of the time dependent Schrödinger equation

SJÄLVSTÄNDIGA ARBETEN I MATEMATIK

MATEMATISKA INSTITUTIONEN, STOCKHOLMS UNIVERSITET

Sobolev norm estimates

of the time dependent Schrödinger equation

av

Erik Melander

2020 - No K2

Sobolev norm estimates

of the time dependent Schrödinger equation

Erik Melander

Självständigt arbete i matematik 15 högskolepoäng, grundnivå

Handledare: Salvador Rodriguez-Lopez

Contents

1 Introduction 2

2 Background 3

2.1 Schwartz space . . . 3

2.2 Fourier transform . . . 4

2.3 Tempered distributions . . . 13

2.3.1 Properties of tempered distributions . . . 14

2.3.2 Fourier transform of tempered distributions . . . 15

2.4 Pseudo differential operators and Sobolev spaces . . . 17

2.4.1 Pseudo differential operators . . . 17

2.4.2 Sobolev spaces . . . 18

2.5 Solving the Heat equation . . . 19

2.6 Solving the time dependent Schr¨odinger equation . . . 21

2.6.1 Regularity estimates for the free time dependent Schr¨odinger 24 2.6.2 The free fractional Schr¨odinger equation . . . 25

3 Regularity estimates for the free time dependent fractional Schr¨odinger equation 26 3.1 Estimate using the Plancherel theorem . . . 26

3.1.1 Replacing|ξ|^a with a differentiable and injective function ϕ 29 3.2 Estimating using Pitt’s inequality . . . 31

3.2.1 Stronger estimate . . . 36

4 Final remarks 37

1 Introduction

This thesis presents regularity estimates for solutions to the free time dependent fractional Schr¨odinger equation with initial data using the theory of Fourier transforms. As such our main focus in this thesis are the functions S^af , where

pS^afq rxsptq  1 2π

»₈

8e^ixξe^i|ξ|atfppξqdξ for a ¥ 2. (1) These functions are solutions to the free time dependent fractional Schr¨odinger equation with initial data f , i.e. they satisfies that

iBtupx, tq  p
B²xq^a{2upx, tq

with upx, 0q  fpxq. As is indicated by the factor pf in the expression for S^af , these solutions are derived by the use of the Fourier transform, which we define as

fppξq :

»₈

8e
^iξxfpxqdx.

The problem at hand is to find regularity estimates for S^af using the initial datum. To tackle this we restrict our focus to problems where we can guarantee the existence of the Fourier transform of our data function. This restriction, while seemingly severe, helps us establish tools and general theory for our esti- mates which we then can expand to hold for less restricted functions.

We assume throughout this text that our data function f is in the Schwartz space on R, denoted S pRq; a space of smooth functions whose derivatives de- crease more rapidly than any polynomial. This space is dense in L^ppRq for 1¤ p   8 and on this space the Fourier transform is a well defined continuous linear operator mapping S pRq onto itself and for such function we can apply the Plancherel theorem.

In the case of a 2, this is a solution to the partial differential equation called the free time dependent Schr¨odinger equation, which states thatBx²u iBtu [14].

We will derive a solution later in section 2.6 to the similar but time reversed equation
Bx²u iBtu.

The idea behind the regularity estimates comes from the fact that if you have an operator T and know thatkT fkB ¤ C kfkAfor some number C we can draw the conclusion that if f P A, then T f P B and as such we have gained knowledge about T f . For our estimates a number of different norms will be used, the main ones which are the L^ppRq-, 9H^spRq- and L⁸pR, L^ppRqq-norms, where

kfkL^ppRq

»₈

8|fpxq|^p 1{p

for 1¤ p ¤ 8,

kfkH9^spRq

»₈

8|ξ|^2s pfpξq²dξ 1{2

,

and

kfkL⁸pR,L^ppRqq sup

xPR

»₈

8|fpx, tq|^pdt 1{p

.

To acquire our result a recurring method throughout this thesis is to observe the effects of dilation of the normed function and different relationships between a function and its Fourier transform, such as the Plancherel theorem and Pitt’s inequality.

2 Background

In this text we will use the Fourier transform and its properties on functions in certain spaces to reach conclusions about differential equations. To make sure that all this makes sense we will start with the space of smooth functions such that they and all of their derivatives vanish faster than any polynomial at the infinities. This space is called the Schwartz space and on this the Fourier trans- form is a well defined bounded linear operator and has some useful properties for us. We will later discuss how to expand to functions beyond this space.

2.1 Schwartz space

The Schwartz space, denoted as S pRq, is the space of all smooth functions f :R Ñ C such that for all natural numbers α and N there is a constant C^α,N for which

|B^αfpxq| ¤ Cα,Np1 |x|q
^N uniformly inR [1, p.13].

Example 2.1. The Gaussian bell function Gpxq  e
^x2{2belongs toS pRq which we see by noting that G is smooth and for all natural numbers M we have that

Gpxq ¤ CMp1 |x|q
^M (2)

for some number CM. And so, given positive integers α, N , there exists numbers Cα from the Leibniz product rule and CN by (2) above such that

|B^αGpxq| ¤ Cα|p1 |x|q^αGpxq|

¤ Cαp1 |x|q^αCNp1 |x|q
^N
α Cα,Np1 |x|q
^N, where Cα,N  Cα CN, showing that GP S pRq.

We call the functions inS pRq Schwartz functions and equip the Schwartz space with the semi norms|f|_m,S for mP N, where

|f|_m,S : sup

|α| |β|¤msup

xPR

x^αB^βfpxq.

These semi norms are defined such as if|f|_m,S   8 for all m P N0then f P S pRq and using them we define convergence inS pRq.

Definition 2.2. Let tfkuk € S pRq for k P N be a sequence of functions in S pRq. We say that tf^kuk converges to f inS pRq iff for all ε ¡ 0 and all m P N there is a KP N such that |fk
f|_m,S   ε whenever k ¥ K.

