Freiling G., Yurko V. Lectures on Differential Equations of Mathematical Physics: A First Course

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82 G.

Freiling and V. Yurko

Using (2.7.8)

and (2.7.13) we calculate

(x,λ) = ϕ

( j−1)

(x,λ)

(x,λ) −Φ

( j−1)

(x,λ)

(x,λ),

(x,λ) = Φ

( j−1)

(x,λ)

ϕ(x,λ) −ϕ

( j−1)

(x,λ)

Φ(x,λ),







(2.7.14)

ϕ(x,λ) = P

(x,λ)

ϕ(x,λ) + P

(x,λ)

(x,λ),

Φ(x,λ) = P

(x,λ)

Φ(x,λ) + P

(x,λ)

(x,λ).

)

(2.7.15)

It follows from (2.7.14), (2.7.7) and (2.7.8) that

(x,λ) = 1 +

∆(λ)

ψ(x,λ)

(

(x,λ

) −ϕ

(x,λ))

−ϕ(x,λ)(

(x,λ) −ψ

(x,λ))

(x,λ) =

∆(λ)

ϕ(x,λ)

ψ(x,λ) −ψ(x,λ)

ϕ(x,λ)

virtue

of (2.2.45), (2.2.46) and (2.2.53), this yields

(x,λ) −1| ≤

|ρ|

, |P

(x,λ)|

≤

|ρ|

, ρ ∈G

, |ρ|

≥ ρ

∗

, (

2.7.16)

(x,λ) −1| ≤

|ρ|

, |P

(x,λ)|

≤C

, ρ ∈G

, |ρ

| ≥ ρ

∗

. (2.7.17)

According to (2.7.7) and (2.7.14),

(x,λ) = ϕ(x,λ)

(x,λ) −S(x,λ)

(x,λ)

M(λ) −M(λ)ϕ(x,λ)

(x,λ),

(x,λ) = S(x, λ)

ϕ(x,λ) −ϕ(x,λ)

S(x,λ)

+(M(λ) −

M(λ))ϕ(x,λ)

ϕ(x,λ).

Thus, if M(λ) ≡

M(λ) , then for each ﬁxed x, the functions P

(x,λ) and P

(x,λ) are

entire in λ. Together with (2.7.16) this yields

(x,λ) ≡ 1, P

(x,λ) ≡ 0.

Substituting into (2.7.15) we get

ϕ(x,λ) ≡

ϕ(x,λ), Φ(x,λ) ≡

Φ(x,λ)

for all x and λ, and consequently, L =

L. 2

Remark 2.7.2. According to (2.7.10), the speciﬁcation of the Weyl function M(λ)

is equivalent to the speciﬁcation of the spectral data {λ

,α

}

n≥0

. On the other hand, by

virtue of (2.7.9) zeros and poles of the Weyl function M(λ) coincide with the spectra of

the boundary value problems L and L

, respectively. Consequently, the speciﬁcation of

Hyperbolic P

artial Differential Equations 83

the W

eyl function M(λ) is equivalent to the speciﬁcation of two spectra {λ

} and {λ

,}.

Thus, the inverse problems of recovering the Sturm-Liouville equation from the spectral

data and from two spectra are particular cases of Inverse Problem 2.7.2 of recovering the

Sturm-Liouville equation from the given Weyl function, and we have several independent

methods for proving the uniqueness theorems. The Weyl function is a very natural and

convenient spectral characteristic in the inverse problem theory. Using the concept of the

Weyl function and its generalizations we can formulate and study inverse problems for

various classes of operators. For example, inverse problems of recovering higher-order

differential operators and systems from the Weyl functions has been studied in [19] and

[26]. We will also use the Weyl function below for the investigation of the Sturm-Liouville

operator on the half-line.

