AN ABSTRACT OF THE THESIS OF
RALPH EDWIN GABRIELSEN for the
(Name)
in
DOCTOR OF PHILOSOPHY
(Degree)
presented on
MATHEMATICS
(Major)
January 14, 1970
(Date)
Title: NUMERICAL ASPECTS OF THE INVERSE STURMLIOUVILLE PROBLEM
Redacted for privacy
Abstract approved
G5jAar Bodvarsson
Redacted for privacy
Arvid Lonseth
By general approximation methods, within the framework of
functional analysis, algorithms are developed for numerically determining the solution K(x, t)
of the integral equation
K(x, t) + F(x, t) +
and the solution dK(x, x)
dx
dK(x, x)
dx
(s, t)K(x, s)ds = 0
of the integro-differential equation
dF(x, x)
+ K(x, x)F(x, x)
dx
s)
F(s,x) aK(x,
ax
ds = 0,
These functions are essential in the solution of typical inverse problems of mathematical physics, In addition, applications are presented of the inverse Sturm-Liouville theory to problems of mathematical physics, Furthermore, asymptotic results are developed for
an approximate spectral function,
Numerical Aspects of the Inverse
Sturm- Liouville Problem
by
Ralph Edwin Gabrielsen
A THESIS
submitted to
Oregon State University
in partial fulfillment of
the requirements for the
degree of
Doctor of Philosophy
June 1970
APPROVED:
Redacted for privacy
I5'rofessor of Mathematics
Redacted for privacy
Professor of Mathematics
in charge of major
Redacted for privacy
Acting Chairman of Department of Mathematics
Redacted for privacy
Dean of Graduate School
Date thesis is presented
Typed by Clover Redfern for
January 14, 1970
Ralph Edwin Gabrielsen
ACKNOWLEDGMENT
wc ild hice to take this opportunity to express my sincere
appreciation and gratitude to my friends, for their sincere encouragement and backing has served as a vital nurture in the culmination of
this dissertation.
In particular, I would like to thank Dr. Arvid Lonseth for his
assistance and compassionate understanding of all the problems en-
countered in the process of completing this dissertation, and
Dr. Gunnar Bodvars son for his willingness and desire to assist on
any difficulty stemming from the dissertation.
This work was supported in part by the Atomic Energy
Commission Contract AT(45-1)-1947,
TABLE OF CONTENTS
Page
Chapter
I.
INTRODUCTION
1
1.0 Objectives
1.1 Definition
1.2 Background
1.3 Scope
4
II. ALGORITHM FOR K(x, t)
7
2. 0 Introduction
2.1 Kantorovich Approach
2.2 Algorithm Development
2.3 Results
III.
ALGORITHM FOR q(x)
3. 0 Definition of Problem
3.1 Simplification
3.2 Algorithm for aloft
3.3 Algorithm for 8K/ax
3.4 Results
IV. APPLICATION OF BASIC THEORY
4.0 Introductory Statement
4.1 Specific Application
4. 2 Indicated Applicability
4.3 Interrelationships Between Indicated Systems
and Basic Theory
1
1
3
7
8
12
21
29
29
30
31
33
38
41
41
41
46
48
V. RESULTS FOR AN APPROXIMATE SPECTRAL
FUNCTION
5. 0 Introductory Remarks
5. 1 Approximation of F(x, t)
5.2 Approximation of K(x, t)
5.3 Approximation of q(x)
57
57
59
67
69
BIBLIOGRAPHY
80
APPENDICES
81
81
Appendix A: Iterative Procedures
Appendix B: Kantorovich Method
Appendix C: Synopsis of Basic Theory
88
93
NUMERICAL ASPECTS OF THE INVERSE
STURM-LIOUVILLE PROBLEM
INTRODUCTION
I.
1. 0 Objectives
The fact that the implications of the Inverse Sturm-Liouville
Theory are far-reaching has been the motivating force behind this
dissertation.
The objectives of this dissertation are, first, to resolve certain mathematical problems which restrict the practical utility of the
Inverse Sturm-Liouville Theory, and secondly, to indicate the signi-
ficance of this inverse theory to other fields.
In Section 1. 1 the inverse Sturm-Liouville problem will be
stated; in Section 1. 2 a brief resume of the historical development of
the problem will be given, and in Section 1. 3 the principal theory of
this dissertation will be formulated.
1. 1
Definition
Determining
(1.0)
of the differential equation
q(x)
(X...(1(
with the initial conditions,
))yz=0,
0 <x <00,
2
y(0) = 1
y'(0) = h,
(1. 0')
for a given spectral function
where
p(X),
is a real number,
h
is referred to as the inverse Sturm-Liouville problem. p(k) is the
spectral function of (1.0)-(1. 0') if p(k) is a nondecreasing function
on
-co <
< 00
such that
oo
ao
E2(k)dp(k)
f2(x)dx
=
_oo
0
where
E(X) = lirn EN(X.),
N-00
and
E (X)
L.--
ts-
f(x)cp(x, X)dx,
0
for any function f,
which is square integrable in the Lebesque
sense on0 < x < co (see Reference [4]), and
cp(x, X)
solution of (1.0)-(1. 0'). For Equation (1.0) on 0 <x < n
is the unique
with the
boundary conditions
(1. 0")
y1(0) - hy(0)
the spectral function
{(Pi(x)}0i.0
p(X)
y(Tr) + Hy1(.7) = 0,
0,
equals
Z
oo
1
a
,
n
and
where {X.},0
i 1=
are the eigenvalues and eigenfunctions, respectively, of
the system (1. 0)-1. 0"), and
Tr
a
n
=
S'
0
cp
2(x)dx
n
(see Gelfand and
3
Levitan, [4]).
1.2 Background
The general theory for solving this problem has evolved during
the last 40 years. In 1929, V.A. Ambartsumyan produced the first
result in this field [1]. He proved the following theorem:
"1 t
X0, X1, .
. .
denote the eigenvalues of the system:
y" + {X-q(x)}y = 0,
(1.1)
0 <x <IT ,
y(0) = y(n) = 0,
(1.2)
where
q(x)
Ifn = n2
is a real, continuous function on the interval.
(n = 0,1, ...),
then
q( x) = 0.
The second important step in the general development of the
theory was made by G. Borg in 1945 [Z, p. 1-98]. His main result is:
"let
X
0
,X
1
denote the eigenvalues of Equation (1.1)
under the boundary conditions,
(h and H are finite,
real numbers),
(1.3)
y'(0) - hy(0) = 0
(1.4)
yt(Tr) + Hy(r)= 0,
and let
110,p.1
.,
be the corresponding eigenvalues of
(1.1) under the boundary conditions (1.4) and
4
(1.5)
h1y(0) = 0,
y;(0)
Then the sequences
{X. n}
uniquely determine the function
h, h1, and H.
h).
and
{j.
q(x)
}, (n= 0,1,... ),
and the numbers
n
In 1950, V. Marchenko made the next important advance in the
theory by showing that if the spectral function of (1. 0)-(1. 0') (or
(1. 0")) is given, then
q( x)
and the constant(s) h
(or h and H)
are unambiguously determined [7].
In 1951 Gelfand and Levitan presented the first effective method
for constructing
q(x)
of (1. 0)-(1. 0') (or (1.0 )) from its spectral
function, as well as giving necessary and sufficient conditions for a
monotonic function
p(X.)
to be the spectral function of (1. 0)-(1. 0')
(or (1. 0")).
In 1964 Levitan and Gasymov [6, p. 1-63] published a paper
which refines the 1951 paper of Gelfand and Levitan and also attains
a new result for a variation of boundary conditions. A brief synopsis
of this work will be presented in Appendix C.
1.3 Scope
In determining
q(x)
of the system (1. 0)-(1. 0') or (1. 0") from
the spectral function
p (X.)
by means of the Gelfand-Levitan theory,
it is necessary to determine the function K(x, t) and its derivative
5
dK(x, x)idx
([4] or [6]), for
q(x) = + 2 dX(x, x)
dx
where
K(x, t + F(x, t) +Sx F(s, t)K(x, s)ds = 0,
F(x, t) =
0 < t < x.
urnyN cos NTX x cos NiKtdcr(X),
N-1-00
-00
and
01X) =
(X) -
2
x.
o.
7T
Accordingly, in Chapters II and III algorithms are constructed
for solving K and
dKidx
numerically. In Chapter IV, the sig-
nificance of this theory is discussed. In particular, an application of
the theory is given in 4.1 which is relevant to the field of medicine, i.e.
an indirect method of determining the elasticity of a flexible tube is suggested. This example is indicative of the truly diverse applicability
of this theory. In 4. 2, applications of the theory to various types of
problems will be indicated, and in 4.3 the interrelationships between
the physical systems of Section 4. 2 and the Gelfand-Levitan theory
are explicitly given. For the inverse problem on
in Chapter V, that if--for sufficiently large
growth conditions--the first
N eigenvalues
[0, Tr],
it is shown
N and appropriate
K.
and normalizing
6
pIT
constants
a.
(a.2(x)dx)
0
1
are known, meaningful results can
be derived. In Appendix A, the integral equation of Chapter II is
solved by various iterative methods. In Appendix B, a brief synopsis of the theory of Kantorovich, upon which Chapters II and III are
based, is presented. In Appendix C, a brief synopsis of the GelfandLevitan theory is given.
7
ALGORITHM FOR K(x, t)
2. 0 Introduction
In this chapter an algorithm is developed for numerically determining the solution K(s, t)
(2.0)
K(s, t) + F(s, t)
in which
F
+s
of the integral equation
F(x, t)K(s, x)dx =
,
0<t<s<R
_< t_< s <R.
is a given continuous function on
in the framework of the Gelfand-Levitan theory,
F
00,
With-
is even abso-
lutely continuous in both variables and K is as smooth as
F
(see
[4] or [6]). Results of this chapter are based upon Kantorovich's
general theory of approximation methods [5].
The most important result to be developed in this chapter, for
our purposes, is as follows:
there exists an equipartition
For a given E > 0,
r, 0 < s < r <R,
[0,R] such that for each
constructed continuous,
that
of
[0, r],
there exists a readily
0 <s < r,
providing the partition
partition L;
if, in addition,
1
F
^in
r such
for each equipartition An
iecewise differentiable function
K(r, s) - rg:(s) I < E,
of
A
An
K
is at least as fine as the
of Equation (2.0) satisfies a
Lipschitz condition in each variable, then
8
n.Jn
K(r, s) = Kr(s) + O(A),
providing the equipartition An of [0, r} is at least as fine as the
of [0, Rj, where A denotes the length of the
equipartition A
1111
subpartitions of
A
.
1/1
The procedure followed in Sections 2.2 and 2.3 in developing an
algorithm for
K
of Equation (2. 0) is briefly sketched in Section
2.1. Sufficient conditions for constructing an algorithm for
of Equation (2.0) for fixed
s
K(s, t)
are developed in Section 2.2. Then
based upon Section 2. 2, the results of this chapter are developed in
Section 2.3.