We note that for p¥ 1 we have that S pRq € L^ppRq. This is a direct result of the definition of the Schwartz space, since for f P S pRq there is a number C such that|fpxq| ¤ Cp1 |x|q
²and so we have

kfk^pL^ppRq

»₈

8|fpxq|^pdx¤ C

»₈

8p1 |x|q
^2pdx  8.

A fundamental property of the Schwartz functions is that they vanish rapidly at infinity and an important subspace ofS pRq is C0⁸pRq which is the space of smooth functions with compact support, i.e. vanish outside a compact set. Here we define the support of a function f , written as supppfq, by

supppfq : tx P R|fpxq  0u.

That is, the support of a function is the closure of the set of points on which the function is nonzero.

Example 2.3. The bump function ϕ defined by ϕpxq 

#e
^1{p1
x2q for|x|   1 0 for |x| ¥ 1

is a smooth function of compact support. Here supppϕq  p
1, 1q  r
1, 1s.

The space C₀⁸pRq has the useful property that it is dense in L^ppRq for 1 ¤ p   8 [15, p.13], and from this and the fact that S pRq € L^ppRq we can draw the following conclusion.

Lemma 2.4. S pRq is dense in L^ppRq for 1 ¤ p   8.

And so, while we will restrict our results to a space of nicely behaving functions, it is feasible to expand our results to a wider space of functions by density.

2.2 Fourier transform

Our main focus on the Schwartz space is as mentioned the Fourier transform.

For functions in the Schwartz space we denote by pf the Fourier transform of a function f and define the transform as [1, p.9]

fppξq :

»₈

8e
^iξxfpxqdx.

We can see that, given f P S pRq, we have a first indication why this definition makes sense by noting that

 pfpξq 

»₈

8e
^iξxfpxqdx

 ¤»₈

8|fpxq| dx   8.

Example 2.5. The Gaussian bell function Gpxq  e
^x2{2is a Schwartz function as seen in example 2.1 and as such we can calculate its Fourier transform.

We have that Gpξq p

»₈

8e
^iξx
x2{2dx

»₈

8e
^{px iξq2{2
ξ2{2}dx

 e
^ξ2{2

»₈

8e
^{px iξq2{2}dx e
^ξ2{2

»

γ

e
^z2{2dz,

where γ is the curve inC parametrized by zptq  t iξ for t P R. The integrand is analytic and vanishes as|z| Ñ 8 and so we get that

»

γ

e
^z2{2dz

»₈

8e
^z2{2dz? 2π, where we use the known fact that ³₈

8e^x2dx?

π. This gives us that pGpξq  e
^ξ2{2?

2π?

2πGpξq.

Now, define the operator D as D
iB. From the definition above we get the following properties for the Fourier transform.

Theorem 2.6. Let ϕP S pRq. Then

1. pαϕ βψqpξq  α p{ ϕpξq β pψpξq for α, β P R 2. {pDⁿϕqpξq  ξⁿϕpξqp

3. Dⁿϕpξq  {p pp
xqⁿϕqpξq 4. ϕpP S pRq

Proof. 1. From the linearity of integration we have that pαϕ βψqpξq {

»₈

8e
^iξxpαϕpxq βψpxqqdx

 α

»₈

8e
^iξxϕpxqdx β

»₈

8e
^iξxψpxqdx

 α pϕpξq β pψpξq.

2. We begin by noting that since ϕP S pRq we have that

ξ^ke
^iξxD^mϕpxq₈

8 0 for all k, mP N⁰ and ξ P R. Using this, integration by parts now gives us that

pD{ⁿϕqpξq 

»₈

8e
^iξxDⁿϕpxqdx  ξⁿ

»₈

8e
^iξxϕpxq  ξⁿϕpξq.p 3. By the Leibniz rule we have that

Dⁿϕpξq  Dp ⁿ

»₈

8e
^iξxϕpxqdx 

»₈

8Dⁿe
^iξxϕpxqdx



»₈

8e
^iξxp
xqⁿϕpxqdx  {pp
xqⁿϕqpξq.

4. From Lemma 2.4 we have that if ϕP S pRq then

| pϕpξq| 

»₈

8e
^iξxϕpxqdx

 ¤»₈

8|ϕpxq| dx  kϕkL¹pRq  8 and furthermore we can note that Dⁿx^mϕP S pRq for n, m P N0. Now, using 2. and 3. above, we have that

| pϕ|_m,S  sup

|α| |β|¤msup

ξPR

ξ^αD^βϕpξqp 

 sup

|α| |β|¤msup

ξPR

ξ^αpp
xq{^βϕqpξq

 sup

|α| |β|¤msup

ξPR

pD^α{p
xq^βϕqpξq

¤ sup

|α| |β|¤m

D^αp
xq^βϕ _L1pRq  8.

Using the following boundedness lemma we can make an important observation about the Fourier transform.

Lemma 2.7. [1, p.14] For ϕP S pRq and m P N there is a constant Cm such that

| pϕ|_m,S ¤ Cm|ϕ|_{m 2,S}

Proof. Let ϕP S pRq, then kϕkL¹pRq

»₈

8|ϕpxq| dx 

»₈

8p1 |x|q
²p1 |x|q²ϕpxqdx

¤ |ϕ|_2,S

»₈

8p1 |x|q
²dx C |ϕ|_2,S. Using this we have

| pϕ|0,S  sup

ξPR| pϕpξq|  sup

ξPR

»₈

8e
^iξxϕpxqdx



¤ sup

ξPR

»₈

8|ϕpxq| dx  kϕkL¹pRq¤ C |ϕ|_2,S. Furthermore we have by theorem 2.6 that

ξ^αD^β_ξϕpξq p pD^αx{p
x^βqϕqpξq and by the Leibniz rule we have that

D^α_xp
x^βqϕpxq ¤ ¸

k¤α

α k

pD^kx^βqpD^α
kϕpxqq ¤ Cα,βx^βD^αϕpxq.