2. Solution of the inverse problem on a ﬁnite interval

In this subsection we obtain an algorithm for the solution of Inverse Problem 2.7.1 and

provide necessary and sufﬁcient conditions for its solvability. For this purpose we use the

so-called transformation operator method ([15], [16]). The central role in this method is

played by a linear integral equation with respect to the kernel of the transformation operator

(see Theorem 2.7.5). The main results of this subsection are stated in Theorem 2.7.6.

We ﬁrst prove several auxiliary assertions.

Lemma 2.7.1. In a Banach space B, consider the equations

(E + A

= f

(E + A)y = f ,

where A and A

are linear bounded operators, acting from B to B, and E is the identity

operator. Suppose that there exists the linear bounded operator R

:= (E + A

)

−1

, this

yields in particular that the equation (E + A

= f

is uniquely solvable in B. If

kA −A

k ≤ (2kR

−1

then there exists the linear bounded operator R := (E + A)

−1

with

R = R

E +

∞

∑

k=1

((A

−A)R

)

and

kR −R

k ≤ 2kR

kA −A

Moreover, y and y

satisfy the estimate

ky −y

k ≤C

(kA −A

k+ kf − f

k),

where C

depends only on kR

k and kf

Proof. We have

E + A = (E + A

) + (A −A

) =

E + (A −A

(E + A

84 G.

Freiling and V. Yurko

Under the

assumptions of the lemma it follows that k(A −A

k≤1/2, and consequently

there exists the linear bounded operator

R := (E +A)

−1

= R

E + (A −A

−1

= R

E +

∞

∑

k=1

((A

−A)R

)

This yields in particular that kRk ≤ 2kR

k. Using again the assumption on kA −A

k we

infer

kR −R

k ≤ kR

k(A −A

1 −

(A −A

≤ 2kR

kA −A

Furthermore,

y −y

= R f −R

= (R −R

) f

+ R( f − f

Hence

ky −y

k ≤ 2kR

kkA −A

k+ 2kR

kkf − f

The following lemma is an obvious corollary of Lemma 2.7.1 (applied in a correspond-

ing function space).

Lemma 2.7.2. Consider the integral equation

y(t,α) +

A(t, s,α)y(s, α)ds = f (t,α), a ≤t ≤ b, (2.7.18)

where A(t, s,α) and f (t,α) are continuous functions. Assume that for a ﬁxed α = α

the

homogeneous equation

z(t) +

(t, s)z(s)ds = 0, A

(t, s) := A(t, s,α

)

has only the trivial solution. Then in a neighbourhood of the point α = α

, equation

(2.7.18) has a unique solution y(t,α), which is continuous with respect to t and α. More-

over, the function y(t, α) has the same smoothness as A(t,s,α) and f (t,α).

Lemma 2.7.3. Let numbers {ρ

, α

}

n≥0

of the form

= n +

πn

, α

, {κ

},{κ

}

∈ l

, α

6= 0 (

2.7.19)

be given. Denote

a(x) =

∞

∑

n=0

cosρ

−

cosnx

, (2.7.20)

wher

(

, n > 0,

π, n = 0.

Then

the

series (2.7.20) converges absolutely and uniformly on [0, 2π] , and a(x) ∈

(0,2π).

Hyperbolic P

artial Differential Equations 85

Proof

. Denote δ

= ρ

−n. Since

cosρ

−

cosnx

cosρ

x −cosnx

−

cosρ

x −cosnx = cos(n + δ

)x −cosnx

= −sin δ

x sin nx −2

sin

cosnx

= −δ

x sin nx −(sinδ

x −δ

x)sin nx −2

sin

cosnx,

a(x) = A

(x) + A

(x),

where

(x) = −

ωx

∞

∑

n=1

sinnx

= −

ωx

π −x

, 0 < x < 2π,

(x)

∞

∑

n=0

−

cosρ

x +

cosρ

x −1

−x

∞

∑

n=1

sinnx

−

∞

∑

n=1

(sinδ

x −δ

x)sin nx −2

∞

∑

n=1

sin

cosnx. (2.7.21)

Since

= O

−

, {γ

}

∈ l

the

series in (2.7.21) converge absolutely and uniformly on [0,2π], and A

(x) ∈W

(0,2π).

