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

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112 G.

Freiling and V. Yurko

= z(λ) −

2πi

r(x,λ,µ) − ˜r(x,λ,µ)

2πi

˜r(x,λ,ξ)r(x,ξ,µ)dξ

z(µ)d

µ.

virtue of (2.7.119) this yields

AAz(λ) = z(λ), z(λ) ∈C(γ).

Interchanging places for L and

L, we obtain analogously A

Az(λ) = z(λ). Thus,

AA =

A = E, where E is the identity operator. Hence the operator

A has a bounded inverse

operator, and the main equation (2.7.118) is uniquely solvable for each ﬁxed x ≥0. 2

Denote

(x) =

2πi

ϕ(x,µ)ϕ(x, µ)

M(µ)d

, ε(x) = −2ε

(x). (2.7.124)

Theorem 2.7.19. The following relations hold

q(x) = ˜q(x) + ε(x), (2.7.125)

h =

h −ε

(0). (2.7.126)

Proof. Differentiating (2.7.118) twice with respect to x and using (2.7.112) and

(2.7.124) we get

(x,λ) −ε

(x)

ϕ(x,λ) = ϕ

(x,λ) +

2πi

˜r(x,λ,µ)ϕ

(x,µ) d

, (2.7.127)

(x,λ) = ϕ(x,λ) +

2πi

˜r(x,λ,µ)ϕ

(x,µ) d

2πi

ϕ(x,λ)

ϕ(x,µ)

M(µ)ϕ

(x,µ) d

2πi

(

ϕ(x,λ)

ϕ(x,µ))

M(µ)ϕ(x,µ)d

. (2.7.128)

In (2.7.128) we replace the second derivatives using equation (2.7.30), and then we replace

ϕ(x,λ) using (2.7.118). This yields

˜q(x)

ϕ(x,λ) = q(x)

ϕ(x,λ) +

2πi

hϕ(x,λ), ϕ(x,µ)i

M(µ)ϕ(x,µ)d

2πi

ϕ(x,λ)

ϕ(x,µ)

M(µ)ϕ

(x,µ) d

2πi

(

ϕ(x,λ)

ϕ(x,µ))

M(µ)ϕ(x,µ)d

After canceling terms with

(x,λ) we arrive at (2.7.125). Taking x = 0 in (2.7.127) we

get (2.7.126). 2

Thus, we obtain the following algorithm for the solution of the inverse problem.

Hyperbolic P

artial Differential Equations 113

Algorithm 2.7.2. Let

the function M(λ) be given. Then

(1) Choose

L ∈V

such that (2.7.110) holds.

(2) Find ϕ(x,λ) by solving equation (2.7.118) .

(3) Construct q(x) and h via (2.7.124) −(2.7.126) .

Let us now formulate necessary and sufﬁcient conditions for the solvability of the in-

verse problem. Denote in the sequel by W the set of functions M(λ) such that

(i) the functions M(λ) are analytic in Π with the exception of an at most countable

bounded set Λ

of poles, and are continuous in Π

with the exception of bounded set

Λ (in general, Λ and Λ

are different for each function M(λ) );

(ii) for |λ| → ∞, (2.7.91) holds.

Theorem 2.7.20. For a function M(λ) ∈ W to be the Weyl function for a certain

L ∈V

, it is necessary and sufﬁcient that the following conditions hold:

1) ( Asymptotics ) There exists

L ∈V

such that (2.7.110) holds;

2) ( Condition S ) For each ﬁxed x ≥ 0, equation (2.7.118) has a unique solution

ϕ(x,λ) ∈C(γ);

3) ε(x) ∈W

, where the function ε(x) is deﬁned by (2.7.124) .

Under these conditions q(x) and h are constructed via (2.7.125) −(2.7.126) .

As it is shown in Example 2.7.3, conditions 2) and 3) are essential and cannot be omit-

ted. On the other hand, in [27, Sec. 2.3] we provide classes of operators for which the

unique solvability of the main equation can be proved.

The necessity part of Theorem 2.7.20 was proved above. We prove now the sufﬁciency.