Using this and taking the supremum over α and β, we have that sup

α β¤msup

ξPR

ξ^αD_ξ^βϕp ¤ sup

α β¤mCα,βD_x^αx^βϕ

2,S

¤ sup

α β¤m 2sup

ξPRCα,βD^α_xx^βϕ ¤ Cm|ϕ|_{m 2,S}. That is, we have that | pϕ|_m,S ¤ Cm|ϕ|_{m 2,S}, and so we have that ϕp P S pRq whenever ϕP S pRq.

That is, for a Schwartz function its Fourier transform is bounded by the function itself which, together with the linearity of the transform, helps us establish that the transform is in fact continuous.

Theorem 2.8. Given tϕku_k € S pRq such that ϕk Ñ ϕ in S pRq for some ϕ P S pRq as k Ñ 8, then xϕk Ñ pϕ in S pRq as k Ñ 8, i.e. the Fourier transform is continuous on S pRq.

Proof. Assume ϕk Ñ ϕ in S pRq as k Ñ 8, then pϕP S pRq and for all k P N we have thatϕpkP S pRq. Furthermore, we have from lemma 2.7 that there is a constant Cmsuch that

 {pϕk
ϕq

m,S ¤ Cm|ϕk
ϕ|_{m 2,S}. Now, given ε¡ 0 and m P N, there is a K P N such that

|ϕk
ϕ|_{m 2,S}   ε Cm

whenever k¥ K. Thus, using the linearity of the transform, we have that

|xϕk
pϕ|m,S  {pϕk
ϕq

m,S ¤ Cm|ϕk
ϕ|m 2,S   ε whenever k¥ K.

And so by Theorem 2.6, Lemma 2.7 and Lemma 2.8 we have that on S pRq the Fourier transform is a continuous linear operator mapping S pRq to S pRq.

Furthermore, it has the important property of transforming differentiation of a function to multiplication of its transform with a polynomial. This is useful in solving differential equations and we will later use this to define other spaces on which the concept of pseudo differential equations makes sense. Before this, however, the Fourier transform has other important properties we need.

We begin with translations, modulations and dilations, which we for a func- tion ϕ write as Tyϕ for a translation of ϕ by a factor y, where

pTyϕqpxq : ϕpx yq, and Myϕ for a modulation of ϕ by a factor y, where

pMyϕqpxq : e^ixyϕpxq,

and finally Dβϕ for dilation of ϕ by a factor β, where pDβϕqpxq : ϕpβxq.

These are connected by the Fourier transform and are useful tools in interpreting the transform and analysing its effect on functions.

Theorem 2.9. Given ϕP S pRq, then

1. {pTyϕqpξq  e^iξyϕpξq  pMp yϕqpξq,p 2. {pMyϕqpξq  pϕpξ
yq  pT
yϕqpξq,p 3. zDβϕpξq _β¹ϕpp _β^ξq  β
¹pDβ
¹ϕqpξq.

Proof. All three parts relies entirely on a change of variables.

1. Setting t x y gives us that pT{yϕqpξq 

»₈

8e
^iξxpTyϕqpxqdx 

»₈

8e
^iξxϕpx yqdt



»₈

8e
^iξpt
yqϕptqdt  e^iξy

»₈

8e
^iξtϕptqdt

 e^iξyϕpξq  pMp yϕqpξq.p 2. Setting t x
y gives us that

pM{yϕqpξq 

»₈

8e
^iξxpMyϕqpxqdx 

»₈

8e
^iξxe^ixyϕpxqdx



»₈

8e
^ixpξ
yqϕpxqdx  pϕpξ
yq  pT
yϕqpξq.p 3. Setting y βx gives us that

pD{βϕqpξq 

»₈

8e
^iξxpDβϕqdx 

»₈

8e
^iβξβxϕpβxqdx

 1 β

»₈

8e
^iβξyϕpyqdy  1 βϕp

ξ β

.

We continue with convolutions. Given ϕ and ψ inS pRq the convolution ϕ  ψ is defined as [12, p. 139]

pϕ  ψqpxq :

»₈

8ϕpx
tqψptqdt.

For the Fourier transform we have that the transform of a convolution of two functions corresponds with a multiplication of their transforms.

Theorem 2.10. Given ϕ, ψ in S pRq, then pϕ  ψqpξq  p{ ϕpξq pψpξq

Proof. Using Fubini’s theorem [7, p. 86] and theorem 2.9 above we have that pϕ  ψqpξq {

»₈

8e
^iξxpϕ  ψqpxqdx 

»₈

8e
^iξx

»₈

8ϕpx
tqψptqdt

dx



»₈

8

»₈

8e
^iξxpT
_tϕqpxqdx

ψptqdt 

»₈

8pM
_tϕqpξqψptqdtp



»₈

8e
^iξtϕpξqψptqdt  pp ϕpξq

»₈

8e
^iξtψptqdt  pϕpξq pψpξq.

As we will see it’s possible when using the Fourier transform to solve differen- tial equations to arrive at a transformed solution that is written as a product of two functions. Using the inverse transform we will soon define we can then use the result in Theorem 2.10 above to arrive at a solution, now written as a convolution.

We can note that while Theorem 2.6 gives us that the Fourier transform maps the Schwartz space to itself the transform does not reverse this action, i.e. in general pϕpxq  ϕpxq. While this is somewhat inconvenient another useful prop-p erty of the Fourier transform is the existence of an inversion formula that gives us the inverse transform so that we can regain ϕ from ϕ, circumventing thisp problem. However, to show this we need the next two lemmas, the first called the adjoint or multiplication lemma.

Lemma 2.11. Let ϕ, ψP S pRq, then

»₈

8ϕpxq pψpxqdx 

»₈

8ϕpxqψpxqdxp Proof. Using Fubini’s theorem we have that

»₈

8ϕpxq pψpxqdx 

»₈

8ϕpxq

»₈

8e
^ixyψpyqdy

dx



»₈

8

»₈

8ϕpxqe
^ixyψpyqdy

dx



»₈

8

»₈

8ϕpxqe
^ixydx

ψpyqdy



»₈

8ϕpyqψpyqdy.p

The second lemma, while not directly connected with the Fourier transform, describes a useful property of convolutions.