2.1 Kantorovich Approach
Let
(2.1)
be the solution of
x(r, s)
x(r, s)
Cr
h(s, t)x(r,t)dt = y(r, s),
for
0 < s < r < R;
0
(or equivalently)
(2. 1)
Kx (s) = xr(s) + Hxr(s) = yr(s)
in the Banach Space Xr of real continuous functions on [0, r].
Now suppose there exists a linear transformation cpn that maps
Equation (2. 1) of Xr
into the system
9
(2.2)
x
h(t., t)x r (tk )
(t.) +
r3
y r(t.),
j = 1, 2, ...,n,
k=1
r. In order to determine
whether the solution (if it exists) of Equation (2. 2) satisfactorily
of the n-dimensional vector space
X
approximates the solution of (2. 1), it is necessary to answer questions of the following type:
Is the "reduced" linear system (2. 2) of algebraic equations
uniquely solvable?
If (2.2) is uniquely solvable, then to what degree of accuracy
does its solution actually represent the solution of Equation
(2. 1) ?
For a given E > 0,
is it possible to determine a linear
system of equations (i. e. , a system (2. 2)) such that its solution does not vary from the exact solution of Equation (2. 1)
by more than
c
uniformly?
Questions of this type are readily answered within the Kantorovich
theory under rather general conditions. Consequently, it will be
necessary to familiarize the reader somewhat with the Kantorovich
framework, and secondly to show how this theory ties in with the par-
ticular problem under consideration.
First, with respect to the Kantorovich framework, suppose 3.a linear operation Pn projecting Xr
on to
^4n
Xr
(a complete
10
subspace of X )
P 2=
P.
consider the
In the space Xr,
equation
(2. 1')
xrr (s) +
H7c
r (s) = Pnyr (s),
where H is a linear operator on Xr.
If the following conditions
are satisfied Equation (2. 1') and its solution "d*
xr will be referred to
as the approximate equation and solution, respectively, of Equation
(Condition that H and H be neighboring operations)
For
r-'n
everyxr
II
E Xr,
Pnr - HXr f<
1137
II
-
(Condition for elements of the form Hx, x Xr,
approximated by elements ofrsinX) For every
to be
xEX
x EX
Hx III. (Condition for close approximation of
min
(2. 1)) .3- Sr)
X
r
r
II Y
where
712
r
-
Yr
of Equation
3-
r
II
< rl 2 IIY r
II
may be dependent on
(see Kantorovich [5]).
11
If these conditions are satisfied, questions of the type mentioned on
page 9 are readily resolved [5]).
Secondly, with respect to the system under consideration (i.e.,
(-inn
Equation (2.1) of Xr and
suppose there exists a
(2.2) of X),
r
where 9 0, nr is the
of
X ,
isomorphic
to
subspace (-)n
r
Xr
Xr
n
function mapping Xr isomorphically onto X r and (pr is a linear extension of cp0, nr mapping Xr onto X nr such that
cp
=nP
0, r n
.
Therefore,
PO,
n -1 n
r
rvn
In view of the isomorphism between
Xr
n
and X r, Equation (2. 1')
can be transformed into an equivalent equation in
X nr
(and vice
versa). This is accomplished by substituting x r = ((p 0, r )
(2. 1'), and applying the operation 0, r to both sides.
xr
in
cp
xr- +
n.
0, r
n - 1x
0, r )
r
=
epPy.
0nr
Letting
H = 9q'0
nr1(49 0, nr -1
xr + H xr
=ryr.
Hence, the Kantorovich conditions on page 10 can be expressed in
terms of the system under consideration (i. e. , Equation (2. 1) of Xr
12
3Enr)
and (2. 2)
n.
0, r
by means of the mapping functions
cpr
and
Therefore, under appropriate conditions (see page 10) involv-
n
evn
ing X, X,r X,
r
the mapping functions
(p
r
,
0,
rn,
and the linear
operator H of Equation (2. 1) it is possible to answer all questions
of the type stated on page 9 (see Kantorovich [51).
Consequently, in
order to satisfactorily approximate the solution of Equation (2. 0), it
is sufficient to properly construct mapping functions and spaces in
the above context such that all conditions of the Kantorovich theory
are satisfied. In Section 2 2, this is accomplished for the solution
K(s, t)
of Equation (2. 0) for fixed
2. 2 Algorithm Development
K(s, t)
Mappings and spaces are constructed in this section for
of Equation (2. 0) for fixed
such that the Kantorovich theory is
s
applicable.
Let
An(r) =
Ai
and
A (=
=
I
Ji= 1
Ti-1< S < T., T.1 - T.1-1
=
1
Ai, (i =
1,
, n).
T
0
= 0},
1,
= 1, 2,
, n.
Let t.
Transform Equation (2. 1)
xr(s) + Hxr(s)
into
n
is the length of the interval Ai, i =
be the midpoint of
where
be an equipartition of [0, r],
yr(s)
13
xr (t.)
+A h(t., t xr(tk
j
(2. 2)
(j = 1, 2,
j
r (t.),
, n),
k=1
and
by
that the integral
(t)dt
be replaced by the finite sum
0
k=1
Equation (2.2) may be expressed in the form:
nr r
n--x
K x = xr +H
r r
=
ry,
r
where
ryr
..... Yr(tn)) E
= (yr(t
and
H
r
=
Ah(ti, t
)
Ah(t2, t
)
h(t n ,t1 )
Ah(ti
.
A h(tn,tn )
Definition. For the given equipartition
subspace of Xr
n -1--
xr = (0 r)
xr
,
for
tE
.
1
such that if
r Eln,
r
t(r) =
then
i}ni,
n
)
be a
x (1)
r k) g
E Xr and xr(t)
where =, n
g
let Xr
g
1
14
Hencemaps
It r
0, r
0.1n
r, and cpn is a
In brief- then, the questions introduced on
isomorphically onto
X
linear extension of 90, r.
page 9 will be resolved within the framework of the spaces
and Xnr,
on
where
r /Inr
is the Banach space of continuous functions
Xr
a Banach space of continuous functions linear on each
[0, r], 3e1
subinterval
X
and X r
A.
,
a vector space isomorphic to
X
fl
r.
Diagrammatically
r
n
where
cp
n
and
cpr
0, r
is a function mapping
is a linear extension of
For the given equipartition
of
A
=A,
onto
i = 1,
Xr,
,n),
P
2
n
=
)Tzl.
isomorphically onto
(0
Xr onto X r.
0, r mapping
(r) of [0, r] (with the length
An
let Pn be a projection operator mapping
P.n
,vn,n
Krxr = x,nr + /vnoin
Hx
rr Py,
nr
x ,
with solution r4n*
will be called the approximate equation and
solution, respectively, of
Kx
r,
n
r = xr + Hxr = yr
15
if
There exists an
such that for each xr E "-)n
Xr,
ri
P-HxiI
r
rr
rJ
n
<1113r r II.
A
such that for each xr E Xr there
6
,..,n 1,
exists an r E fvn
X
r such that I Hx r - x r II <1 1,r II x r
',al 5F
III. For each yr E X r, there exists a y
r such that
r
A
tvn
Yr - Y r < n2 Ar II Y r , where n2, r may depend. on yr.
e-ill
r4n
It should b e noted that H r i. s a linear operator on Xr,
There exists an
1,r
S'
II
II
II
and if x E Xr
if
II .
x EX,
or
then
Xr ,
114 =
then
114 =
max
i= 1, .
gi
I
. .
max
,n
I x(s) I
0 < s <r
and
,
where
,
1'2'
grd
A
A
In accordance with the Kantorovich theory, if 11,
11 , r and
x=
11
exists, and if the
can be made sufficiently small, if K-1
2, r
three conditions (I, II, III) are satisfied, then xr (the approximate solution, see page 14) exists [5]. Furthermore, if 1rA 111, Ar
A
and-92 , r converge to zero as n
the equipartition of [0, r],
0),
n
then
A (r)
II
(or equivalently,
xr -rII
°
is the solution of Kxr = yr [5].
Since the problem of interest directly involves
^in
xr
in
as
where x r
0,
than Xr,
of
co
rather
it will be necessary to interpret Condition I in the setting
To this end, we proceed as follows:
r
Writing (pnr = 0, rPn , we see that Pn =
X
Hrxr
Pnyr,
_n n
there exists an xr E X
((p
0,
n -1 n . Since
r
r
)
such that
16
xr
Hence
= (cp
n -1n
n
x r,
xr
0, r
nx +Hnnx =q,y,
rr
so
r r
nes-ln
nn
H
((p
x
r = q'rYr
r
0,r r
+
H
nrJn
n -1
0, r H r(q/ 0, r
=
If we replace Condition I by the Condition
nH cpn,on
x
n ,n
r 0, rr r r
cpr Hxr
then by letting
LIP
ri A
r
=
n
II
II
0, r
Hx = H x II
rr
n r
=
=
1r
x
Condition I is satisfied, for
n -1 nHx - H x
rr
r r
0, r
II (
1r
II (
II
< 11
0,
-1 n
r
r
-
-1 (HI n Hx
((Po, 7.)-11117rAll
n -1n
(PO,
r 0, r r
r
nv nell
;jcrri
r 0, r r
II Vcrti, II
In summary we have the following:
Problem. Kx(r, s) = x(r, s)
pr
h(t, s)x(r, t)dt = y(r, s);
0
(equivalently)
Domain.
Kxr = xr(s) + Hxr(s) = y (s).
0 <s <r < R .
Assumption. r fixed
(0 <r <R) and equipartition
(r) =
17
of
A = length of A
[0, r];
i = 1, 2,
, n.
tk the midpoint of
Reduced Problem (Algebraic System).
x(r, t.) +
A h(t.,
tk
Ak,
)x(r, t) = y(r, t.), (j = 1, 2, ...,n);
nn
n+Hnnx
K x =x
(equivalently)
r r
where
r r = yr
r
nxE Xn
r
,
yE
r X r.
Hr : An n x n matrix whose element in the i-th
j-th column is of the form A h(t., t.).
row,
1
Xr : Banach Space of continuous functions on [0, r].
rvn
X
:
Banach Space of continuous functions linear on each interv
ial
= 1, 2,
where
)r'cjerk) =
and
, n,
Z(t) =
for
x=
such that if x E rs-ri
Xr,
n-1
(90, r
)
x=
x,
l'
then
)
EX
t E1.