Let a function M(λ) ∈ W , satisfying the hypothesis of Theorem 2.7.20, be given, and let

ϕ(x,λ) be the solution of the main equation (2.7.118). Then (2.7.118) gives us the analytic

continuation of ϕ(x, λ) to the whole λ - plane, and for each ﬁxed x ≥ 0, the function

ϕ(x,λ) is entire in λ of order 1/2. Using Lemma 2.7.1 one can show that the functions

(ν)

(x,λ), ν = 0,1, are absolutely continuous with respect to x on compact sets, and

|ϕ

(ν)

(x,λ)| ≤C|ρ|

exp(|τ|x). (2.7.129)

We construct the function Φ(x, λ) via (2.7.123), and L = L(q(x),h) via (2.7.125)-

(2.7.126). Obviously, L ∈V

Lemma 2.7.9. The following relations hold

`ϕ(x,λ) = λϕ(x,λ), `Φ(x,λ) = λΦ(x,λ).

Proof. For simplicity, let

∞

∗

ρ|

V (λ)|dλ < ∞

(the general case requires minor modiﬁcations). Then (2.7.129) is valid for ν = 0,1,2.

Differentiating (2.7.118) twice with respect to x we obtain (2.7.127) and (2.7.128). It

follows from (2.7.128) and (2.7.118) that

−

(x,λ) + q(x)

ϕ(x,λ) = `ϕ(x,λ) +

2πi

˜r(x,λ,µ)`ϕ(x,µ)d

114 G.

Freiling and V. Yurko

2πi

ϕ(x,λ),

ϕ(x,µ)i

M(µ)ϕ(x,µ)d

−2

ϕ(x,λ

)

2πi

(

ϕ(x,λ)

ϕ(x,µ)

)

M(µ

)ϕ(x,µ) dµ.

Taking (2.7.125) into account we get

ϕ(x,λ) = `ϕ(x,λ) +

2πi

˜r(x,λ,µ)`ϕ(x,µ)d

2πi

ϕ(x,λ),

ϕ(x,µ)i

M(µ)ϕ(x,µ)d

. (2.7.130)

Using (2.7.123) we calculate similarly

(x,λ) −ε

(x)

Φ(x,λ) = Φ

(x,λ)

2πi

Φ(x,λ),

ϕ(x,µ)i

λ −µ

M(µ)ϕ

(x,µ) d

, (2.7.131)

Φ(x,λ) = `Φ(x,λ) +

2πi

Φ(x,λ),

ϕ(x,µ)i

λ −µ

M(µ)`ϕ(x,µ)d

2πi

Φ(x,λ),

ϕ(x,µ)i

M(µ)ϕ(x,µ)d

. (2.7.132)

It follows from (2.7.130) that

ϕ(x,λ) = `ϕ(x,λ) +

2πi

˜r(x,λ,µ)`ϕ(x,µ)d

2πi

(λ −µ)˜r(x,λ,µ)ϕ(x,µ)d

Taking (2.7.118) into account we deduce for a ﬁxed x ≥ 0,

η(x,λ) +

2πi

˜r(x,λ,µ)η(x,µ)d

µ = 0

, λ ∈γ, (2.7.133)

where η(x,λ) := `ϕ(x,λ) −λϕ(x, λ). According to (2.7.129) we have

|η(x,λ)| ≤C

|ρ|

, λ ∈γ. (2.7.134)

By virtue of (2.7.133), (2.7.112) and (2.7.113),

|η(x,λ)| ≤C

1 +

∞

∗

V (µ)|

|ρ −θ|+ 1

|η(x,µ)|d

, (2

.7.135)

λ ∈ γ, θ > 0, Re ρ ≥0.

Substituting (2.7.134) into the right-hand side of (2.7.135) we get

|η(x,λ)| ≤C

1 +

∞

∗

V (µ)|

|ρ −θ|+ 1

, λ ∈γ, θ > 0

, Re ρ ≥ 0.

Hyperbolic P

artial Differential Equations 115

Since

ρ(|ρ −θ|+ 1)

≤ 1 for θ,ρ ≥ 1,

this

yields

η(x,λ)| ≤C

|ρ|, λ ∈γ. (2.7.136)

Using (2.7.136) instead of (2.7.134) and repeating the preceding arguments we infer

|η(x,λ)| ≤C

, λ ∈γ.