Theorem 2.12. Given ϕ P L¹pRq such that ³₈

8ϕpxqdx  1, let ϕεpxq  ε
¹Dε
¹ϕpxq. Then for any f P L^ppRq, 1 ¤ p   8 we have that

f ϕεÑ f in L^ppRq as ε Ñ 0.

Proof. Using the change of variables τ  ε
¹t, Fubini’s theorem and Minkowski’s integral inequality [5, p. 101] we have that

kf  ϕ^ε
fkL^ppRq

»₈

8|pf  ϕεqpxq
fpxq|^pdx 1{p



»₈

8

»₈

8fpx
tqϕεptqdt
1  fpxq

^pdx 1{p



»₈

8

»₈

8fpx
tqϕpε
¹tqε
¹dt
1  fpxq

^pdx 1{p



»₈

8

»₈

8fpx
ετqϕpτqdτ

»₈

8ϕpτqdτ  fpxq

^pdx 1{p



»₈

8

»₈

8pfpx
ετq
fpxqqϕpτqdτ

^pdx 1{p

¤

»₈

8|ϕpτq|

»₈

8|fpx
ετq
fpxq|^pdx 1{p

dτ



»₈

8|ϕpτq| kT
ετf
fkL^ppRqdτ.

Now sincekT
ετf
fkL^ppRq¤ 2 kfkL^ppRq andkT
ετf
fkL^ppRq Ñ 0 as ε Ñ 0 we have by Lebesgue dominated convergence theorem thatkf  ϕ^ε
fkL^ppRqÑ 0 as εÑ 0.

And so we can now state the Fourier inverse formula which gives us a tool to retrieving a transformed function.

Theorem 2.13. Let ϕP S pRq, then ϕpxq  1

2π

»₈

8e^iξxϕpξqdξ.p Remark: We denote in this formula the action of _2π¹ ³₈

8e^iξxϕpξqdξ on ϕ in the right hand side above by the symbol ^_ so that the formula above can be written as

ϕpxq  p pϕq^_pxq.

Proof. Define ϕεby

ϕεpxq  1 2π

»₈

8e^{iξx
ε2ξ2{2}ϕpξqdξp

and let

gpξq  e^ixξe
^ε2ξ2{2 pMxDεGqpξq,

where G is the Gaussian bell function Gpξq  e
^ξ2{2as of example 2.5. Then, by Theorem 2.9 on G and since pGpξq ?

2πGpξq (by example 2.5), we have that

pgpζq  ε
¹pT
xDε
¹Gqpζq p ?

2πε
¹e
^pζ
xq2{2ε2. Since

ϕεpxq  1 2π

»₈

8gpξq pϕpξqdξ we now have by Lemma 2.11 that

ϕεpxq  1 2πε

»₈

8pgpξqϕpξqdξ



?2πε
¹ 2π

»₈

8e
^pξ
xq2{2ε2ϕpξqdξ





ϕ ε
¹Dε
¹

?G 2π

pxq.

Since³₈

8?1

2πGpxqdx  1 we now have by Lemma 2.12 that in L^ppRq ϕεpxq Ñ ϕ as ε Ñ 0.

Finally, Lebesgue dominated convergence gives us that ϕεÑ 1

2π

»₈

8e^iξxϕpξqdξ as ε Ñ 0p and so we have that

ϕpxq  1 2π

»₈

8e^iξxϕpξqdξ.p

With this inversion formula, we now have that the Fourier transform is a con- tinuous bijective mapping on S pRq onto S pRq [12].

The next theorem is one of the major theorems for the Fourier transform and one of our main tools when establishing our future results.

Theorem 2.14 (Plancherel theorem). Given ϕP S pRq, then

?2πkϕkL²pRq  kpϕkL²pRq.

Proof. We prove this by first proving a more general case, i.e. that

»₈

8fpxqgpxqdx  2π

»₈

8

fpξqpgpξqdξ.p

Using Fubini’s theorem and the Fourier inversion formula we have that

»₈

8

fppξqpgpξqdξ 

»₈

8fpxq

»₈

8e
^iξxpgpξqdξ

dx



»₈

8fpxq

»₈

8e^iξxpgpξqdξ

dx 2π

»₈

8fpxqgpxqdx.

and so for gpxq  fpxq we get the Plancherel theorem.

This gives us a useful tool in analysing the solutions to differential equations as it allows us to switch between a function and its transform somewhat when analysing the regularity of the function. Furthermore we can make a direct observations from the Plancherel theorem and the fact that S pRq is dense in L²pRq.

Corollary 2.15. There is a unique continuous extension of the Fourier trans- form from S pRq to L²pRq, and as such the Plancherel theorem is extended to functions in L²pRq.

Proof. Given f P L²pRq, by Lemma 2.4 there is a sequence tϕku_k in S pRq converging to f in L²pRq. We have by Lemma 2.6 that tpϕku_kis also a sequence in S pRq and by linearity of the transform and the Plancherel theorem we have that

k pϕm
pϕnkL²pRq ϕm{
ϕn _L2pRq?

2πkϕ^m
ϕnkL²pRq,

and so we have that k pϕm
pϕnkL²pRq Ñ 0 as m, n Ñ 8. By the completeness of L²pRq we now have that there is a function g^ϕin L²pRq such that pϕk Ñ gϕ

in L²pRq.

Now, given another sequence tψku_k in S pRq such that ψ^k Ñ f in L²pRq we have by the same argument as above that there is a function gψ in L²pRq such that pψkÑ gψ in L²pRq. Furthermore, by the triangle inequality and Plancherel theorem, we have that

pϕn
pψn

_L2pRq ?

2πkϕⁿ
ψnkL²pRq

¤?