1
Lemma 1. For each equipartition An(r) = i }n
of [0, r] there
1
rA
(\in
r =r
exists an -A
x E Xr
such
that
for
each
r
t 2
()
cpnHZ
r r
rA
where CJ(-)
t
H 119
r 0, rr
n)"c
r
<11All3c) r
II
is the modulus of t- continuity of h(t, s):
r
18
r
wt 2 = sup {1h(t+6, s)-h(t, s)I, 0<s <r, 0 <t <r,
is the length of the interval Ai, i = 1,2, ... , n; h,
A
and H
I6
<-2
n
,r' "'
'r'
are discussed on page 17.
is a function in Xr, whose modulus of continuity does not exceed w(6). Let x E Xr,
Proof. Suppose
z
/A ztk xr t)
k
z(t)xr(t)dt -
[z(t)-z(t k
r (t)dt + A
Sjc.
k
Ak
rkjz(t.)(t)dt
r (t)dt-X' (t
k
A
= xr(tk ).
z(t)1 (t)dt
=>
=
kAk
Let
z(t) = h(t., t)
-
Az
[z(t)-z(tk)]Vcr(t)dt1
r(tk )1
tX)
rco()11371..II
=>
3
r
0
=>
h(t.,3 t).;\er (t)dt -
Ah(tj, tk )Vcr tk
11;jc
II Con/4)T
r r -H -nv
r 0, r r
1
< rwr(-4-11;
17:113cr 11
t
r
r
19
L(r)
= {A 1 }ri1 of [0, r],
n
Lemma 2. Given an equipartition
x
each
il Hx -Vc
r
EX
r
r
11
.
run
E Xr
rO
r
< ri
there exists an xr
A
1, r
II x
r
such that
where
,
II
for
wr(A) = sup {11-1(t,s+6)-h(t,s), 0 <t <r, 0<s <r, I5I < A},
A
1,r
and
=
Hx (s) = Crh(t,$)x(t)dt,
r
rwr(A),
s
j0
is the length of Ai, i = 1, 2, ...,n.
A
Proof. Suppose
z(s) E
Xr;
z(s) E Xr, where
let
n.)
z
(i.e.,
Tl'
coincide with the corresponding values of the
wise linear function whose values at the points
T.-T.
T
=
function
= 0),
s <T.+l, j = 1, 2,
-ri
)-2)(s)1
=
ri
A
z(s) =
T.3+1
3
sSince
,
,
[(T.+i-s)z(Ti)
1 z(s) 1
+1
3
Therefore
n-1.
(s-I.j)z(Ti+i
-s)(z(s)-z(T.)1
)z(s) + s-T.)z(s)]
(s)-1(s)I<
If
-r-
z.
Suppose
where
is a piece-
1(s--r.) z(s)-z(T.3+1 )1
(<
1
A
(T. ,-T.)W(A) =
(A
w(A) = the modulus of continuity of z(s).
0<s<T ,
then
A
20
IZ(S)-Z(S)I
z(s) - z(s'),
Consider
for
I Z(S)-Z(Ti)I <W(A)
=
z = Hx, x E
z(s)-z(s')I <
h(st, t) II x(t)Idt.
Ih(s, t)
0
Therefore
lz(s)-z(s')I <rwr(6)II
which implies for arbitrary
s,
lz(s)--V(s)I <rwrs(A)11x11
=>
< 11
r
II X
r
each
y
r
EX
<
rAII
z
Hx, x E X
A
ri1,
r
An(r) = {a.}n
11
r
rco (b.)
[0, r],
for
ron
yEX
r such that
r there exists a
II Yr -34r II
where
11
Lemma 3. Given an equipartition
for
yr
0<
,
<
where
112,
A
1
r
11Yr 11
(r)
(4
(A),
(r) (A) = sup {I y(r, s+6)-y(r, s)I, 0 < s <r,
61 < A
is the length of the partition
, n.
w
and
A
Ai, i
1, 2,
Proof. In the proof of Lemma 2, it was shown that for each z E Xr,
there exists a Nz EX
such that
I z(s)-z(s)I <(t), where
w
was
the modulus of continuity of z.
implies that
By.
ly(s)-(i)r(s)1 < (:(r)(A)
Hence,A
II Y-rir)11
< 12, r
letting
1
A
2
112,16:11Y°
21
1r
-
co (A)
,r
for arbitrary s.
II Y II
2.3 Results
Theorem 2.1. There exists an equipartition
[ 0, R] such that for each equipartition An(r) =
0 < r <R,
at least as fine as
A
(i.e.,
,
nn
K x nr
nn
= {A
A
'ft
11
of
{A.}n
1
of
[0,
A < A')
r r =y
is uniquely solvable for all y r E Xr ;
Ilx(r, s)-(cp
where
AT
i = 1, 2,
is the length of
n;
P
also,
x
0rn -1--n
r
< P Ilx(r,$)II,
)
Ali,
i = 1, 2,
n;
.
A
is the length of
is defined below, and the remaining notation
is discussed on page 17.
=
P= (1+ cR)rriR
A
(1+ 11 1, R )M 1
1-q=A
M M(1+
+
cR(1+
R
where M >
,,
sup {(1K
0 <r <R
q
M>
},
1
=
r
R
( 1 + ii
sup
{li
+j
< r <R
A
1,R
1, R
A
nl, R)
)
1-=q°
R
22
A
ER
R
1,R
+ ri
A
2, R
M.
for 0 <r <R, since
Proof of Theorems 1 and 2. Ar <11
A
rA
for 0 < r <R, since
<
1
Also,
1, R'
r
Tir =t (-2).
rwr (s). In applying the Kantorovich theory [5], it is neces11
11
1, r
s
A
sary to make
aA
where
< 1,
qr
r(1+11, r) + 1 1, r II ((P 0, r
-`r
K-
- 1 (pnic
A
=
r
11
ill
Therefore
qrA <
(1
A
+111
,
R
A
)+
1, R M)M1
Let
A
=A
qR
=
(11R
(1+11
)+
,R
1, RAM)M1 ,
therefore
<q R
ci&
r
Let
=
.6
P-
th,'.}n'
11
of the subinterval
Therefore
for
0 < r <R.
be an equipartition of [0, R],
AI = AI, (i = 1, 2,
=AT
q". < 1
for
n),
such
with the length
=A' < 1.
thatR
A =>
n'
KR xR
is solvable for all
partitions of [0, R].
for all
yr-
n(A ,)
E Xr 4 ,
E
-X-r1R1
Hence
where
(see Appendix B), and all finer equi-
-n(e1)--n(.6. 1)
[K
xr
A
,
la
=
yr]
is solvable
is an equipartition of [0, r] at
23
least as fine as A .
Let
<p)
w(A) = sup Ily(r, s+5)-y(r, s)I, 0<s <r <R,
s, r
and
INTH
min
min { max (I y(r, s) I),
0 <r <R 0 <s <r
such that
deleting those valuse of
where
r E (r0 -5, r0+6),
5
is a given,
positive, aribtrarily small number and Y(r0' s) =
for 0 <s < r0 , which implies, x (r0' s), the solution of the equation Kx(r , s) 7- y(ro, s), 0 <s <r0,
is
identically zero.
In particular, since x(r, s) is a continuous function, for a given
E
>0,
there exists a
0 < s < r.
5 >0 such that
I x(r, s) I < E
for I r-r 0 <5,
1
We will refer to this deleted set as the "exceptional set."
Therefore
(r) (A)
A
712,
providing
r
w
=
yr(s)II
n
A<
2, r
0, (A)
Y II min
,R
r I exceptional set for 0 <r <R.
Noting that the two theorems, plus the corollary and theorems to follow in this section must be qualified with respect to the "exceptional
set."
24
In accordance with the Kantorovich Theory [5], (see Appendix B), for
with
a given equipartition A (r) = {A,}n1 of [0, r],
of
P
A.,
= (1+Er)r
-1
n
(
r
cp0,
112,
111,
A
A
"E. A
R
+
-1
r, A< (1+ER)
Suppose
A
p,
(Exceptional set).
-
IIn
1111K11).
is the equipartition of [0,R] with the
= {A i}
i 1
is at least as fine as
r ql
-1
+ E A 1+
r
length of the subinterval
,
(Kr) (PrKil)'
therefore
2, R'
1, R
r
4. Er(1+11(`PO,
A
II <1
11
n -1 n -1 n
A
Kr)
A
E
Let
the length
I., 2,,
i.
A
r, A
A
Al,
A
A .,
=
and that
If 2,
i.e., A < A/.
A
(r)
Through use of the
Kantorovich Theory (see Appendix B), we directly obtain the following
estimate of
(K
)
-1
:
-1
Kr
<
II KII
Also,
ER<E R
Therefore
(1+71
(1+i1,R
A)M 1
1
=A
1-qR
1-qA
for
.5- M1
since
A)M
1, r
A
11
1, R
0<r<
<11
A1
1, R
25
(,9
1+
1, R)M
A
A1
p
r, A
since
< (11-E
1
r)
R
1
+
(1+
(1+11 , R )M1
1-qR
qR => Pr
q
1
M
=A
1-qR
A
=.-
< P r, A x
xr(s)- ((Po, nr)-1- x-
< pA
from which Theorem (1) directly follows.
Since
II x r
(90, :)- 1 -;--cr
II
<
II
(q)o,rn)-1;_c
n-1
=-'-""
Ilxr-(g9 ,r)
xr-('Po, nr)-17cril
Therefore
(IIx -(49 0, n)-117
r
r)(1-7):A
By selecting the equipartition
II(v
1)
such that
A
1111
ni-17cr II
r
P1
< 1,
(2) directly follows.
(9
- 1 Tc
PA
-:-
II ((Po,
-1;r II
Corollary. Under the hypothesis of Theorem 2. 1 or 2. 2
n-1--n
(q)
xr
A
< (1-Fi 1, R )
M1 ily r
=A
1-qR
Theorem
26
Proof. Since
Kr x r =yr=q)ny
rr
and
n
n -.1--n
x = (K)
r
n -1(Kr)
n -1n
n -1--n
(
xr
0, r )
r
r
= ((PO, r)
Therefore
II
lix
fl
II
< II (R1-11)-111 liCr-
where
I
A
II (Rnri
(1+11, It)M1
=46
II
cIR
Theorem Z. 3.
For the solution
Kr x =ny
r r,
n*
x
of
r
0 < r < R,
0 < s < r,
which is an approximate solution of the equation
Kxr(s)
whose solution is
yr(s),
0 < r <R,
xr
lirn
n
'Ix
00
n -1 n*
r - ((P0, r ) xr = 0;
in addition, there exist constants
Q,
Q1'
and
Q2
such that
27
n -1n*
<011R + Q 11
1, A
R
1
xr
Ilx (s)-((P0,r)
2, R
where the symbols have exactly the same meaning as given in Theorem 2.1.
Proof. From the results of Theorem 2.1, these results readily follow since
A
2, R
0
0
TIR
as
T1
as i0
0
1,R
as
A
0,
and
of concern is actually the function
Since the kernel h(s, t)
F(s, t)
A
0,
of Spectral theory, which is at least absolutely continuous in
each variable, let us now assume
h(s, t)
is absolutely continuous
in each variable. Furthermore, let us assume h(s, t) is lipschit-
zian in each variable This implies that there exists an a such that
Ih(s+o-, t)-h(s,
< alcr
for all s, t in
Ih(s,t+o-)-h(s, 01 <al cr I , for all
ws(6)<
( 6) < a 61
[0, 11].
s, t in [0, R].
al 61
<
A
a.,A 1<RaA,
and
A
2,R
for partition
A
w(A)
IlI min
< i3A
with the length of its subintervals
A. =A.
Theorem 2.4. If in addition to the hypothesis of Theorem 2.3, the
28
kernel h(s, t)
satisfies a lips chitz condition in each variable, then
fun
-x
for all r
such that
least as fine as
Proof.
P4
.
al
.