According to Condition S of Theorem 2.7.20, the homogeneous equation (2.7.133) has only

the trivial solution η(x,λ) ≡ 0. Consequently,

`ϕ(x,λ) = λϕ(x,λ). (2.7.137)

It follows from (2.7.132) and (2.7.137) that

Φ(x,λ) = `Φ(x,λ) +

2πi

Φ(x,λ),

ϕ(x,µ)i

λ −µ

M(µ)µϕ(x,µ)d

2πi

(λ −µ)

Φ(x,λ),

ϕ(x,µ)i

λ −µ

M(µ)ϕ(x,µ)d

Together with (2.7.123) this yields `Φ(x, λ) = λΦ(x, λ). 2

Lemma 2.7.10. The following relations hold

ϕ(0,λ) = 1, ϕ

(0,λ) = h. (2.7.138)

U(Φ) = 1, Φ(0,λ) = M(λ), (2.7.139)

Φ(x,λ) = O(exp(iρx)), x → ∞. (2.7.140)

Proof. Taking x = 0 in (2.7.118) and (2.7.127) and using (2.7.126) we get

ϕ(0,λ) =

ϕ(0,λ) = 1,

(0,λ) =

(0,λ) −ε

(0)

ϕ(0,λ) =

h + h −

h = h,

i.e. (2.7.138) is valid. Using (2.7.123) and (2.7.131) we calculate

Φ(0,λ) =

Φ(0,λ) +

2πi

M(µ)

λ −µ

, (2.7.141)

(0,λ) =

(0,λ) −

Φ(0,λ)ε

(0) +

2πi

M(µ)

λ −µ

Consequently,

U(Φ) = Φ

(0,λ) −hΦ(0,λ) =

(0,λ) −(ε

(0) + h)

Φ(0,λ) =

(0,λ) −

Φ(0,λ) =

Φ) = 1.

116 G.

Freiling and V. Yurko

Furthermore, since hy, z

i = yz

−y

z, we rewrite (2.7.123) in the form

Φ(x,λ) =

Φ(x,λ)

2πi

(x,λ)

ϕ(x,µ) −

Φ(x,λ)

(x,µ)

λ −µ

M(µ)ϕ(x,µ)d

, (2.7.142)

where λ ∈ J

. The function ϕ(x, λ) is the solution of the Cauchy problem (2.7.137)-

(2.7.138). Therefore, according to (2.7.104),

|ϕ

(ν)

(x,µ)| ≤C|θ|

, µ = θ

∈ γ, x ≥ 0, ν = 0,1. (2.7.143)

Moreover, the estimates (2.7.104)-(2.7.105) are valid for

ϕ(x,λ) and

Φ(x,λ), i.e.

(ν)

(x,µ)| ≤C|θ|

, µ = θ

∈ γ, x ≥ 0, ν = 0,1, (2.7.144)

(ν)

(x,λ)| ≤C|ρ|

ν−1

exp(−|Im ρ|x), x ≥0, ρ ∈Ω. (2.7.145)

By virtue of (2.7.91),

M(λ) = O

, |ρ|

→ ∞, ρ ∈ Ω. (2

.7.146)

Fix λ ∈J

. Taking (2.7.143)-(2.7.146) into account we get from (2.7.142) that

|Φ(x,λ) exp(−iρx)|≤C

1 +

∞

∗

dθ

θ|λ −µ|

≤C

i.e.

(2.7.140)

is valid.

Furthermore, it follows from (2.7.141) that

Φ(0,λ) =

M(λ) +

2πi

M(µ)

λ −µ

µ.