2πpkϕⁿ
fkL²pRq kf
ψⁿkL²pRqq Ñ 0 as n Ñ 8 and so

kg^ϕ
gψkL²pRq¤ kgϕ
pϕmkL²pRq

pψn
gψ

_L2pRq

pϕm
pψm

_L2pRq. Letting m, nÑ 8 now gives us that

kg^ϕ
gψkL²pRq 0

i.e. gϕ gψin L²pRq. We now have that given f in L²pRq there exists a unique functionFf in L²pRq such that for every sequence tϕku_k in S pRq converging to f in L²pRq the sequence tpϕku_k inS pRq converges to Ff in L²pRq.

Finally, we have that

kFfkL²pRq¤ kpϕk
FfkL²pRq k pϕkkL²pRq

 kpϕk
FfkL²pRq ?1

2πkϕ^kkL²pRq

and so, by letting kÑ 8 above we have that kFfkL²pRq¤ 1

?2πkfkL²pRq.

Furthermore

kfkL²pRq¤ kϕk
fkL²pRq kϕ^kkL²pRq

 kϕk
fkL²pRq

?2πk pϕkkL²pRq.

and so, by once again letting kÑ 8 we have that kfkL²pRq¤?

2πkFfkL²pRq. That is, for f in L²pRq we have that

kfkL²pRq?

2πkFfkL²pRq.

This gives us that there is a extensionF for the Fourier transform from S pRq to L²pRq such that for ϕ P S pRq we have that Fϕ  pϕ. As such we will from now on also useF to indicate the Fourier transform on functions in L²pRq.

While we now have a space of nicely behaving functions for which the Fourier transform makes sense and has a number of useful properties, not all functions are nicely behaving, e.g. we can’t calculate a Fourier transform of or even dif- ferentiate all functions, and so to make the transform a bit more useful the restriction toS pRq need to be relaxed, why the next step is needed.

2.3 Tempered distributions

To generalize the idea of the Fourier transform, we introduce the space of tem- pered distributions. This space, denotedS¹pRq, is the dual space of S pRq, i.e.

the set of all linear and continuous mappings T :S pRq Ñ C in the sense of that f is a tempered distribution if its linear and there exists a constant C and an interger k such that|fpϕq| ¤ |ϕ|_k,S for all ϕP S pRq [2, p. 18]. This allows us to define the Fourier transform on objects in a wider sense than before while retaining and expanding on the core properties of the transform.

Example 2.16. The Dirac delta function, δ0, is classically defined to be such that δ0pxq  0 if x  0, and ³₈

8δ0pxqdx  1, i.e. it spikes at zero enough to

give its integral a value of 1. This is not actually a function, but it is a tempered distribution and as such defined as

δ0pϕq : ϕp0q for all ϕ P S pRq .

To see that δ0 is indeed a tempered distribution we have that for all ϕ P S pRq and natural numbers k we have that

|δ0pϕq|  |ϕp0q| ¤ |ϕ|_k,S.

There is a more general case δx, the translation of δ0 by a factor x, which gives us

δxpϕq : ϕpxq for all ϕ P S pRq .

A core property of this space is that a function f that behaves nicely with all functions in S pRq (called test functions) can be identified with a tempered distribution Tf by the duality product [1]

Tfpϕq  xTf, ϕy :

»₈

8fpxqϕpxqdx, for all ϕ P S pRq .

This is linear by the linearity of integration and continuous by the Lebesgue con- vergence theorem. The important niceness of f here is that this integral above need to make sense. Indeed if f P L¹pRq is at most of polynomial growth (i.e.

p1 |x|q
^Nfpxq P L¹pRq for some N P N), then Tf is a tempered distribution[1].

Remark: For functions f we will write xf, ϕy instead of xTf, ϕy when iden- tifying f as a tempered distribution and there is no risk of ambiguity.

To see that this duality product does indeed give us a tempered distribution we can note that given f P L¹pRq of at most polynomial growth of degree k (as described above), then we have that there is a number D such that p1 |  |q^kf _L1pRq ¤ D. Furthermore, given ϕ P S pRq we have that for k P N there exists a number Ck such that sup

xPRp1 |x|q^kϕpxq ¤ Ck|ϕ|_k,S. We now have that

|xf, ϕy| 

»₈

8fptqϕptqdt

 

»₈

8p1 |t|q
^kfptqp1 |t|q^kϕptqdt



¤ sup

xPRp1 |x|q^kϕpxq

»₈

8p1 |t|q
^kfptqdt



¤ sup

xPRp1 |x|q^kϕpxq p1 |  |q^kf _L1pRq¤ D  Ck|ϕ|_k,S for all ϕP S pRq.

2.3.1 Properties of tempered distributions

The tempered distribution behaves linearly [6, p. 6], i.e. for λ, µ P C, f, g P S¹pRq and ϕ P S pRq we have

xλf µg, ϕy  αxf, ϕy µxg, ϕy

and we define what we mean by differentiation of a tempered distribution by:

Definition 2.17. If f P S¹pRq, then the derivative of order m of f, B^mf , is xB^kf, ϕy  p
1q^kxf, B^kϕy

Example 2.18. This definition coincides with the classical differentiation of differentiable functions. Let f be a differentiable function such that its derivative is polynomially bounded. Then, using integration by parts, we get that

xBTf, ϕy 
xf, Bϕy 

»₈

8fpxqϕ¹pxqdx



hkkkkkkkkikkkkkkkkj0

f¹pxqϕpxq₈

8

»₈

8f¹pxqϕpxqdx  xTf¹, ϕy

and so we se that the differentiation of the tempered distribution defined by f correspond to the tempered distribution defined by the derivative of f , as we would expect if we want to avoid ambiguity.

This definition of differentiation has the benefit of moving the differentiation to the smooth test functions, thus not only giving us a way to differentiate tem- pered distributions but also expand up on which functions that are differentiable (as identified as a tempered distribution). As such, since they relax the criteria of differentiation, these derivatives are called distributional derivatives.

Example 2.19. The function

fpxq 

#

1, if n¤ x ¤ n 1 0, else

has the distributional derivative δn
δn 1 since we have that xBf, ϕy 

»₈

8fpxqϕ¹pxqdx 

»n 1 n

ϕ¹pxqdx  ϕpnq
ϕpn 1q

 xδn, ϕy
xδn 1, ϕy  xδn
δn 1, ϕy for all test functions ϕ.