Consequently, a(x) ∈W

(0,2π). 2

Now we go on to the solution of the inverse problem. Let us consider the boundary value

problem L = L(q(x), h,H). Let {λ

, α

}

n≥0

be the spectral data of L, ρ

√

. W

shall

solve the inverse problem of recovering L from the given spectral data {λ

, α

}

n≥0

It was shown in Section 2.2 that the spectral data have the properties:

= n +

πn

, α

, {κ

},{κ

}

∈ `

, (

2.7.22)

> 0, λ

6= λ

(n 6= m). (2.7.23)

Consider the function

F(x,t) =

∞

∑

n=0

cosρ

x cos ρ

−

cosnx cos n

, (2.7.24)

where

(

, n > 0,

π, n = 0.

Since F(x,t)

(a(x +t) +a(x −t))/2, then by virtue of Lemma 2.7.3, F(x,t) is continu-

ous, and

F(x, x) ∈ L

(

0,π).

86 G.

Freiling and V. Yurko

Theorem

2.7.5. For each ﬁxed x ∈ (0,π], the kernel G(x,t) appearing in representa-

tion (2.7.4) the linear integral equation

G(x,t) + F(x,t) +

G(x,s)F(s,t)ds = 0, 0 < t < x. (2.7.25)

This equation is the Gelfand-Levitan equation.

Thus, Theorem 2.7.5 allows one to reduce our inverse problem to the solution of the

Gelfand-Levitan equation (2.7.25). We note that (2.7.25) is a Fredholm type integral equa-

tion in which x is a parameter.

Proof. One can consider the relation (2.7.4) as a Volterra integral equation with respect

to cos ρx. Solving this equation we obtain

cosρx = ϕ(x,λ) +

H(x,t)ϕ(t, λ)dt, (2.7.26)

where H(x,t) is a continuous function. Using (2.7.4) and (2.7.26) we calculate

∑

n=0

ϕ(x,λ

)cos ρ

∑

n=0

cosρ

x cos ρ

cosρ

G(x,s) cosρ

∑

n=0

ϕ(x,λ

)cos ρ

∑

n=0

ϕ(x,λ

)ϕ(t, λ

)

ϕ(x,λ

)

H(t, s)ϕ(s,λ

This

yields

(x,t) = I

(x,t) + I

(x,t),

where

(x,t) =

∑

n=0

ϕ(x,λ

)ϕ(t, λ

)

−

cosnx cos n

(x,t)

∑

cosρ

x cos ρ

−

cosnx cos n

(x,t)

∑

n=0

cosn

G(x,s) cosns

(x,t) =

∑

n=0

G(x,s)

cosρ

t cos ρ

−

cosn

t cos ns

(x,t) = −

∑

n=0

ϕ(x,λ

)

H(t, s)ϕ(s,λ

Let f (x) ∈ AC[0,π]. According to Theorem 2.2.7,

lim

N→∞

max

0≤x≤π

f (t)Φ

(x,t)dt = 0;

Hyperbolic P

artial Differential Equations 87

furthermore, uniformly

with respect to x ∈ [0, π],

lim

N→∞

f (t)I

(x,t)dt =

f (t)F(x,t)dt,

lim

N→∞

f (t)I

(x,t) dt =

f (t)G(x,t) dt,

lim

N→∞

f (t)I

(x,t)dt =

f (t)

G(x,s)F(s,t)ds

dt,

lim

N→∞

f (t)I

(x,t)dt

= − lim

N→∞

∑

n=0

ϕ(x,λ

)

ϕ(s,λ

)

H(t, s) f (t)d

s = −

f (t)H(t, x)dt.