Cauchy‘s integral formula

M(λ) =

2πi

M(µ)

λ −µ

, λ ∈int γ

Since

lim

R→∞

2πi

|µ|=R

M(µ)

λ −µ

µ = 0

we get

M(λ) =

2πi

M(µ)

λ −µ

µ, λ ∈J

Consequently

, Φ(0,λ) =

M(λ) +

M(λ) = M(λ), i.e. (2.7.139) is valid. 2

Thus, Φ(x,λ) is the Weyl solution, and M(λ) is the Weyl function for the constructed

pair L(q(x),h), and Theorem 2.7.20 is proved. 2

Example 2.7.3. Let ˜q(x) = 0 and

h = 0. Then

M(λ) =

iρ

. Consider

the

function

M(λ) =

M(λ) +

λ −λ

Hyperbolic P

artial Differential Equations 117

where a and λ

are comple

x numbers. Then the main equation (2.7.118) becomes

ϕ(x,λ

) = F(x)ϕ(x,λ

where

ϕ(x,λ

) = cos ρ

x, F(x) = 1 + a

cos

t dt, λ

= ρ

The solvability condition for the main equation takes the form

F(x) 6= 0 for all x ≥0,

and the function ε(x) can be found by the formula

ε(x) =

2aρ

sin2ρ

F(x)

cos

(x)

Case

1. Let λ

= 0

. Then

F(x) = 1 +ax,

and the solvability condition is equivalent to the condition

a /∈ (−∞,0).

If this is fulﬁlled then M(λ) is the Weyl function for L of the form (2.7.30)-(2.7.31) with

q(x) =

(1 + ax)

, h = −a,

ϕ(x,λ)

= cos ρ

x −

1 + ax

sinρx

, e(x,ρ)

= e

xp(iρx)

1 −

iρ(1 + ax)

∆(ρ)

= i

ρ, V (λ) =

πρ

, ϕ(x, 0)

1 + ax

If a < 0, then

the

solvability condition is not fulﬁlled, and the function M(λ) is not a Weyl

function.

Case 2. Let λ

6= 0 be a real number, and let a > 0. Then F(x) ≥1, and the solvability

condition is fulﬁlled. But in this case ε(x) /∈ L(0,∞), i.e. ε(x) /∈W

for any N ≥ 0.

The Gelfand-Levitan method. For Sturm-Liouville operators on a ﬁnite interval, the

Gelfand-Levitan method was presented in subsection 2. For the case of the half-line there

are similar results. Therefore here we conﬁne ourselves to the derivation of the Gelfand-

Levitan equation. For further discussion see [15], [16] and [25].

Consider the differential equation and the linear form L = L(q(x),h) of the form

(2.7.30)-(2.7.31). Let ˜q(x) = 0,

h = 0. Denote

F(x,t) =

2πi

cosρx cos ρt

M(λ)dλ, (2.7.147)

118 G.

Freiling and V. Yurko

where γ is the

contour deﬁned above (see ﬁg. 2.7.1). We note that by virtue of (2.7.111)

and (2.7.111),

2πi

cosρx cos ρt

M(λ)dλ =

∞

∗

cosρx cos ρt

V (λ)dλ < ∞.

Let G(x,t) be

tak

en from (2.7.4), and let H(x,t) be the kernel of the operator E + H :=

(E + G)

−1

, i.e.

cosρx = ϕ(x,λ) +

H(x,t)ϕ(t, λ)dt. (2.7.148)

Theorem 2.7.21. For each ﬁxed x, the function G(x,t) satisﬁes the following linear

integral equation

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

G(x,s)F(s,t)ds = 0, 0 < t < x. (2.7.149)

Equation (2.7.149) is the Gelfand-Levitan equation.

Proof. Using (2.7.4) and (1.7.148) we calculate

2πi

ϕ(x,λ) cosρt

M(λ

)dλ =

2πi

cosρx cos ρt

M(λ

)dλ

2πi

G(x,s) cosρs

cosρtM(λ)dλ,

2πi

ϕ(x,λ) cosρt

M(λ

)dλ =

2πi

ϕ(x,λ)ϕ(t,λ)M(λ) dλ

2πi

H(t, s)ϕ(s,λ) d

ϕ(x,λ

)M(λ)dλ,

where γ

= γ ∩{λ : |λ| ≤ R}. This yields

(x,t) = I

(x,t) + I

(x,t),

where

(x,t) =

2πi

ϕ(x,λ)ϕ(t,λ)M(λ) dλ −

2πi

cosρx cos ρt

M(λ)dλ,

(x,t)

2πi

cosρx cos ρt

M(λ)dλ,

(x,t)

x,s)

2πi

cosρt cos ρs

M(λ)dλ

(x,t) =

2πi

cosρt

G(x,s) cosρs

M(λ)dλ,

(x,t) = −

2πi

ϕ(x,λ)

H(t, s)ϕ(s,λ) d

M(λ

)dλ.