2.3.2 Fourier transform of tempered distributions

One of the main reasons for us to introduce tempered distributions is to expand the use of the Fourier transform. To do this we define the Fourier transform of f P S¹pRq as the distribution identified through passing the transform over to the test functions. That is, given f P S¹pRq, we define the Fourier transform of f , by [1]

xFrfs, ϕy : xf, Frϕsy, for all ϕ P S pRq and, similarly, for the inverse

xF
¹rfs, ϕy : xf, F
¹rϕsy, for all ϕ P S pRq .

Since our test functions are Schwartz functions this definition lets us keep prop- erties of the Fourier transform while allowing us to calculate a Fourier transform of any tempered distribution [1, p .24], which contains functions that are not necessarily compatible with the first idea of the Fourier transform, such as the constant functions.

Example 2.20. The Fourier transform of the Dirac delta function δ0: We have that

xFrδ0s, ϕy  xδ0,Frϕsy  Frϕsp0q 

»₈

81 ϕpxqdx  x1, ϕy and soFrδ⁰s is identified with the constant function 1.

Example 2.21. The fourier transform of the constant function fpxq  1:

We have that

xFr1s, ϕy  x1, Frϕsy 

»₈

8ϕpξqdξ  ϕp0q  xδp 0, ϕy and soFr1s is the δ⁰ distribution.

The Fourier transform keeps the property of being a linear and continuous operator in S¹pRq in the sense of the following lemma given by Bahoiuri et al [2, prop. 1.23].

Lemma 2.22. Given a sequencetfkuk inS¹pRq that converges to f in S¹pRq, thentFrfksuk converges toFrfs.

As stated before, an important tool of the Fourier transform is that of convo- lutions and for tempered distributions we define them by once again passing along the operation to the test functions. As such we can define a convolution between a distribution and a Schwartz function as follows.

Let f P S¹pRq and g P S pRq, then f  g P S¹pRq is defined by xf  g, ϕy : xf, ˜g  ϕy for all ϕ P S pRq where ˜gpxq : gp
xq for all x P R.

Example 2.23. Convolution with δ0. Let gP S pRq then for all ϕ P S pRq pδ0 gqpϕq  δ0p˜g  ϕq  p˜g  ϕqp0q 

»₈

8gpx
0qϕpxqdx  gpϕq and so as distributions we have that δ0 g  g.

As we will see later, if we use the Fourier transform to acquire solutions to a differential equation it, it is possible to get a solution in the form of a convolution with certain functions (called good kernels [12, p. 139]). To help us establish these solution we observe a property of some sequences of distributions that has a delta-like behaviour which, together with the property of δ0 we saw in the example above, will be of use for us.

Lemma 2.24. [6, p.12] Let βpxq be a nonnegative function with³₈

8βpxqdx  1 and let

βεpxq  1 εβpx

εq.

Then βεpϕq Ñ δ0pϕq as ε Ñ 0 for all ϕ P S pRq.

Proof. We have that

βεpϕq
δ0pϕq 

»₈

8βεpxqϕpxqdx
ϕp0q

1 ε

»₈

8βpx

εqϕpxqdx
ϕp0q



»₈

8βpxqϕpεxqdx

»₈

8βpxqdxϕp0q



»₈

8βpxqpϕpεxq
ϕp0qqdx

Now since |ϕpεxq
ϕp0q| is bounded and ϕpεxq Ñ ϕp0q as ε Ñ 0 for all x P R we have from Lebesgue convergence theorem [7, p. 54] that

»₈

8βpxqpϕpεxq
ϕp0qqdx

 ¤»₈

8βpxq |ϕpεxq
ϕp0q| dx Ñ 0 as ε Ñ 0.

2.4 Pseudo differential operators and Sobolev spaces

A function f onR is said to be locally L^ppRq, 1 ¤ p ¤ 8, written as L^plocpRq, if

»

K

|fpxq|^pdx 1{p

  8

for all compact K€ R [1, p. 204]. Clearly every function in L^ppRq is in L^plocpRq while the reverse is not true.

Example 2.25. On R, every nonzero constant function is in L^plocpRq but not in L^ppRq.

Example 2.26. OnR, every continuous function is in L^plocpRq by the existence of upper and lower bound on compact intervals for such functions.

Example 2.27. The function fpxq  x
¹ is in L^p_locpRzt0uq.

2.4.1 Pseudo differential operators

We defined the differential operator D acting on a function f as Df :
iBf and as such we had the property zpDϕq  ξ pϕ for ϕP S pRq. Using this we can draw

the conclusion that Dϕ  pξ pϕq^_ by the Fourier inversion theorem (Theorem 2.13). Extending on this, for the linear differential operator Ppx, Dq °

kakD^k we define the symbol Ppx, ξq by replacing D^k with ξ i.e.

Ppx, ξq ¸

k

akξ^k

and as such we have that

pP px, Dqϕqpxq  pP px, ξq pϕq^_pxq

That is, we can represent a differential operator with a polynomial by the use of the inverse Fourier transform. To further extend upon this we can define the class of operators called pseudo differential operators Ppx, Dq by using a wider set of signs Ppx, ξq than polynomials, where as before

Ppx, Dqϕpxq : 1 2π

»₈

8e^iξxPpx, ξq pϕpξqdξ.

Using this idea of pseudo differential operators we can create an understanding of our results in this thesis by applying this to the following spaces.

