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European Journal of Applied Sciences – Vol. 12, No. 6

Publication Date: December 25, 2024

DOI:10.14738/aivp.126.17880.

Ayala, Y. S. S. (2024). Existence of the Local Solution of a Non-Homogeneous Schrödinger Type Equation. European Journal of

Applied Sciences, Vol - 12(6). 153-168.

Services for Science and Education – United Kingdom

Existence of the Local Solution of a Non-Homogeneous

Schrödinger Type Equation

Yolanda Silvia Santiago Ayala

ORCID: 0000-0003-2516-0871

Universidad Nacional Mayor de San Marcos,

Fac. de Ciencias Matemáticas, Av. Venezuela Cda. 34 Lima-PERU

ABSTRACT

In this article, we prove that initial value problem associated to the Schrödinger

type non homogeneous equation in periodic Sobolev spaces has a local solution in

[0, T] with T > 0, and the solution has continuous dependence with respect to the

initial data and the non homogeneous part of the problem. We do this in a intuitive

way using Fourier theory and introducing a Co- group inspired by the work of Iorio

[1] and Santiago [7]. Also, we prove the uniqueness solution of the Schrödinger

type homogeneous equation, using its conservative property, inspired by the work

of Iorio [1] and Santiago [6]. Finally, we study its generalization to n-th order

equation.

Keywords: Uniqueness solution, Schrödinger type equation, non homogeneous

equation, n-th order equation, periodic Sobolev spaces, Fourier Theory, calculus in

Banach spaces.

INTRODUCTION

First, we want to comment that from Theorem 3.1 in [5], we have that the homogeneous

problem is globally well posed and, in addition to the equality (3.2) in [5], we have the

continuous dependence of the solution of homogeneous problem respect to the initial data.

In this work, in Theorem 3.2 we will prove the existence and uniqueness of the local solution

for the non homogeneous problem and from inequality (3.8) we will get the continuous

dependence of the solution with respect to the initial data and respect to the non

homogeneous part.

Thus, in both homogeneous and non homogeneous cases, the estimatives are made from the

explicit form of the solution, that is, by applying the Fourier transform to the respective

equation. Another result in this work is about the conservative property of the homogeneous

problem and some estimates of it, using differential calculus in Hper

s

. This is included in

Theorem 3.3 which we will develop in subsection 3.2. So, using Theorem 3.3, we deduce the

results of continuous dependence and uniqueness of solution for homogeneous problem.

Finally, we get to generalize the results obtained.

We cite some works about Schrödinger equation [1], [5] and for dissipative properties of

systems [2].

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Services for Science and Education – United Kingdom 154

European Journal of Applied Sciences (EJAS) Vol. 12, Issue 6, December-2024

Our article is organized as follows. In section 2, we indicate the methodology used and cite the

references used. In section 3, we proved the main results for the non homogeneous

Schrödinger type equation. Also, in this section we manage to generalize the results to the n- th order equation.

Finally, in section 4, we give the conclusions of our study.

METHODOLOGY

As a theoretical framework in this article we use the existence and regularity results of [5].

Also, we use the references [1], [5], [6], [7], [8], [9] and [3] for the Fourier theory in periodic

Sobolev spaces, and differential and integral calculus in Banach spaces.

MAIN RESULTS

First, using the Fourier transform, we will prove that the non homogeneous problem has a

unique solution and it continuously depends respect to the initial data and the non

homogeneity in compact intervals.

Second, we will study the uniqueness of the solution for homogeneous case using another

technique that involves the conservative property of the problem.

Finally, we will get to generalize the results to n-th order equation.

The Non Homogeneous Problem (Q3

F

) is Locally Well Posed

Theorem 3.1:

Let s a fixed real number, F ∈ C([0, T], Hper

s

), where T>0, {S(t)}t∈R the unitary group of class

Co of homogeneous case (F = 0), introduced in the Theorem 4.1 from [5], and

up

(t): = ∫ S(t − τ)F(τ)dτ

t

0

.

Then up ∈ C([0, T], Hper

s

) ∩ C

1

([0, T], Hper

s−2

) and satisfies

|

∂tup

(t) − iμ∂x

2up

(t) + iα up

(t) = F(t) ∈ Hper

s−2

up

(0) = 0

(3.1)

with the derivative given by

lim

h→0

‖

up(t+h)−up(t)

h

− iμ∂x

2up

(t) + iα up

(t) − F(t)‖

s−2

= 0. (3.2)

Proof:

We remark that S(t − τ)F(τ) ∈ Hper

s

,∀τ ∈ (0,t) and τ → S(t − τ)F(τ) is continuous in [0, t]

then exists

∫ S(t − τ)F(τ)dτ t

⏟0

up

(t) =

∈ Hper

s

.