Hyperbolic P

artial Differential Equations 119

Let ξ(t), t ≥ 0 be a

twice continuously differentiable function with compact support. By

Theorem 2.7.15,

lim

R→∞

∞

ξ(t)Φ

(x,t)dt = 0, lim

R→∞

∞

ξ(t)I

(x,t)dt =

∞

ξ(t)F(x,t)dt,

lim

R→∞

∞

ξ(t)I

(x,t)dt =

∞

ξ(t)

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

dt,

lim

R→∞

∞

ξ(t)I

(x,t)dt =

ξ(t)G(x,t)dt,

lim

R→∞

∞

ξ(t)I

(x,t)dt = −

∞

ξ(t)H(t, x)dt.

Put G(x,t) = H(x,t) = 0 for x < t. In view of the arbitrariness of ξ(t), we derive

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

G(x,s)F(s,t)ds −H(t,x) = 0.

For t < x, this gives (2.7.149). 2

Thus, in order to solve the inverse problem of recovering L from the Weyl function

M(λ) one can calculate F(x,t) by (2.7.147), ﬁnd G(x,t) by solving the Gelfand-Levitan

equation (2.7.149) and construct q(x) and h by (2.7.27).

Remark 2.7.9. We show the connection between the Gelfand-Levitan equation and the

main equation of inverse problem (2.7.118). For this purpose we use the cosine Fourier

transform. Let ˜q(x) =

h = 0. Then

ϕ(x,λ) = cos

√

λx. Multiplying

(2.7.149)

by cos

√

λt

and

inte

grating with respect to t, we obtain

G(x,t) cos

√

λt

t +

cos

√

λt

2πi

cos

√

µx cos

√

M(µ

)dµ

cos

√

λt

G(x,s)

2πi

cos

√

t cos

√

µs

M(µ)d

s = 0.

Using (2.7.4) we arrive at (2.7.118).

Remark 2.7.10. If q(x) and h are real, and (1+x)q(x) ∈L(0,∞) , then (2.7.147) takes

the form

F(x,t) =

∞

−∞

cosρx cos ρt d

σ(λ),

where

−

and

are the spectral functions of

and

respectively.

5. The generalized Weyl function

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

`y := −y

+ q(x)y = λy, x > 0,

U(y) := y

(0) −hy(0).

120 G.

Freiling and V. Yurko

In this

subsection we study the inverse spectral problem for L in the case when q(x) is a

locally integrable complex-valued function, and h is a complex number. We introduce the

so-called generalized Weyl function as a main spectral characteristic.

For this purpose we deﬁne a space of generalized functions (distributions). Let D be

the set of all integrable and bounded on the real line entire functions of exponential type

with ordinary operations of addition and multiplication by complex numbers and with the

following convergence: z

(ρ) is said to converge to z(ρ) if the types σ

of the functions

(ρ) are bounded ( sup σ

< ∞ ), and kz

(ρ) −z(ρ)k

L(−∞,∞)

→ 0 as k → ∞. The linear

manifold D with this convergence is our space of test functions.

Deﬁnition 2.7.2. All linear and continuous functionals

R : D →C, z(ρ) 7→ R(z(ρ)) = (z(ρ), R),

are called generalized functions (GF). The set of these GF is denoted by D

. A sequence of

GF R

∈ D

converges to R ∈ D

, if lim(z(ρ),R

) = (z(ρ),R), k → ∞ for any z(ρ) ∈ D.

A GF R ∈ D

is called regular if it is determined by R(ρ) ∈ L

∞

via

(z(ρ),R) =

∞

−∞

z(ρ)R(ρ)dρ.