2.4.2 Sobolev spaces

A function f :R Ñ R is called homogenous of degree s P R if fptxq  t^sfpxq for all t¡ 0 and x  0 [1, p. 34]. By this definition, the function p1 |ξ|q^s is not homogenous (also called inhomogenous) while the function|ξ|^s is homogeneous of degree s. From this, for s P R, the space of tempered distributions f such that pf P L²_locpRq and

kfk²H^spRq:

»₈

8p1 |ξ|²q^s pfpξq²dξ  8

is called the inhomogenous Sobolev space H^spRq and the space of tempered distributions f such that pf P L¹_locpRq and

kfk²H9^spRq:

»₈

8|ξ|^2s pfpξq²dξ  8

is called the homogenous Sobolev space 9H^spRq[2, p. 38,25]. These two spaces, while quite similar in their definitions, are not the same space but are linked in that for positive s we have that H^spRq € 9H^spRq and for negative s the reverse is true i.e. 9H^spRq € H^spRq. We will in this thesis concentrate our findings to Schwartz functions, which are in all 9H^spRq, and use the homogenous Sobolev norm to acquire our results. As such we will try to understand how to interpret the norm.

We can begin by noting that by Plancherel theorem we have that kfk²H9^spRq

»₈

8|ξ|^2s pfpξq²dξ ξ^sfp

L²pRq¤?

2πkD^sfkL²pRq

where D^s is the pseudo differential operator represented by the symbol ξ^s, i.e.

D^sf  F
¹rξ^sfps, and so we have that

f P 9H^spRq ðñ D^sf P L²pRq .

As such, we can interpret this in the following way. If s k is a positive integer and f P 9H^kpRq then the derivative of order k of f is in L²pRq. So for positive integers k we have that 9H^kpRq is the space of distributions whose kth derivative is in L²pRq.

Now, what happens if s is negative? We start with the case s 
1 and by noting that

f P 9H
¹pRq ðñ pF : |ξ|
¹fpP L²pRq .

To get an understanding of what this means we need to understand F , the inverse of pF defined above. Firstly we note that

DFpxq  pξ pFq^_pxq  pξ |ξ|
¹fpq^_pxq  psgnpξq pfq^_pxq.

Now define σpξq : sgnpξq and let G : σpDqF . Then G P L²pRq and DGpxq  pξσpξq pFq^_pxq  pσpξq²fpq^_pxq  fpxq.

And so 9H
¹pRq consists of those distributions that are a distributional deriva- tive of a function in L².

This reasoning can be expanded on for when s is a negative integers.

If we want to understand what happens when s in a non-integer number then the idea is to expand on these results to pseudo differential operators. This is, however, outside the scope of this thesis.

2.5 Solving the Heat equation

We have now established a theory we can apply to find solutions to some partial differential equations. At times this requires assuming that the function and its initial data are in S¹pRq. To help with some calculations of our solutions we have the following useful lemma.

Lemma 2.28. [2, p.23] Let z be a nonzero complex number with nonnegative real part and let fzpxq  e
^zx2 for xP R. Then fzP S pRq if Repzq ¡ 0 and

F rf^zs pξq 

?π

?ze
^ξ24z

where?

z |z|^1{2e^{iparg zq{2} and arg zP r
π{2, π{2s.

For z above with Repzq  0, the result holds in a distributional way.

Proof. Firstly, assume that Repzq ¡ 0 and let z  a bi with a ¡ 0. Then fzpxq  e
^ax2e
^ibx2 and so |fz| ¤ |Gapxq|, where Gapxq  e
^ax2, a Gaussian bell function. Since Ga P S pRq by Example 2.1 and |B^αf| ¤ Cαp1 |x|q^α|fz|

we have that|fz|_m,S ¤ |Ga|_m,S   8, and so fzP S pRq.

If Impzq  0 (i.e. z is a positive real number), then we have that F rf^zs pξq 

»₈

8e
^iξxfzpxqdx 

»₈

8e
^iξx
zx2dx

 e
^ξ24z

»₈

8e
^{zpx i2zxq2}dx e
^ξ24z

»

γ1

e
^zw2dw,

where γ1 is the line in C parametrized as wpsq  s i^ξ2z²,
8   s   8. The function e
^zw2 is analytical onC and vanishes as |w| Ñ 8. Cauchy’s integral theorem [10, p. 187] now gives us that we can change the integration over γ1 to integrating over γ2, the curve parametrized by wpsq  s,
8   s   8 (i.e. the real line inC), giving us that

»

γ1

e
^zw2dw

»

γ2

e
^zw2dw

»₈

8e
^zs2ds

?π

?z, and thus we have that

F rf^zs pξq 

?π

?ze
^ξ24z.

Both F rf^zs and ^??π_ze
^ξ24z are analytical (w.r.t. z) on D  tz P C : Repzq ¡ 0u and since they are equal on the part of D that are strictly real then they are equal on all of D [10, p. 297].

Now let z s.t. Repzq  0, i.e. z  it for t P R, and let tznun € D be a sequence s.t. zn Ñ it as n Ñ 8, then by Lebesgue’s dominated convergence theorem [7, p. 54] we have for all ϕP S pRq that

nlimÑ8

»₈

8e
^znx2ϕpxqdx 

»₈

8e
^itx2ϕpxqdx and that

nlimÑ8

»₈

8

?π

?z_ne
^4znξ2 ϕpxqdx 

»₈

8

?π

?ite
^4itξ2ϕpxqdx.

Now, sinceF

 e
^znx2

pξq ^??_z^π

ne
^4znξ2 for each znwe have by Lemma 2.22 that as tempered distributions

F rf^zs pξq 

?π

?ze
^ξ24z.

The heat equation is a partial differential equation that describes how heat disperses over time and is what inspired Joseph Fourier to establish the theory that grew into Fourier analysis. In one space dimension with initial heat given by f P S¹pRq the equation is as follows [12, p. 146].

#Btupx, tq  B²xupx, tq upx, 0q  fpxq

To solve this assume that for all t we have upx, tq P S¹pRq. We can now use the Fourier transform on this equation (with respect to x) and get

FrB^tuspξ, tq  FrBx²uspξ, tq

and so, using the theory of differentiation and the Fourier transform from The- orem 2.6 on the right hand side and Leibniz integral rule on the left hand side we have that

Btpupξ, tq 
ξ²pupξ, tq and thus that

pupξ, tq  Apξqe
^ξ2t.

This gives us that pupξ, 0q  Apξq and from the initial condition we have that pupξ, 0q  pfpξq, so our solution becomes

pupξ, tq  e
^ξ2tfppξq.