O(A)
0 < r <R providing the equipartition is at
29
III. ALGORITHM FOR q(x)
3. 0 Definition of Problem
In determining
of the Sturm-Liouville differential equa-
q(x)
tion
y" - q(x)y +
(3.0)
,
0 <x <R
from its spectral function, it has been shown by Gelfand-Levitan (see
[4] or [6]) that
q(x) = +2
dK(x, x)
dx
where
K(x, t) + F(x, t) +j F(s, t)K(x, s)ds = 0,
0 <t <x <
0
In this chapter an algorithm for numerically solving for
dK(x, x) /dx
is developed, where
(3.1)
dK(x, x)
dx
dF(x, x + F(x, x)K(x, x) +
dx
0
= O.
xF(s,x)(x,$)dx
ax
This algorithm is based primarily upon Chapter II. However, this
time it will be necessary to assume that F of Equation (2. 0) is in
C
1
rather than just in
C
0
.
The main result to be developed in this chapter is the following:
30
there exists an equipartition
c > 0,
Given
such that for each x, 0 <x <R,
of
A
(0, R]
there exists a readily constructed
continuous, pie cewise differentiable function GX(
)
on
0<s<x
such that
I
dK(x, x) _ Gn(x) I <E
dx
providing the equipartition An
is at least as fine as
A
of
(0, x]
associated with G'1(s)
.
1171
In Section 3.1, it is shown to be sufficient to develop algorithms
for
and
BK/Bx
8K/8t rather than for dK(x,x)/dx of Equation (3.1).
In Sections 3.2 and 3.3 algorithms are developed for
BK/8x,
aloft
and
respectively, based primarily upon Chapter II. In Section
3.4 the results of this chapter are obtained by combining the results
of Sections 3.2 and 3.3.
3. 1 Simplification
Since
dK(x, t)
BK(x, t)
=
(
8x
dx
aK dt
at dx/
t = f(x),
along
it follows that
K(x,
x
dx
(K(x, t))1 t=x
= (
aK
ax
+
8K,,
ot
)1
x
Therefore, determining dK(x,x)/clx of Equation (3.1) reduces to
31
determining
8K/ax I
and
t=x
3K/3tt=x
Differentiating Equation (2. 0) yields:
8F(s, t ).1.c x, s ds,
at
8F(x, t)
81C(x, t)
(3. 2)
at
at
and
aF(x, t) - F(x, t)K(x, x) - 51xF(s, t) 8K(x, s) ds.
ax
8K(x, t)
ax
(3. 3)
0
axiat of Equation (3.2) is unique and con-
It follows directly that
tinuous. Also, since
8F/3x is continuous by hypothesis, it fol-
lows from the spectral theory [6, p. 14] that 3K/3x
is unique and
continuous.
3. 2 Algorithm for moat
Let
aK'/at be the unique solution of Equation 3. 2).
ax (x, t)
at
=
aF(x, t)
at
aFts, t) K* (x, s)ds.
at
8F(x, t)
at
8F(s, t)
at
-
For fixed x,
Arn*
x, s)-Kx (s))ds
x 8F Kn*ds
x
let
n
8F(x, t)
at
Sx 8FK n* (s)ds,
at x
0
32
where
n*
Kx
(s)
is the approximate solution of K(x, s)
(2. 0), for fixed x, 0 <s <x <R,
of Equation
as developed in Chapter II (see
page 7), and we denote the unique solution of Equation (2.0) by K.
Therefore,
8K
Ai n*
at - Kx
&K (x, t)
at
'
=
-
x 8F(s, t) (Kx(s)
rdn *
-K* x, sflds ;
at
o
K(t)i*,
(^On
<e1M4 = c,
for 0<t<x,
where
M4 =
Hence, for a given
(vn
K (t)
tc x
max
0 <t <x <R 0
c1
> 0,
Fa (s, t)
at
there exists a continuous function
for each x, 0 <x <R,
aK*(x, t) Pen *
- Kx(t)
at
such that
I
<E2,
0 <t <x, c= e M4,
1
and
ax (x, t) .-K
"in *,
(t)1 < c1M4 =
at
,
for
0 < t < x.
In particular, for a given el > 0 there exists an equipartition
of [0, IQ such that for each equipartition
of [0, x] at
An(x)
least as fine as
ruin
Kx(s)
A
a continuous piecewise differentiable function
0 <s <x, can be constructed (as shown in Chapter II) such
that for each
x,
0 < x <R,
K -Kx
1-
< E0 <t < x,
and
33
I
aK (x, t) - Kn* ()
ti
at
x
<e
'
0 <t <x'
< x <--- R,
2
=
1
M
4
3.3 Algorithm for MK/3x
For Equation (3.3)
aK(x, t)
ax
(3. 3)
let
G
=
aF(x, t
ax
- F(x, t)K(x, x) -
0
s)
F s, t aK(x,
ax
s;
= ax /at denote its unique solution. Therefore
*
t)
G = aF(x,
ax
Fix x,
5-1
t K*(x,
F(s, t)G (x, s)ds.
-
let
an(t) = -F
(3.4)
rvn*
t)[K (x, x)-K
(x)]
xF(s, t)Gn(s)ds.
0
For fixed x,
(3.5)
consider the equation:
"in *
G(t) = - aF - F(x, t)Kx(x)
F(s, t)G (s)ds
0
In Equation (3. 4), divide through by K (x, x)
Gn(t)
A-'n*
K (x, x)-Kx (x)
--F
, t)
*
6-) n*
Kx (x):
1
A.'n*
(K (x, x)-K
(x)) 0
F(s,t)Gn(s)ds
Noting that Equation (3.3) cannot be numerically solved directly for
alciax,
since
K(x, x)
is typically unknown. Therefore
34
"Nn
GX(t)
*
"In
G
(s)
x
- -F(x, t) - S F(s, t)[
Nn*
K (x, x)-Kx (x)
x
Iv
Ids
,,,
.
K (x,x)-Knx '(x)
o
However
K(x, t) = -F(x, t)
F(s, t)K(x, s)d
for each fixed x, has the unique solution K (x, t).
Hence,
= K (x, t).
Let
"n*
denote the unique solution of Equation (3. 4) for the given
Gx
equipartition
i(x)
of
[0, x].
An
exists a unique function G(t)
appropriate equipartition
*
Therefore for each x,
satisfying Equation (3.4) for each
i(x) = {A }nI of [0, x] (i.
see page 7); it will be assumed that
Nn*
K (x, x) = Kx
(t),
then
there
Ain*
K*(x, x)
Kx
(x),
. ,
A<
for if
Ain
G(x, t) = G(t) of (3. 5). Hence, the follow-
ing analysis holds.
For fixed
x,
consider
G* (x,
t) - G:(t),
where
*
G*(x, t) =
ax is the solution of Equation (3. 3):
ax
(3.3)
G (x, t) = aF(x,
(3.4)
^n*
G (t) = -F(x, t)[K (x,
ax
Therefore,
t)- F(x, t)K (x, x) n*
x)-Kx (x)
F(s, t)G (x, s)ds;
F(s, t)G;( )ds
0
35
An *
*
G(t)
G (x, t)
- F x, t)Kx(x) -
=
* "n
F(x, t)[G -Gx]ds.
Let
An
G (t) * = G (x, t) - Gx(t).
n*
Therefore
Gx
(t)
Adn(o*
x
(3.5)
is the unique solution of Equation (3. 5):
t)
ax
A
there exists an equiparti-
> 0,
e3
such that for each x,
of [0, R]
441
x
0
We will now show that for a given
tion
is)n
F(s, t)G
(s)*ds.
(x)F(x, t) -
-K
"ni,
< c,
G
x
G*-G
(i.e.,
< E ),
for all equipartitions of [0, x] at least as fine as
A
Since
Gx(t)
- K (x, t
NI1*
K (x, x)-K(x)
G(t)
x
fin
1
I
*
K x, x)-K nx(x) *
*
- K(t
x
<E
'
0 <t < x.
Hence,
^n (t)-K
"in
(t) (K *(x,
x
An
x
/N-m*
*
*
)
x)-Kx( )I<
xG
n
I Gx(t)-Kx(t)(K (x, x) K (x)
Therefore
)
"../n*
I K (x, x)-K
E1
<C
.
1
(x) I.
36
I 8n(t) I - I kn (t)I
*
* x)-K(x)
K(x,
n
kin(t)*1 =E
+
*
2
j
<Ei
ki:(0*1).
For fixed x,
IIG*(x, t)- Gn*(t) II
where
G (x, t) = 81c (x,
ax
8K(x, t)
ax
<E
II Ir11),
(E
t)is the solution of (3.3):
OF(x, t)
ax
F(s, t aK(x' s) ds
ax
F
t)K(x, x) = 0,
and Gr4n* is the solution of (3. 5):
G:(t) = -
Ox
t)
F(x, t)Kin(x)*
F(s t)61.1(s)ds
From the above result, the assertion given on page 30 directly follows, since
K(x, 011,
<Jj 1((x, t)
and
C1
is inde-
pendent of x.
By letting
y(t) = -
OF(x, t)
ax
- F(x, t)K:(x)*,
Equation 3.5 can
be written in the form:
'-ajn(t)
F(s, t)Gx(s)ds = y (t).
This is the same type of integral equation whose numerical solution
37
was considered in Chapter II; the only difference is that in Chapter II
we were primarily interested in the case where y (t) = F(x, t);
whereas in this case
t
y(t)
x
ax
, t)Kx(x)
j:
*
.
Therefore, in order to apply the results of Chapter II to Equation
(3.5), it will be sufficient to show that the three conditions cited on
page 15 are satisfied. However, since the equations are the same
it is sufficient to check
except for the nonhomogeneous term y (t),
the conditions that involve yx(t). Therefore, it suffices to show that
for each y(t) there exists a 3r) (t) such that
Yx,
Yx
Since
x
rj
t( ) = -
<
2x
,
aF(x,
ax
II
-.t
0 < x < R,
Yx II
F
,
(see page 20).
n
t)K(x)*
which is continuous, the above condition is shown by following
exactly the same line of reasoning as presented in Chapter II on this
item.
Hence, for a given
A
m
of
[0,11],
c4
> 0,
and a sufficiently fine equipartition
for each equipartition
at least as fine as
A
esJ
entiable function Gx
m,
A
n
of
[0, 4 0 < x <R,
there exists a continuous piecewise differon
[0, x]
such that
38
x4
"in* m*
G
-G
x
<e
II
.
3.4 Results
For each fixed
X:
8K (x, t)
at
8K (x, t)
ax
= (
ax (x, t) - Kx(t)*) + '4n
K (t) *
at
;
ax (x, t)
*
ax
e4n
fsdn*
(t)) + Gx(t)
Gx
.