Deﬁnition 2.7.3. Let a function f (t) be locally integrable for t > 0 (i.e. it is integrable

on every ﬁnite segment [0,T ] ). The GF L

(ρ) ∈ D

deﬁned by the equality

(z(ρ),L

(ρ)) :=

∞

f (t)

∞

−∞

z(ρ)exp(iρt) dρ

dt, z(ρ) ∈ D, (2.7.150)

is called the generalized Fourier-Laplace transform for the function f (t).

Since z(ρ) ∈ D, we have

∞

−∞

z(ρ)

dρ ≤ sup

−∞<ρ<∞

z(ρ)

∞

−∞

z(ρ)

dρ,

i.e. z(ρ) ∈ L

(−∞,∞). Therefore, by virtue of the Paley-Wiener theorem [4], the function

B(t) :=

2π

∞

−∞

z(ρ)e

xp(i

ρt)dρ

is continuous and has compact support, i.e. there exists a d > 0 such that B(t) = 0 for

|t| > d, and

z(ρ) =

−d

B(t) exp(−iρt)dt. (2.7.151)

Consequently, the integral in (2.7.150) exists. We note that f (t) ∈ L(0,∞) implies

(z(ρ),L

(ρ)) :=

∞

−∞

z(ρ)

∞

f (t)exp(iρt) dt

dρ,

i.e. L

(ρ) is a regular GF (deﬁned by

∞

f (t)exp(iρt) dt ) and coincides with the ordinary

Fourier-Laplace transform for the function f (t). Since

∞

−∞

1 −cosρx

xp(i

ρt)dρ =

x −t, t < x,

0, t > x,

Hyperbolic P

artial Differential Equations 121

the follo

wing inversion formula is valid:

(x −t) f (t)dt =

1 −cosρx

, L

(ρ)

. (2.7.152)

Let

w u(x,t) be the solution of (2.6.1)-(2.6.2) with a locally integrable complex-

valued function q(x). Deﬁne u(x,t) = 0 for 0 < t < x, and denote (with λ = ρ

)

Φ(x,λ) := −L

(ρ), i.e.

(z(ρ),Φ(x,λ)) = −

∞

u(x,t)

∞

−∞

z(ρ)exp(iρt)dρ

dt. (2.7.153)

For z(ρ) ∈ D, ρ

z(ρ) ∈ L(−∞, ∞), ν = 1,2, we put

(z(ρ),(iρ)

Φ(x,λ)) := ((iρ)

z(ρ),Φ(x, λ)),

(z(ρ),Φ

(ν)

(x,λ)) :=

(

z(ρ),Φ(x,λ)).

Theorem 2.7.22. The following relations hold

`Φ(x,λ) = λΦ(x,λ), U(Φ) = 1.

Proof. We calculate

(z(ρ),`Φ(x,λ)) = (z(ρ),−Φ

(x,λ) + q(x)Φ(x,λ))

= −

∞

−∞

(iρ)z(ρ)exp(iρx)dρ −u

(x,x)

∞

−∞

z(ρ)exp(iρx)dρ

∞

(x,t) −q(x)u(x,t)

´³

∞

−∞

z(ρ)exp(iρt) dρ

dt,

(z(ρ),λΦ(x,λ)) = −

∞

−∞

(iρ)z(ρ)exp(iρx)dρ −u

(x,x)

∞

−∞

z(ρ)exp(iρx)dρ

∞

(x,t)

∞

−∞

z(ρ)exp(iρt) dρ

dt.

Using u

(x,x) + u

(x,x) =

(x,x) ≡ 0, we infer (z(ρ),`Φ(x,λ) −λΦ(x,λ)) = 0. Fur-

thermore, since

(z(ρ),Φ

(x,λ)) =

∞

−∞

z(ρ)exp(iρx)dρ

−

∞

(x,t)

∞

−∞

z(ρ)exp(iρt) dρ

dt,

we get

(z(ρ),U(Φ)) = (z(ρ),Φ

(0,λ) −hΦ(0,λ)) =

∞

−∞

z(ρ)dρ.

Deﬁnition 2.7.4. The GF Φ(x,λ) is called the generalized Weyl solution, and the GF

M(λ) := Φ(0,λ) is called the generalized Weyl function (GWF) for L(q(x),h).