Now for t¡ 0, by Lemma 2.28 above with z  t we have H^tpxq :  F
¹

e
^ξ2t

pxq  1 2π

»₈

8e^iξxe
^ξ2tdξ

 1 2π

»₈

8e
^iηxe
^η2tdη 1 2πF

e
^ξ2t

pxq  e
^x24t

?4πt

and so, by Theorem 2.10, we get our solution as upx, tq  pHt fqpxq

and we are left with confirming that upx, tq Ñ fpxq as t Ñ 0.

Now, we note that if βpxq  ^?1_πe
^x2 then ³₈

8βpxqdx  1 and since H^tpxq 

?1

4tβp^?x_4tq we have that Htpxq Ñ δ0as tÑ 0 by lemma 2.24. And so we have, for fixed x, that inS¹pRq

u Ht f Ñ δ0 f  f as t Ñ 0.

2.6 Solving the time dependent Schr¨ odinger equation

Our main focus in this thesis is the equation proposed by Erwin Schr¨odinger in the early 20th century to describe a wave equation for matter. The need for this equation came from the discoveries in particle physics where, under certain conditions a particle has a wavelike behaviour[4]. The equation presented by Schr¨odinger was

i~B^tupx, tq  p ~²

2m∆ Vpx, tqqupx, tq

where x P Rⁿ, t ¡ 0 [3]. Solutions to this equation describes probabilities of particles being located in certain regions. We will now consider the case of one space dimension so that the Laplacian ∆ of the space variables is B²x and we

will omit the positive physical terms ~ and 2m (for clarity) and assume that the potentiality term V is zero. As such the free time dependent Schr¨odinger equation in one space dimension [13] with initial datum fP S¹pRq states that

#iBtupx, tq 
B²xupx, tq

upx, 0q  fpxq. (3)

To solve this we begin a similar approach as for the heat equation above and using the Fourier transform (with respect to x) gives us that

Btpupξ, tq 
iξ²pupξ, tq and so our solution becomes

pupξ, tq  pgtpξq pfpξq wheregptpξq  e
^itξ2.

First we note that we have for finite t¡ 0 and a P N that |Bξ^agpt| ¤ Cap1 |ξ|q^a and given hP S pRq then for any α, N P N there is a C^α,N such that

|Bξ^αh| ¤ Cα,Np1 |ξ|q
^N and so there is a Cα,N such that

¸

a¤α

|B^aξh| ¤ Cα,Np1 |ξ|q
^N
α

thus, for any α, N P N, Leibniz rule gives us

|B^αξp pgthq|  ¸

a bα a,b¥0

Ca,b|pB^aξgptqpB^bξhq|

¤ p1 |ξ|q^α ¸

0¤b¤α

Cb|Bξ^bh| ¤ Cα,Np1 |ξ|q^αp1 |ξ|q
^α
N

¤ Cα,Np1 |ξ|q
^N

and so for all hP S pRq we have pgthP S pRq.

We have that

| pgt
1| e
^itξ2
1 

»1 0

e
^itsξ2dspitξ²q

 ¤ t|ξ|²

and, for t   1, we have for α ¥ 1 that |B_ξ^αgpt| ¤ Cαtp1 |ξ|q^α. Given m P N

then, using the same method as above for the sum, we have that

| pgth
h|m,S  sup

ξPR sup

α β¤m||ξ|^βB^αξp pgtpξqhpξq
hpξqq|

 sup

ξPR sup

α β¤m|ξ|^β|p pgt
1qB^αξh ¸

a bα a¡0, b¥0

Ca,bpBξ^agptqpBξ^bhq|

¤ sup

ξPR sup

α β¤m|ξ|^βt|ξ|²|Bξ^αh| tp1 |ξ|q^α ¸

0¤b¤α

Cb|B^bξh|

¤ sup

ξPR sup

α β¤mtCα,β|ξ|^{β 2}p1 |ξ|q
^β
3 tCαp1 |ξ|q^αp1 |ξ|q
^α
1

¤ sup

ξPR sup

α β¤mtCα,βp1 |ξ|q
¹ tCαp1 |ξ|q
¹¤ tD and so for all hP S pRq we have | pgth
h|_m,S Ñ 0 as t Ñ 0.

Now, given f P S pRq then pf P S pRq and since pupξ, tq  pgtpξq pfpξq and pupξ, 0q  fppξq we have from our previous result, with h  pf , that pupξ, tq Ñ pfpξq in S pRq as t Ñ 0. Since F
¹ is continuous, we have upx, tq  F
¹r pgtfpspxq Ñ F
¹r pfspxq  fpxq as t Ñ 0. That is, u Ñ f in S pRq as t Ñ 0.

Let ˜fpxq  fp
xq, then ˜gt gt and, from the results above, for all ϕ P S pRq we have gt ϕ Ñ ϕ in S pRq as t Ñ 0 and so, using convolutions in S¹pRq, we have

pgt fqpϕq  fp˜gt ϕq  fpgt ϕq.

So by the continuity of f and the results above we have that fp˜gt ϕq Ñ fpϕq as t Ñ 0

i.e.

gt f Ñ f as t Ñ 0 in S¹pRq . Now, if f P L²pRq, then, using Plancherel

kukL²pRq p2πq
^1{2kpukL²pRq  p2πq
^1{2 pgtfp _L2pRq

 p2πq
^1{2 pf

L²pRq  kfkL²pRq

and so for t¡ 0 we have u P L²pRq.

Finally, using Plancherel again, we have

ku
fkL²pRq p2πq
^1{2 pu
pf _L2pRq

 p2πq
^1{2 p pgt
1q pf

L²pRq

and so, since gpt
1 Ñ 0 as t Ñ 0, Lebesgue dominated convergence theorem gives us thatku
fkL²pRqÑ 0 as t Ñ 0, i.e. u Ñ f in L²pRq as t Ñ 0.

We can now summarize our solution as upx, tq  pgt fqpxq 

»₈

8gtpx
yqfpyqdy (4)