Therefore, since
dK(x, lt)
dx
dK (x, t
=
dx
(8K
OK (x, t
t=x
at
t)
ax
we obtain
dK (x, t)
dx
I
-(Kn(t)+G (t) )1
t=x
x
x
ax
(x(t)
)t=x
at
t
=x
aK*
,
+
IdK (x, x)
I
dx
"n* (x) <c M
-(Kx(x)+Gx
4+
uniformly in x,
tion
A n(x)
32) and
4%
providing for each
11P)ri*II
1(£
1
x,
+jj K
)
x
NJ*
1
its associated equiparti-
is at least as fine as both equipartitions
(see page
(see page 35), where
* ^in
II K -K
and
*
)
-G
(t)
t=x
x
8x
X-
<E
aF(x, t)
at
t K"xn(x)*ds
39
ron*
Since
rs-in*
OF(x, t)
*
F(x,
t)Kx(x)
8x
rdn(tI
x
(3.5)
Zn*
is readily determined. However,
K
is known,
K
evn(s)ds.
F s, t)Gx
can only be obtained by solving the integral Equation (3. 5).
As mentioned earlier, Equation (3.5) can be numerically solved
by the method developed in Chapter II. In this light, (see page 37),
for fixed x:
8K (x, t)
ax
18K (x, t
I
x
8x
t),
8x
IAK(x
<
I
'0*w'
aK (x, t)
I
(t)*
G
ax
ax (x
r-)m. *
- Gx
(t)
t)
8x
< E (E
I
- Gx (t)) + (G--(t)
-G x (t) );
x
- G--(t)
,
I*
+
I
G(t)44*(t)I
+ II 11:11
X4
1,-,n* 7-6m *
1G
X
-G
(t) I < E
For fixed x:
,
IdK (x, t)Nin*
- (Kx (t) + Gx (t) )1
ax
I(
ax (x, t - Kx
Zn*(t)
at
8K* (x, t)
I
at
-
+
Krin(t)"1/4I
'1(6 +
+
II
I
I
(
ax (x, t) - G
rs/s'm
*
(t)
ax
aK(X, t)
"-in*
ax
x
G
+
,-Im*
(t)1
Gnx (t)-G x
40
idK(x, x)
dx
I
x
e
x
ME1
+e
for
4
0 <x <R.
The results of Sections 3.2 and 3.3 were based upon the
assumptions that
there does not exist an r0 such that
0 < s < r0 < R, and
that there does not exist an xo
(9F(xt)
0'
0
for
such that
A.)n*
(x)= 0 for
- F(x , t)K
x0
F(ro' s)
0 < t < x0 <R.
If (a) does not hold, one must proceed as in Chapter II (see page 23).
If (b) does not hold, then in determining Gx the same type of
qualifications must be imposed as were imposed in constructing
^Jrn
Kx
under similar conditions in Chapter II (see page 15).
41
IV. APPLICATION OF BASIC THEORY
4. 0 Introductory Statement
The significance, in the sciences, of the inverse theory will be
indicated in this chapter. In Section 4. 1 an application of the theory
is given, which is relevant to the field of medicine; in particular, an
indirect method of determining the elasticity of a flexible tube is suggested. In section 4. 2 the application of the inverse theory to a few
problems of mathematical physics is indicated, and in Section 4.3
the interrelationships between the equations describing the systems
of Section 4. 2 and the inverse theory are given.
4.1 Specific Application
Consider a one-dimensional flow through a slightly flexible tube
of an inviscid liquid with constant density and velocity v(x, t), which
is constant over each cross sectional area. Let F(x, t) denote the
total flow through the cross section, and
f(x, t)
the cross sectional
area. Furthermore, assume that f(x, t) is a linear function of the
pressure P(x, t),
i.e. ,
f(x, t) = f (l+k(x)P(x, t)) = fo + fok(x)P(x, t),
where
k(x)
is the proportionality constant of elasticity.
42
In this section it will be assumed that all variables are sufficiently smooth, in particular, twice continuously differentiable.
A first order linearized theory will be assumed, in particular,
with respect to the equation
F(x, t) = pv(x, t)f( , t) = pvfo
pvfok(x)P(x, t),
the magnitude of the second order term vP will be assumed to be
negligible in comparison to the magnitude of v. Hence,
F(x, t) = pvfo
By conservation of mass
ar.
af
= H(x' t),
TX-
where
H(x,t) = source density (fluid mass added per unit length per
unit time).
8P
8v
Pfo
(4. 0)
-aTc
Pfok(x)
av
ap
ax
at
H
t).
H
pf
0
By conservation of momentum (one dimensional Euler Equation),
p
Dv
Dt
ap
ax
.
Dv_ av
av
at
v" ax
Dt
43
Disregarding the second order term
(4. 1)
v
8P
av
-aTc
P at -
av
ax
implies
A
From Equations (4. 0)-(4. 1) it follows that
a
(4. 2)
2
8v
l ay
k ax
- H = 0,
8t2
where
H=
H
ax pf k`
0
The elasticity of the tube is expressed by the function k(x).
Therefore, the problem of determining the elasticity by indirect
means entails determining the function k(x) by indirect means. The
function k(x)
can be determined by indirect means through utiliza-
tion of the Inverse Sturm-Liouville Theory. In this regard, let us
consider a specific example.
Now suppose we wish to determine
length
Tr,
where the end (x = 0)
from the other end (x
density
H(x, t)
k(x)
for a flexible tube of
is closed and the liquid flows
into a reservoir. Also, let the source
Tr)
be of the form G(x) sin wt.
Therefore, from
Equation (4. 2), the flow in this tube is described by the equation
(4.3)
where
(iv)
1
pvtt
=
+ J(x) sin wt
44
G(x)
J(x)
)
= dx
0pk(x)
At x=
(4. 4)
v(0, t) = 0.
It will be assumed that an incremental change in the internal
pressure of the reservoir is proportional to an incremental change in
the volume of the reservoir (i. e.
dt
dP = CdV
,
(volume)
).
Hence,
t) = f C (TT, t),
0v
since
dV(volume)
f dt
1T, t)
Therefore, by Equation (4. 0),
vx(ri,t) + hv(Tr, t) = 0,
where
h = k(Tr)Cfo.
Suppose
v(x, t)
U(x) sin cot,
then Equation (4. 3) reduces to
(4. 5)
(UT +pUz + J = O.
(4. 6)
U(0)
1
0,
U'(lr) + hU(Tr) = 0.
For (4. 5)-(4. 6) there exists the eigenvalues
ized real-valued eigenfunctions
00
{491 (X) }0
oo
{x.
]. }0
such that
and the normal-
45
1
)=
a.(p.- (x),
U(x) =
b
a. =
yo- l(Tr) + hcp.(
7r(P.(-
) = 0.
T.- ).
b.v.- (x).
(----, 9104
,
0) = 0,
+kipcoi(x) = 0,
w2p
1
i k(x)
../
a.go.- (x) =
1
Therefore,
[b.(-X..p+w2p) + a.p cp.(x) = 0.
=>
b. X.
- w2
vi)(pi(x)
U(x) =
X.-w
2
The
co
(J, vi)(pi(x) sin wt
x, t) =
i=0
wX.-2
Noting that this mathematical solution is singular at
as a result of linearizing the equations of the actual physical system.
By varying the frequency
w
of the driver mechanism it is
46
possible to determine the resonances of the system (in particular,
the eigenvalues
Exactly how this example ties into the Gelfand-
Xd.
Levitan Theory will be clarified in 4.3.
This example is applicable to the field of medicine, in particu-
lar, in determining indirectly the elasticity of an artery, which is a
measure of the degree of arteriosclerosis.
4. 2 Indicated Applicability
Now let us briefly look at a few additional mathematical models,
which describe numerous problems of mathematical physics, to which
the Gelfand-Levitan Theory can be applied.
Determining
(4.7)
U
=
xx
f(x)
(assuming
fE
(4. 8)
f(x)
of the equation
+ f(x)U(x,t), (with suitable boundary conditions),
reduces, by separation of variables
f(x)
C°)
(U
to determining
X(x)T(t)),
of the differential equation
Xii(x) - f(x)X + X X = 0, (with appropriate boundary conditions).
can be determined, for example, if in addition to knowing the
eigenvalues
k.(i 7-- 0, 1, 2, ... ),
1
the constants
c.
1
and
d.
1
(i = 0, 1, 2, ...).1-/can be determined (see Gelfand and Levitan, [4] or
[6]), where c.i
(%) 2
=Scp. 1S( )d, ,
and
ev
,
.s
'Pi
are the eigenfunctions such
1/a i of references [4] and [6] is determined from
C.
1
and d.,
1
47
that
A'ci(0) =
di,
(i = 0, 1, ... )
or if, in addition to knowing the eigenvalues
k.
under one set of boundary conditions, one boundary
of the altered sys-
condition can be changed and the eigenvalues
tem determined (Theorem C8, Appendix C).
Determining p(x)
(4. 9)
=W
p
(x)Wtt
(assuming
XX
p E CZ)
of
with suitable boundary conditions),
,
can be reduced (by separation of variables,
W = XT)
to determining
of the differential equation
p(x)
(4. 10)
X" + X. p X(
0, (with appropriate boundary conditions
)
which, in turn, can be transformed into a differential equation of the
form
(4.11)
y"(t) - q(t)y(t) + ky(t) = 0,
to which the Gelfand-Levitan Theory is applicable. For example,
p(x)
of (4.9) can be determined indirectly by means of the Gelfand-
Levitan Theory if, in addition to knowing the eigenvalues
(4.10),
K.
of
p(0), p'(0),
p
(g)a
=S1/2
pEC,
can be determined, where
2l,
(0),
and
and
LP n's
are a set of
48
eigenfunctions of (4.10), or if addition to knowing the eigenvalues
of (4.10),
p(0),
p,(0),
and
p E C2,
.
it is possible to
change one boundary condition and determine the new set of eigenvaides
for the system.
Similarly, for example,
p(x)
(assuming
p E C2)
of the
equations
p(x)U
Uxx'
Ut =
etc.
(p(x)Ux)x,
can be determined indirectly by means of the Gelfand-Levitan Theory.
4.3 Interrelationships Between Indicated Systems and
Basic Theory
The physical systems as described by the above equations will
now be explicitly connected to the Gelfand-Levitan Theory.
The differential equation
(4.12)
X" + kp(x)X(x) = 0,
0 <x < Tr,
with the boundary conditions
(4.13)
X'(0) + hX(0) = 0,
X'(IT) + HX(Tr) = 0,
(4.12)-(4.13), can be reduced to the system:
(4.14)
Z"(r) + q(r)Z(r) + AZ(r) = 0,
with the boundary conditions
0 <r
49
(4.15)
where
Z'(0) + gZ(0) = 0,
g
Z'(.12) + GZ(1) = 0
ale constants, by the appropriate transforma-
and
tion [3]; noting that the Gelfand-Levitan Theory is directly applicable
to (4.14)-(4.15).
The following theorem clarifies the relation between (4.12)(4.13) and (4. 14)-(4. 15).
Theorem 4.3.1.
If
is known and
p
exists a transformation setting up
a
1 correspondence between the
1
eigenfunctions of the system (4.14)-(4.15), where
related to
then there
p "(x) E L(0, Tr),
q
of (4.14) is
of (4.12) in the following manner:
p
1
q(r) = -
d2
p3/4( ) dx
and the constants
and
g
G
1
(
p/4(x)
1
of (4.15) are related to h and H
of (4.13) in the following way:
(h-
g
P")
4p(0)
1
1
G = (H- PI(Tr))
4p(v)
)
pl /2(0)
1/2
P
(1T)
(In [2], similar results are used but never proven explicitly.)
