(10)
Therefore,
That is,
b)
. That is, we will prove that
In effect, let
and
, we will prove that
We know that if
then
. Using (5) we have
Let
, from (8) we get
(11)
Using (11) and that
we obtain
since
.
Using the Weierstrass M-Test we conclude that the series
converges absolute and uniformly. Then it is possible to take limit and obtain
Since
was taken arbitrarily, then we can conclude that
c)
. That is, we will prove that
In effect, let
and
, using item a) we have
(12)
when
, since item b) is valid with
for
.
From b) and c) we have that
.
3) Now, we will prove that the solution depends continuously respect to initial data. That is, if
we will prove that:
We know that if
then
, that is
(13)
For
fixed and arbitrary, we want to prove that
Thus, let
be fixed and
, using the generalized Parseval identity, we obtain the following equalities:
(14)
(15)
From (14) and (15) we obtain:
when
, since
and (13) holds.
Corollary 3.1 The unique solution of (
) is
where
,
.
3.2. Group of Operators in
In this subsection, we will introduce families of operators
in
, with
,
and m even not multiple of four; and we will prove that these operators are continuous in the weak sense. That is,
is continuous from
to
with the weak topology of
, which we will call the weakly continuous operator.
Furthermore, we will prove that
satisfies the group properties.
For simplicity, we will denote this family of operators by
.
Theorem 3.2 Let
, we define:
then the following statements are satisfied:
1)
.
2)
is
—linear and weakly continuous
. That is, for every
, if
then
.
3)
,
.
4)
when
,
.
That is, for each
fixed, the following is satisfied
Proof.- Let
then
. Then, from (3) we have
taking the inverse Fourier transform, we obtain
That is,
is well defined for all
.
1) We easily obtain:
2) Let
, we will prove that
is
-linear. In effect, let
and
, we have
Now, for
we will prove that
is weakly continuous. That is, if
then we will prove that
. Note that the case
is obvious.
We know that if
then
, that is,
That is,
(16)
We want to prove that:
Thus, let
fixed and
, using the generalized Parseval identity, we obtain the following equalities
(17)
(18)
From (17) and (18) we get
when
, since
and (16) holds, that is
when
.
3) Let
, we will prove that
. In effect, let
,
(19)
Since
, using (3) we have that
(20)
Then, taking the inverse Fourier transform, we get:
Thus, we define:
That is,
(21)
Taking the Fourier transform to
we get:
that is,
(22)
Using (22) in (19) and from (21) we have:
So we have proven,
(23)
If
or
then equality (23) is also true, with this we conclude the proof of
(24)
4) Let
, we will prove that:
That is, we will prove that
In effect, for
and
, we have
(25)
Since
, from (8) we get
(26)
From (26) we obtain
(27)
From (27) with
, we have
(28)
Then using (28) and that
, we obtain
since
.
Using the Weierstrass M-Test we conclude that the
series converges absolute and uniformly. So,
Thus, we have proved
Theorem 3.3 For each
fixed and the family of operators
from Theorem 3.2, then the application
is continuous in
. That is,
(29)
(is the continuity at t).
That is, (29) tell us that for each
fixed, the following is satisfied
And if
, we have the continuity of M at 0, which is item 4) of Theorem 3.2.
Proof.- Let
, arbitrary fixed and
then
, using item 4) of Theorem 3.2, we have that
when
. That is,
where we use item 3) of Theorem 3.2.
Remark 3.1 The results obtain in Theorems 3.2 and 3.3 are also valid for the family of operators
, defined as
for
. Its proof is similar.
3.3. Version of Theorem 3.1 Using the Family
We improve the statement of theorem 3.1, using a family of weakly continuous Operators
.
Theorem 3.4 Let
and the family of operators
from Theorem 3.2, defining
,
, then
is the unique solution of (
). Furthermore, u continuously depends on f. That is, given
with
implies
,
, where
,
(that is,
is a solution of (
) with initial data
).
Proof.- It is analogous to the proof of Theorem 3.1.
Corollary 3.2 Let
be fixed and the family of operators
from Theorem 3.4, then
,
and the mapping
is continuous at
. That is,
(30)
(30) tells us that for each
fixed, it holds:
Proof.- Indeed,
when
, due to Theorem 3.3 with
, for
.
Corollary 3.3 Let
be fixed and the family of operators
from Theorem 3.4, then the solution of (
):
,
, satisfies
.
Proof.- It comes out as a consequence of Corollary 3.2.
4. Conclusions
In our study of the generalized Schrödinger type homogeneous model in the periodic distributional space
, we have obtained the following results:
1) We prove the existence, uniqueness of the solution of the problem (
). Thus we also prove the continuous dependence of the solution respect to the initial data.
2) We introduce families of operators in
:
and we prove that they are linear and weakly continuous in
. Furthermore, we proved that they form a group of weakly continuous operators in
.
3) With the family of operators
we improve Theorem 3.1.
4) In contrast to what was obtained in
with what has already been studied in
, we see that the weakly continuous operators are not unitary due to the topology of
.
5) It is mathematically enriched, since we generate families of operators.
6) We must indicate that this technique can be applied to other evolution equations in
.
7) Finally, for future work we want to emphasize that the results obtained will allow us to apply computational methods to determine the solution with a degree of approximation that is required and with a lower error rate.