Extend G(x,t) = H(x,t) = 0 for x < t. Then, in view of the arbitrariness of f (x), we

derive

G(x,t) + F(x,t) +

G(x,s)F(s,t)ds −H(t,x) = 0.

For t < x, this yields (2.7.25). 2

The next theorem, which is the main result of this subsection, gives us an algorithm

for the solution of the inverse problem as well as necessary and sufﬁcient conditions for its

solvability.

Theorem 2.7.6. For real numbers {λ

, α

}

n≥0

to be the spectral data for a certain

boundary value problem L(q(x),h,H) with q(x) ∈ L

(0,π), it is necessary and sufﬁcient

that the relations (2.7.22)−(2.7.23) hold. The boundary value problem L(q(x), h,H) can

be constructed by the following algorithm:

Algorithm 2.7.1. (i) From the given numbers {λ

, α

}

n≥0

construct the function

F(x,t) by (2.7.24) .

(ii) Find the function G(x,t) by solving equation (2.7.25) .

(iii) Calculate q(x), h and H by the formulae

q(x) = 2

(x,x), h = G(0,0), H = ω −h −

q(t) d

t. (

2.7.27)

The necessity part of Theorem 2.7.6 was proved above, here we prove the sufﬁciency.

Let real numbers {λ

, α

}

n≥0

of the form (2.7.22)-(2.7.23) be given. We construct the

function F(x,t) by (2.7.24) and consider equation (2.7.25).

Lemma 2.7.4. For each ﬁxed x ∈ (0,π], equation (2.7.25) has a unique solution

G(x,t) in L

(0,x).

Proof. Since (2.7.25) is a Fredholm equation it is sufﬁcient to prove that the homoge-

neous equation

g(t) +

F(s,t)g(s)ds = 0 (2.7.28)

has only the trivial solution g(t) = 0.

88 G.

Freiling and V. Yurko

Let g(t) be a

solution of (2.7.28). Then

(t)dt +

F(s,t)g(s)g(t) dsdt = 0

(t)dt +

∞

∑

n=0

g(t) cos ρ

−

∞

∑

n=0

g(t) cos n

= 0.

Using Parseval‘s equality

(t) dt =

∞

∑

n=0

g(t) cos n

for the function g(t), extended by zero for t > x, we obtain

∞

∑

n=0

g(t) cos ρ

= 0.

Since α

> 0, then

g(t) cos ρ

t dt = 0, n ≥0.

The system of functions {cos ρ

n≥0

is complete in L

(0,π) (see Levinson‘s theorem

[27, p.118]). This yields g(t) = 0. 2

Let us return to the proof of Theorem 2.7.6. Let G(x,t) be the solution of (2.7.25). The

substitution t → tx, s → sx in (2.7.25) yields

F(x, xt) + G(x,xt) + x

G(x,xs)F(xt,xs)ds = 0, 0 ≤t ≤1. (2.7.29)

It follows from (2.7.25), (2.7.29) and Lemma 2.7.2 that the function G(x,t) is continuous,

and has the same smoothness as F(x,t). In particular,

(x,x) ∈ L

(0,π).

We construct ϕ(x,λ) by (2.7.4) and the boundary value problem L(q(x), h,H) by

(2.7.27). Then one can verify that ϕ(x,λ) satisﬁes (2.7.1), and the numbers {λ

,α

}

n≥0

are the spectral data for the constructed problem L(q(x),h, H) (see [17, Sect.1.6]). Thus,

Theorem 2.7.6 is proved. 2

Example 2.7.1. Let λ

= n

(n ≥0), α

(n ≥1), and

let α

> 0

be an arbitrary

positive number. Denote a :=

−

. Let

use Algorithm 2.7.1:

1) By (2.7.24), F(x,t) ≡ a.