Proof. Let
t
p
1/2
(0d.
/
=p1/2( )4
0
0
For each eigenfunction
and
4J.(x)
of (4.12)-(4.13) with its eigenvalue
Xi,
50
1/4
consider the mapping(pi(r) = p
where
(x)Lpi(x),
r is the running
dummy variable in the transform space. By direct substitution, it
can be shown that
0 <r
X.q.(r) = 0,
V."(r) - cl(r)(P.(r)
and that
+ g.(0) =
where
cp! (1) + G .( ) = 0,
,
iyo!(0)
and G are given by the above formulas. Conversely,
g
suppose we have the eigenfunction
eigenvalUe
X.;
of (4.14)-(4.15) with its
p.(r)
that is,
q(r)v.(
+ X.v.(r) = 0,
rcp!'(r)
0 < r <I,
J
and
cp!(0) + g(p.(0)
q(i) + Gcp .(e ) = 0,
0,
where
g
and
q
P"
(h- 4p(0))
1
)
p
1/2
is constructed from
by the formula:
d2
1
(x) dx2
P3
Letting
co .(r)
P
11/4
)
P
p
q(r) =
j
(H- 217T)
4p(Tr)
G
(0)'
(x)
I
1
1/4
p(x))
1
1/2
(1r)
51
we see by direct substitution that
satisfies (4.12)-(4.13).
Liii(x)
Theorem 4.3.2. If there exists a unique solution
q(p) =
d2
1
y(0)
,
o<
of
y
< Tr
00)
q E L,
0
such that
is strictly positive, then there exists a unique solution
y
to the nonlinear differential equation
q(
S
10
d2Z
cg) = -Z 3(x)---2-Z(0) = Go,
Proof. For
0<x <I
,
dx
Z
Z'(0) = po
it is well known (see Theory of Differential
q E L,
Equations by Coddington and Levinson) that there exists a unique
solution to the differential equation
dy(x)
1
q(x)
y(x)
00) = 430.
y(0) =
dx
ao
(This proof is based upon ideas developed by Borg 12].)
Let
1
r
0
y(
and
2
Definition of the function
I
*1
y(
0
)
p(r):
)
dx
p(r) = +(---)1/2
.
dr
52
Noting that
dr
1
dx
y*(x)2
and
p(r) = y(x),
where
2d.
*
=Sx1
0 y (t)
r
d
dp(r) dr=
d
dxY'x' = dxP`ri - dr
dx
d.p(r)
1
dr
2
p(r)
Also,
d2y(x)
dx
d(
p (r) dr
2
dp(r) 2
p (r) dr
d2p
1
2,
P kr)
1
' -
2
2
p (r) dr
[d(
1
dp(r))]
2
dr
p (r)
p (r) dr
dp(r)1
1
dx
2
2
2
3
Since
2
Or)
dr
p"(r)
p 2( r)
p2(r)
dy(x)
d2
1
1
p(r)
p2(r) dr2
dx2
2pI(r)2
p
3
(r)
)
Also,
y(x)q(x)
d
2
dx
q(x)
-
>
y'(0)
y(0) =
y(x)2 ,
0
1
d2
1
1
p3(r) dr2 p(r)
),
P(0) = a
,
0
V)) = -
0
ao
53
where
1
r
*
y (t)
0
Since
p2 ()d
x
and
2td .
p2(r)d.r = dx,
0
p2(t)dt) =
d2
-
p (r) dr
0
Letting
Z(r)
1
q(
0 Z2()
p(0) =
,
p(0)=
0
P
0
a0
1
p(r)
2
d
)d)= -Z3(r) Z(r)
Z(0) -= a0,
Z > 0.
Z1(0) = 13
dr2
Therefore, it remains to show that the solution of
2
q( tcl
0
p2(t)dt) = -
1
3
p (r) dr
2
(
),
p(r)
p(0) =
1,
a0
00) = a0
is unique.
Suppose two solutions,
p
1
Construct a function yi,
dx 1 /2
(x)
such that
f-)
where
x=
C
0
r
and.
p12 ()d.
p 2(x),
exist.
54
it directly follows that
p1 >0,
Since
dx
2
rv
y1(x)
= P1 (1.),
dr - dx
AA.
y1(x)
pl dr
d
dg
A)
y1(g)
1
2
(
1
pi (0) =
(r) ),
p
2
dp1
2
P11(0) = 0
1
1
(x) =
71 3-c
=
0
d2
1
q(x) =
is well-defined.
dx
2
r-
2 '
VI
PO
a0
(r)
dr
P1 (r)
d2
dx
r
1
_
2y11 -
1
2
2
p1 (r)
d
p1
2
psi
(r) dr
dp(r)12.1
2
1
2
-
dr
3
p (r)
2(..)
d yi (x)
= _
dx
2
d2
1
2
p1
(r) dr
1
2
q(x)Yi(x)=yd2 (x),
r-'
1
p1 (r)
y1(0)
=
1
1
With respect to
q(x) =
1
p2
d2 eNJ
Ti y2(x) ;
rv
1
y:(0)
,
-po.
ao
y2
ni
y2( ) =
Ya dx
1,1
1
;
yi (0) =
2
0
Hence
P2 = P1
By combining the results of Theorems 4.3.1 and 4.3.2 with the
55
Gelfand-Levitan Theory, we obtain the following theorem.
Theorem 4.3.3.
If
The hypotheses of Theorem 4.3.2 are satisfied.
The eigenvalues
00
taili=0
oo
{X.}
i=0
and the normalizing constants
2
of (4.14)-(4.15) are known, where a.1
1
0
(04
and
4,i's
are the eigenfunctions such that 4,i(0) = di.
p(0)
and
p'(0)
po = -p1(0)
of (4.12)-(4.13) are known (a 0
of Theorem 4.3.2).
Then
1
=
p(0)
p(x), h, and H
of (4.12)-(4.13) are uniquely determined.
The following facts pertaining to Theorem 4.3.3 should be
noted:
p(x)
of (4.12)-(4.13) can be explicitly determined by means
of the Gelfand-Levitan Theory combined with the results of
Theorems 4.3.1 and 4.3.2.
The smoothness of p(x)
p
E L(0,71-)
of (4.12)-(4.13) (i. e .
,
for some m) can be determined by means of
the Gelfand-Levitan Spectral Theory and Theorem 4.3.3
could have been phrased accordingly.
In the hypotheses of the theorem, it was assumed that p(0)
and p'(0)
of (4.12)-(4.13) were given; however, the same
results could have been attained if
PI(z) were known instead of
p(0)
p(1)and
and
p1(0),
p( 2)
or
where
56
E
[0,
[0, w].
E
4. Also, if one boundary condition can be changed and the
(
14.
ieigenvalues
0, 1, 2,
determined for this altered
)
system, then it is not necessary to know the
s
and
Similarly,
0 <x <ir,
(k(x)Xl(x))1 + XX =
,
X1(0) + hX(0) = 0,
Xl(ir) + HX(r) =
may be transformed into
0<P<
Un(P) - q(P)U(P) + XU(P) = 0,
100) )U(0) = 0,
4 k(0)
+ (h-
W OO + (11- Weir)
47(7)
)TTi
=
by the transformation
U(P) =
P=
1/4 (x)X(x),
s,
1
0k
1 /2
k
d2
k 1/2 (t)
0
(g)
1 /4
1
=
'
1
q(P) = -
Tr
1
/4(x))
dt
.
(x) dP2
Hence, analogous results of Theorems 4.3. 1, 4.3. 2, and 4. 3. 3
directly follow.
d
57
V. RESULTS FOR AN APPROXIMATE SPECTRAL FUNCTION
5. 0 Introductory Remarks
For the inverse Sturm-Liouville problem
y" - q(x)y + X y = 0,
on the finite interval
hy(0) = 0, y'(rr) + Hy( ) =
y1(0)
it has been assumed in the preced-
0 <x < Tr,
ing chapters that the complete infinite set of eigenvalues
i = 0, 1, 2, ...
and normalizing constants
{xi},
i = 0, 1, 2,
{a.},
are
,
known. However, for practical considerations, it is important to
F, K,
know whether
and
q(x)
of chapters two and three can be
uniformly approximated if only the first
normalizing constants
shown that if
X
0
,X
X' 1
Nan = n
+
a0
k.
and
are known. In this chapter it will be
a.
.
eigenvalues
N
N-1' a 0' a 1
al
+
n3
+ O()
4
, 1
..
a
N-1
for
-1
are known,
n,
and
0
Tr
an
+
then for sufficiently large
GN(x, t), KN(x,
t),
F=G
and
1
+
3N,
for all
n,
there exist continuous functions
q(x) such that
1
+ O(--),
N2
K = KN + 0(
1
2
)
58
and
1
q(x) = q(x) +
where
ft
F(x, t) + K(x, t) +
F(s, t)K(x, s)ds = 0, 0 < t< x
0
and
cl(x) - +
ZdK(x, x)
dx
F, K and
The functions
will be approximated in Sections
q
5. 1, 5. 2 and 5. 3, respectively.
Results of this Chapter have neces-
sitated the estimation of certain asymptotic series. Similar techniques have been used by B. M. Levitan, lzv. Akad. Nauk SSSR Ser.
Mat. 28 (1964), 63-78.
Condition A.
X,
for
n = 0,
, N- 1,
a n,
for
n
,
0,
and
are known.
N-1,
The asymptotic conditions
ao al
n+
n
n
+
(-1)
+
n3'
n4
and
=
0, 1
IT
2
2
,
n3
are satisfied for arbitrary
There exist constants
C2 <n 3
2
+
b0
a2
cl
n.
and
c
2
such that
for all n > N,
59
<c2,
0
iT
and
brXn -
5.1 Approximation of F(x, t)
Theorem 5.1. Under the hypotheses of Condition A, there exists a
continuous symmetric functionGN(x ' t)
F(x, t)
such that
1
GN(x, t) +
where
1
GN = F(x, t) + 2 N (x+t)+f (x-td,
01-
N-1
cosNIX. x cosNIKt
FN(x,
0
t) =
a
+
0
cosNa xcosN.rTnt
2
Lian
[
Tr
cosnxcosnt
n=1
and
4b
00
cos nx
0
x
a0 a1
4b0
cos nxk--+)
n
3
,
2
2b0
n=N 2n (1+
Tr2
2
,
n=N
Trn
2
Trn2
00
a
x
n=N
Tan
a
0
sin nx(+---)
3
n
1
n
2b0
4b
0
Trn2(1+
-
2b
2
Tr
20
Trn
Proof. For the inverse Sturm-Liouville problem on
10, Tr
,
the
60
function F
can be expressed in the form see [4] or [6]):
(5. 0)
00
1
F(x, t) =
1
cosNI-Rox cosN1T ot -
cosNrX x cosNTX t
n
n
+
a0
a
-
2
cosnxcosnii,
n=
0 <t <x <it, an > 0.
where
Let
00
cosNI
aN(x) =
Xnx
-
2
cos nx).
n=N
Therefore,
F(x, t) = FN(x, t) +
(5. 1)
[aN(x+t)+ a
where
(5. 2)
N-1
FN(x,
t) =
cosNTox cox\/"K0t
n
it
a0
an
n=1
There exist functions
and
gn
a
(5.3)
c o s NIXx co s Nrfi *t
1
----+
Nrxn = n +
0
n
a
+
1
g
3
and
a
(5.4)
where
n
-n
n
=
Tr
2
+
b0
Z
such that
hn
- hn
,
n
n
2
cosnxcosnt
61
and
0
Tr
(,)
Let
an
and
".)
a
(5. 5)
= ++
a
n
2
hn
be defined as follows:
bn
".,
7
,
a
0
1
n
n
n
3
and
e`i
(5. 6)
b
n
=
n
0
2
- h n.
Since, by Equation (5.1
F(x, t)
to prove that
F=
x+t)+aN(x-t)],
, t) +
F
can be expressed in the form
F(x, t)
1
1
(x+t)+f (x-t)]) + 0( 2)
2
(FN(x, t)+fN
reduces to appropriately estimating the terms of the series
00
aN(x) =
cost\TX x
-
a
2
Tr
a
:
cos nx).
n=N
In this regard, the factor
1
an
be expressed in the form:
of the
nth
term of a
can
62
2
1
a
Tr
n
4b
0
hn
1
22
Tr n
2b
Tr
b0
2
Trn
Also,
C2
lit b0 2
nz
n31T
( 2 + b02 )2
1-
2+
n
0
Tr
C2
1n
b
3
0
I
n2
since
<1
for all n > N
by hypothesis. Therefore,
(5.7)
cosNrrnx
=
an
2
it
cos,\TX-x n
4b0cosNrrnx
2b0
it2n2(1+-2-)
Trn
hncosNr7 x
2 hn
b0
-7)
n
1-
b0
Since
NT-V
=n+
n
by Equations (5.3) and (5. 5),
cos NiXn
can be expressed in the
form
r.)
cosNiTnx = cos nx cos anx - sin nx sin anx.
63
(-1) IN
(anx)
(2i)!
oo
cosN/Tnx = cos nx + cos nx
2i
- sin nx sin 'ajx.
n
i=1
Therefore the
term of the series aN can be expressed
nth
in the form:
(5. 8)
cost\TX-nx
(
an
2
- -cos
nx) = w
4b0 cos nx
it
22 k i -t 2b0,
,
11
wn
2
)
-1)(a x)
4b0
+ cos nx
22
Zb
0
n (1+-2)
wn-
4b0
+ sin nx sin a".)nx
7
7
2
2b
\ir-n-(1+-0-)
wri
h
cos'./XT x
b
(.7,4"-)
n
2(
b
(
= Iln +1 2n
+
2
2
I3n + I4n
respectively.
Now, let us consider
2n
of Equation (5.8)
./
64
(-I) i(a x) 2i
co
(5. 9)
I2n
= cos nx
4b0
(2i)!
2
rr n (1+
i=1
2b0
_y
rrn
Since
2i
(a_x)
v (-1)ir.i
a x2
00
es,
"
_
(20!
Z.,
2
i=2
and by the definition of
2
(a)
n
A.J
(Equation (5.5))
= a0n( 7al 2
gn
[ngn + 2(a + -2)],
4.
it follows that
I
Zn
=-
a 12
x2a 0 +--)
cos nx
2
n
2
n3
4b0
2b
0
rrn2 (1+-2-)
rrn
2
x
- ---z cos nx [gn
(ng n +2(a0
n
4b0
2
)
I
IT
2,
-
2b0,
rrn ki-r-3-)
7rn-
oo
-1)i(2j x)2i
.
+ cos nx
(2i)!
i=2
(2
4b0
2
2b
rrn (1+-2°0
irn
00
)
(2i)!
i=2
since
2i
(-1) (anx)
65
A, 4
(a)
0(4),
and
co
i
(-1) (anx)
2i
= 0(1).
(2i+4)!
i=0
Furthermore,
gn
(ngn+ 2(a0+
n
Hence,
x
2n
2
2
a
a0
n
al
--))
4b0
i
2
/
it
n3
1
2b0
irn2(1+---2-)
\trn
n3
l ,2
= 0(7 )
4b
0
.
n
1
cos nx
+0(;.)
1
2b0
irn 2(1+-7)
\
irn-
Similarly, let us consider the term I3n of Equation (5.8).
I3n =
sin nx sin ax
n
4b0
2
cm2(1+ 2b0
Trn
Since
2i
sin (s)anx
.
=
A.)
anx
+
i+1
(-I)
anx)
PJ 3 3
anx
(2i+3)!
i=0
rv 3
an
and
1
n3
66
co
n.)
(a
x) 21 (-1) i+1
= 0(1),
(2i+3)!
1=0
it directly follows that
a
I
3n
a
4b
= x(n0+--)
3
1
0
1
sin nx + 0(7)
2b0
Trn
The term I4n of Equation 5.8),
hn cosNIXnx
I4n
(1r+0)2
2
n2
(-
hn
ir
0
n2
reduces to
I4n =O(
for it was shown on page 62 that
Therefore, by letting,
67
00
co
fN(x) =
cos nx
4b0
/2
\
2130
n=N n (1+---2-)
co
a
cos nx(
0
n
a
+
1
n3
(
41a
4b0
2
2b
)
0
2
0
)
3
2
wn (1+--
n=N
a
x sin nx( n0
a
x2
2bA
n=N
urn
we directly obtain the result
aN
=f
1
+
It directly follows from this result that
F
t = G (x, t) + 0
1
2
where
G (x, t) = -F
, t) -
1
[fN (x+t)+f (x-t)J
2
5.2 Approximation of K(x, t)
Theorem 5. 2. Under the hypotheses of Condition A, there exists a
continuous function
KN(x,
t)
such that
K(x, t) = K (x, t) +
Proof. The function
F
1
I
0 <t <x <
of Section 5.1 is related to the function
by means of the integral equation
68
x
(5.10)
S F(s, t)K(x, s)ds = 0, 0 < t < x Sir.
K(x, t) + F(x, t)
0
In Section 5.1 it was shown that
GN
a known continuous symmetric
function, is related to the function F by means of the equation
1
F(x, t) = G (x, t) + O() .
N2
Let
HN = F - GN,
and
GN = FN + RN,
where
RN =
Let
KN(x,
t),
(x+t )+f (x-t)].
2
N
for large
N,
N
denote the unique solution of the equa-
tion
(5.11)
GN(s, t)KN(x, s)d = 0 (see [5, p. 547]).
GN(x, t) + KN(x, t) +
0
Let
GN
denote the bounded linea operator such that
Nf(x,
t) =
0
GN(s,
Hence
KN =
where
)
t)f(x, s)ds.
69
(1+-a- N)-111 < M.
By subtracting Equation (5.11) from Equation 5.10) we obtain the
equation
(5.12)
(F-G ) + (K-KN)
+.S1
0
(FK-GNKN)
Therefore
)(K-KN
)=
-(HN
0
K).
HN
Hence
K
KN
=N)-1(-HN-510
HNK),
which implies
K-K
< MII-HN -51
0
NKII.
Consequently
K=
KN
1
+
)
N2
or
a
I
ISxH,K1
0
a
K-KN
is replaced by
N2
IS
HN(K-KN)I + I5HKI,
0
IN
then
0
can be explicitly calculated for sufficiently large
N.
5.3 Approximation of q(x)
Theorem 5.3. Under the hypotheses of Condition A, there exists a
continuous function
q(x) such that
70
1
q(x) = q(x) + O()
.
Proof. By Chapter III, it is sufficient to show that there exist continuous functions
and
4,1
such that
LIJ2
ax
= 4'1 +
and
aic
+2 O("R)
=
By point 2 of Condition A (see [6]) it follows that
aK(x, t)
at
4"
aF(x, t) 4.S x ar(s, t)K(x, s)ds =
at
o
at
and that
OF
at
co
OF
= _. +
at
n=N
8FN
at
+
1
2
-Nrrn
an
cow./ X.x
sin'/1nt +
n
2n
a.
cos nx sin nt],
[a'N(x-t)-al (x+t)1,
where
Ni-Xn
a' (x) =
an
sinNrrn x - 2n sin nx].
TT
Noting that
a
aT. aN(x-t)'
a' (x+t) = -
a
a
a,
(x+t).
71
As in Section 5.1,
1
a
n
2
4b0
w
2
im
1
hn
+
213 0
1+-7
Tr
b
(2+.7
n
wn
h
OZ(
n
1-
b
TT
0
2n )
Since
a
+
Nan =
0
a1
+
n
+g
3
1
n
n4
it directly follows that
n
2n
+ II +gnII2 +hn3
II,
11'1
=
an
where
2a
II1 =
4b
2a
0
wn
1
+
Trn
_
3
w
0
2
1
a
0
a
1
7+7)
n
n(1+
1
2bn \
Trn2j
2
112 =
4b0
Tr2n2
and
a
n+
113 =
1T
0
2b0\
1+ ---2Trn i
+-a
g
(
hn
1
n
b 02
n
1
(
n
)
Tr
(+-0
2
2
)
NTT
Determination of
__sin
NITx
-2nsin nxi:
a
n
n
sinNITnx = sin nx cos anx + cos nx sin a x.
72
-1) (aN x) 21
cos ax = 1 +
(2i) !
A.,2
sin nx cos anx = sin n
i=1
a
a
g
2
n
-- (ng
+2(a
- x sinnx [( n0+) +
n
3
n
1
a
n
+ sin nx
i=2
As in Section 5.1.,
00
(-1)i (a x) 2i
1
4
(21) !
1=2
and
Also,
2i
sin r-vM
a x = ax +N33\
ax
n
anx) (-1)
n
1+1
(21+3) !
1=0
Hence,
sin Nrrn x =
a a
a a
0
0
12
2.
sin nx - x sin nx(
+) + x cos nx(
+)
3
n
3
n
1
n
oo
+ sin nx
L
(-1)iAJ
(ax) 2i
(
2i) !
- x2 sinnx
gn
al ))
(ng
+2(a0
+--n
n
n2
i=2
I
rv
(a
,v3 3
+ xgn cos nx +(cos nx)anx
TI(-1)
(21+3) !
1=0
Therefore,
2i
1+1
I
I
73
-
=
2n
it
nsinN/Vnx
sin nx +
a
a1 +
2
---+)
(sinnx - x2sinnx( a0
n
3
III
II
-1) (anx)
sinnx
al
a0
x cosnx(
n +))
3
i+
2i
(.3! x)(-1)1"\
2i
tv3 3
n
+cosnx)anx
(2i+3 ) !
i=0
2
a
2
1
ngn+2(a0 +)+
x cosnx)] + gnII2 sin Nrr x
gnII
n2
2x2
+h nII3 sin Nax
n
Trn
al
.
sin
nx a
-
2
2
2
Tr
--nx
Tr
gncos
2
cos nx(a
1
sinnxgn(ngn+2(va+7))
n
N3 3
2
+
a
00
2
2
nx + Trn cos nx(anx
1+1
rv
2i
(a
x)(-1)
(2i+3)!
i=0
Let
N
II1
=
2a0
Tr
+
2a1
4b0
--- -(1 -r2
2
,
Tin
it
a
n
2
a1
n
4
1
(
'
2b0
Tin
rs.)