2) Solving equation (2.7.25) we get easily

G(x,t) = −

1 + ax

(2.7.27),

q(x) =

(1 + ax)

, h = −a, H =

1 + aπ

aα

Hyperbolic P

artial Differential Equations 89

By (2.7.4),

ϕ(x,λ

) = cos ρx −

1 + ax

sinρx

Sturm-Liouville

operators on the half-line

In subsections 3-5 we present an introduction to the inverse problem theory for Sturm-

Liouville operators on the half-line. First nonselfadjoint operators with integrable complex-

valued potentials are considered. We introduce and study the Weyl function as the main

spectral characteristic, prove an expansion theorem and solve the inverse problem of recov-

ering the Sturm-Liouville operator from its Weyl function. For this purpose we use ideas of

the contour integral method and the method of spectral mappings presented in [17], [18].

Moreover connections with the transformation operator method are established. Then lo-

cally integrable complex-valued potentials are studied. In this case the generalized Weyl

function is introduced as the main spectral characteristic. We prove an expansion theorem

and solve the inverse problem of recovering the Sturm-Liouville operator from its general-

ized Weyl function.

We consider the differential equation and the linear form L = L(q(x), h) :

`y := −y

+ q(x)y = λy, x > 0, (2.7.30)

U(y) := y

(0) −hy(0), (2.7.31)

where q(x) ∈ L(0,∞) is a complex-valued function, and h is a complex number. Let

λ = ρ

, ρ = σ +iτ, and let for deﬁniteness τ := Imρ ≥ 0. Denote by Π the λ -plane with

the cut λ ≥0, and Π

Π\

}; notice that here Π and Π

must be considered as subsets

of the Riemann surface of the squareroot-function. Then, under the map ρ →ρ

= λ, Π

corresponds to the domain Ω = {ρ : Imρ ≥0, ρ 6= 0}. Put Ω

= {ρ : Im ρ ≥ 0, |ρ| ≥ δ}.

Denote by W

the set of functions f (x), x ≥0 such that the functions f

( j)

(x), j =

0,N −1

are

absolutely

continuous on [0,T ] for each ﬁxed T > 0, and f

( j)

(x) ∈L(0,∞) , j =

0,N.

ost

and Birkhoff solutions. Let us construct a special fundamental system of solutions

for equation (2.7.30) in Ω having asymptotic behavior at inﬁnity like exp(±iρx).

Theorem 2.7.7. Equation (2.7.30) has a unique solution y = e(x,ρ), ρ ∈ Ω, x ≥ 0,

satisfying the integral equation

e(x,ρ) = exp(iρx) −

2iρ

∞

xp(

iρ(x −t))

−exp(iρ(t −x)))q(t)e(t,ρ)dt. (2.7.32)

The function e(x,ρ) has the following properties:

) For x → ∞, ν = 0, 1, and each ﬁxed δ > 0,

(ν)

(x,ρ) = (iρ)

exp(iρx)(1 + o(1)), (2.7.33)

90 G.

Freiling and V. Yurko

uniformly in Ω

. F

or Im ρ > 0, e(x,ρ) ∈L

(0,∞). Moreover, e(x,ρ) is the unique solution

of (2.7.30)( up to a multiplicative constant ) having this property.

) For |ρ| → ∞, ρ ∈ Ω, ν = 0,1,

(ν)

(x,ρ) = (iρ)

exp(iρx)

1 +

ω(x)

iρ

+ o

´´

, ω(x) := −

∞

q(t) d

t, (

2.7.34)

uniformly for x ≥0.

) For each ﬁxed x ≥0, and ν = 0,1, the functions e

(ν)

(x,ρ) are analytic for Im ρ > 0,

and are continuous for ρ ∈Ω.

) For real ρ 6= 0, the functions e(x, ρ) and e(x,−ρ) form a fundamental system of

solutions for (2.7.30) , and

he(x,ρ),e(x,−ρ)i = −2iρ, (2.7.35)

where hy,zi := yz

−y

z is the Wronskian.

The function e(x,ρ) is called the Jost solution for (2.7.30) .

Proof. We transform (2.7.32) by means of the replacement

e(x,ρ) = exp(iρx)z(x,ρ) (2.7.36)

to the equation

z(x,ρ) = 1 −

2iρ

∞

(1 −e

xp(2

iρ(t −x)))q(t)z(t,ρ)dt, ≥ 0, ρ ∈ Ω. (2.7.37)

The method of successive approximations gives

(x,ρ) = 1, z

k+1

(x,ρ) = −

2iρ

∞

(1 −e

xp(2

iρ(t −x)))q(t)z

(t, ρ)dt, (2.7.38)

z(x,ρ) =

∞

∑

k=0

(x,ρ). (2.7.39)

Let us show by induction that

(x,ρ)| ≤

(x))

|ρ|

, ρ ∈Ω, x ≥ 0, (2.7.40)

where

(x) :=

∞

|q(t)|d

Indeed,

for k = 0, (2.7.40) is obvious. Suppose that (2.7.40) is valid for a certain ﬁxed

k ≥0. Since |1 −exp(2iρ(t −x))| ≤ 2, (2.7.38) implies

k+1

(x,ρ)| ≤

|ρ|

∞

|q(t)z

(t, ρ)|d

t. (

2.7.41)

Substituting (2.7.40) into the right-hand side of (2.7.41) we calculate

k+1

(x,ρ)| ≤

|ρ|

k+1

∞

|q(t)|(Q

(t))

t =

(

(t))

k+1

|ρ|

k+1

(k + 1)!

Hyperbolic P

artial Differential Equations 91

It follo

ws from (2.7.40) that the series (2.7.39) converges absolutely for x ≥ 0, ρ ∈ Ω,

and the function z(x, ρ) is the unique solution of the integral equation (2.7.37). Moreover,

by virtue of (2.7.39) and (2.7.40),

|z(x,ρ)| ≤ exp(Q

(x)/|ρ|), |z(x,ρ) −1| ≤ (Q

(x)/|ρ|)exp(Q

(x)/|ρ|). (2.7.42)

In particular, (2.7.42) yields for each ﬁxed δ > 0,

z(x,ρ) = 1 + o(1), x →∞, (2.7.43)

uniformly in Ω

, and

z(x,ρ) = 1 + O

, |ρ|

→ ∞, ρ ∈ Ω, (

2.7.44)

uniformly for x ≥0. Substituting (2.7.44) into the right-hand side of (2.7.37) we obtain

z(x,ρ) = 1 −

2iρ

∞

(1 −e

xp(2

iρ(t −x)))q(t) dt + O

, |ρ|

→ ∞

, (2.7.45)

uniformly for x ≥0.

Lemma 2.7.5. Let q(x) ∈ L(0, ∞), and denote

(x,ρ) :=

∞

q(t) exp(2iρ(t −x))dt, ρ ∈ Ω. (2.7.46)

Then

lim

|ρ|→∞

sup

x≥0

(x,ρ)| = 0. (2.7.47)

Proof. 1) First we assume that q(x) ∈W

. Then integration by parts in (2.7.46) yields

(x,ρ) = −

q(x)

2iρ

−

2iρ

∞

(t) e

xp(2

iρ(t −x))dt,

and consequently

sup

x≥0

(x,ρ)| ≤

|ρ|

Let

now q(x) ∈ L(0,∞). Fix ε > 0 and choose q

(x) ∈W

such that

∞

|q(t) −q

(t)|dt <

Then

(x,ρ)|

≤

(x,ρ)|+ |J

q−q

(x,ρ)| ≤

|ρ|

Hence,

there

exists ρ

> 0 such that sup

x≥0

(x,ρ)| ≤ ε for |ρ| ≥ ρ

, ρ ∈ Ω. By virtue of

arbitrariness of ε > 0 we arrive at (2.7.47). 2