=>
Let
II1
II1
n
74
e
II1
a
2
(sin.nx- x sin nx( 0+
n
n
n
-
2x
IT
2
sin
fl
Let
a
a1
n3
n3
a0
1)2+ x cos nx(--+))
n
nx(aal+)2 + 2x
02
oo
EZ e
_
n
IT
a
l
cos nx().
ax
0
2a x
Tr
Tr
n2
cos nx
n=N
Therefore,
Ia
N - EN
I
<
III1
+
1112 + 1113 + III4 + III5
+
III6
+
where
rv
00
j,'/
00
- 1 ) (a x)
Ill
n sin nxi=2
1111 =
n=
(20!
rv00,v
2i
- cos nx
00
Ill
1112 =
(ax)
3x3
II
1113 =
n=N
x2
L(ng +2(a
+=)+x
cos nx)]
0
nn
n
n
n2
1r
oo
gnII2
sin NTT x
n=N
co
hnII3
n=N
i+1
i= 0
g
III5 =
( -1)
(2i+3)!
n
n=N
1114 =
2i
sin \Tr
xIl
7
+
III8
+ III
75
/
00
1116
00
2n
- 1 )i(rgnx)2i
sin nx
(2i)!
n=N
00
III
7
sinnx)gn(ngn+ 2(a0+ 7)
x
=
Tr
,
n=N
oo
III 8
2
=
ngn cos nx
it
n=N
00
1119
oo
n cos nx( 3x3
=
n=NW
n
fai x)21(- ni+1
n
( Zi+3) !
(i0
=
It can be readily shown that
+
III1
III2
+
1113 + 1114 + 1115 + III + 1117 +
1118
+ 1119 =
0(1)
N
Hence,
1
-z
ra
2-a (x4t)]-aNt)+(xat
N
[EN(x-t)-E (x+01 + 0(1-)
Let
aF
FN
at
N +
1
2
[EN (x-t)-EN(x+td
Therefore,
aF
at
Since
FN
+
(N) .
.
76
at
aF(s, t) K(x, s)d
at
aF(x, t)
at
aK(x, t)
-
it directly follows that
+
x aF
at
I
aF
at
where K. is the function constructed in Section 5. 2 such that
K=
KN
+
(-1)
2
.
Consequently,
t)
Determination of aK(x,
ax
aK(x, t)
ax
F(x,
aF(x, t) - F(x, t)K(x, x) -
_
=F
aF(x, t)
ax
N
+
1
2
0
rN(x+t) +a (x-t)].
OF (x, t)
N
ax
F(s, t)
ax
(x+t)
aN
1
-4-1
2
ax
aaN(x-t)
ax
00
cosq7n(x+t)
aN(x+t) =
U
n=N
-
2
7T
cos n(x+t))
.
aK(x, s)
ax
77
00
Nrk-
aaN(x+t)
ax
n
sin \In(x+t) +
an
n=
00
cosNrr (x-t)
2
-
rr
cos n(x -t)) .
n .2n
00
aaN(x-t)
ax
si n(x+t))
-Tr
an
n=N
2n
sin NrX"
(
-t) +
sin n(x-t)) .
Let
NTX- sin Nrr x
N(x) =
an
n=N
aa (x+t)
N(xax
-aN(x+t);
ax
a, (x) = E (x) + 0(
aaN(x+t)
ax
- 2nsin
nx].
it
t)
-t).
1
1
- -EN(x+t) + 0()
.
1
aaN(x-t) ax
-EN(x-t) + 0(N)
1 r
(x-t)
ax
aa (x+t)
r,.)
aFN
&aN
=-
ax
1
(x+t)+EN(x-t)] + 0
Let
FN = -( ax
+
1
2
[E
x+t)+EN(x-t)}).
78
Therefore
aF
at
ax = FN + o(-1),
F=
aF
+
GN
ax =
,..,t\>
N
H
N'
1
FN + 0(N).
=
1FN = 0(
).
Let
Hence
aF(x, t)- F(x, t)K(x, x).
ax
F(s, t)L(x, s)ds -
L+
Consider
LN
+51
N(s, t)LN(x,
0
s)ds = FN - GN (x, t)K(x, x).
Hence,
(LL N)
N
=-
0
GN
(s, t)(L(x, s)-L (x, sflds
0
N(s,
t)L(x, s)ds
- F(x, t)K(x, x) + G(x, t)KN(x, x).
Therefore
L-LN =N
+ [GF(x,t)K(x)x)+GN(x,t)K(x,x))-G
11 I-1-
II <
Therefore,
II
+II
x,t)K(x,x)+GN(x,t)KN(x,x))
VN II +II K II F-G
II
N II II K-KN II
79
L=
LN
+ 0(
1
).
Letting
ciN(x) = 2(LN(x,
x) +
(X
s, x)KN(x, s)ds),
+
0
it directly follows that
)
80
BIBLIOGRAPHY
SI
1. Ambarzumjan, W. A. Uber eine Frage der Eigenwerttlleorie.
Zeitschrift frir Physik 53:690-695. 1929.
2
Borg, G. Eine Umkehrung der Sturm-Liouvilleschen Eigenwertaufgabe. Bestimmung der Differential-Gleichung clurch die
Eigenwe rte. Acta Mathematica 78:1-96. 1945.
Courant, R. and D. Hilbert. Methods of mathematical physics.
Vol. 1. New York, Interscience, 1953. 561 p.
Gelfand, I.M. and B. M. Levitan. On the determination of a differential equation by its spectral function. Izvestiya Akademii
Nauk SSSR, Ser. Matematicheskikh 15:303-360. 1951.
(Translated in American Mathematical Society Translations, ser.
2,
1:253-304.
1955)
Kantorovich, L. V. and G. P. Akilov. Functional analysis in
normed spaces. New York, Pergamon, 1964. 773 p.
Levitan, B. M. and M. G. Gasyrnov. Determination of a differential equation by two of its spectra. Russian Mathematical Surveys
19:1-63. 1964. (Translated from Uspekhi Matimaticheskikh Nauk)
Marchenko, V.A. Some problems in the theory of second-order
differential operators. Doklady Akademiia Nauk SSSR. 72:457460. 1950. (Cited in: Levitan, B. M and M.G. Gasynov. Determination of a differential equation by two of its spectra.
Russian Mathematical Surveys 19:62. 1964. (Translated from
Uspekhi Matimacheskikh Nauk))
APPENDICES
81
APPENDIX A
Iterative Procedures/
The problem we wish to consider being:
çX
(A.1)
K(x, s)F(s, t)ds = -F(x, t).
K(x, t) +
0
For practical considerations, let us restrict our attention to the finite
region
0 < t < x< a, where
will be assumed sufficiently
a
large.
Domain
T = {x10
x {t10_<t <_ x < a}
x<
Equation (A.1) in the operator form:
Af(x, t) = -
f - Af = g,
Fcg, Of(x, 0
where
g = -F.
f = K,
0
We will assume F(x, t)
is the kernel of the said Integral
Equation discussed in Chapter II. Consequently, at least continuous.
Let us now consider (A.1) in the B-space
the Gelfand Theory, we know that
f
E
CT [61.
A maps
CT
into
f
CT.
from the fact it is an integral operator.
CT.
g
E
CT,
is as smooth as g,
and from
hence,
A is linear, which follows
A
is a bounded linear
operator, for
2/Similar interative techniques have been used by V. A.
Marchenka, Trudy Moskov, Mat, Obsc. 2, 3-83 (1953).
82
$F(t, t)f(x, t)dt .
= max
t, xET
T
Af
0
< max $
cT -t,xET 0
11fIIc
t)
I
max
T t,xET
f(
a
S
I F(t, t) I dt
,
0
where
ItflI
cT
max I f(x, t)(.
=
x, t E T
=>
s
c T -< 0 max
<t<a
II
a
F(t, t) I dt .
0
Therefore A is a bounded linear operator from CT -6. CT.
Since by hypothesis a solution exists (a consequence of the Gelfand
Theory) =>
(I-A)1
exists;
if
Since
11 All
c
<
(IA)1g.
n = 1, 2, 3, ...
g + Afn-1
f
11 f-fn
f=
(I-A)- 111 c
c< q < 1,
then
11 An))
11f-f
CT
11 fi 4011
c
n CT -1-q Ilf -f 0
f(x, t) = K(x, t); g(x, t) = -F(x, t),
11
1
we have that
11
CT
83
+F(g
c3c
t) + F , t)
Kn(x,
t)Kn-1(x, g)cll = 0,
0
ill CT
K-Kn11CT <
and if
11Allc
CT
11
11
n = 1, 2, 3,
(x, t)-K (x,t)I1
0
<q<1,
then
II K(x, t)
t) II
-Kn(x,C
1- q
II K (x, t)-K x,
1
in particular, if
S
max
q=
0<t<a
vg,
dg <
0
Now, let us consider the problem in the space
f + Af =
L,..,.
f and g E LT.
;
Let
max sa
0<x<a
0
I F(x, t) dt = M'
A
LT -* LT
pa r
...c
f
dx
=
LT
ad x
Af
I f(x, t) I dt
0
0
0
x
0
s, x
a
I
S' F(g, t)f(x, g)dg I dt
0
s.x
dx
<
0
F(g,
0
0
)1
I f(x, g) I dg)dt
,
CT
.
[5],
84
F(t, t) I I f(x, t)I
oI
=
F(t,
I f(x, t) clt
=
I f(c, g)idt
I dt
stx
t)dg
.
0
a
=>
Aill
dxs
LT
x
0
< M'
=>
.
.
fk,
I f(x, t))dt =
M' < 1,
if
then
In
K(x, t)-K (x, t) II
n
LT
Now, consider the problem in
1_,M1
I f(x, 01 2dt
dx
=T
i42
sx
a
PH 2
.
L2.
fEL2 ,
2
II K1(x, t)-K(x, t)
0
0
a
Af
Af(x, t) 2d;
dx
2
0
a
SdS
0
0
F(g, t)f(x, g)dg
Af(x, t) =
0
0
x
[I si F(t, t)f(x, t)dt112dt
0
,)j2dJdt <
85
f(x, g)idt)dx
F(g, t) I 2d.td.t
S'
0
0
a
$'
<IIfIIz
T
I
t) I
2dO.t.
0
0
pa [la
.)
II All <
2dWt)1
I F(g, t
/2
= q.
0
.
if
q<
,
then
K(x, t)-K(x, t) L
1-q
II
K1
For the three spaces considered
strictly less than
1
(x, t)-K x, t) II L2
q
can always be made
by restricting the domain T (i. e., by re-
stricting a).
Assuming
F(x, t) (EC].)
is once continuously differentiable
1
with respect to both variables, let us consider the problem in C.
F(s, t) + K(s, t) +S F(
t)K(s, g)dg
0
f
Af = g;
F(t, t)f(x, t)c:1
Af
0
86
Af
= max
,
s, t ET
S,t E T
13
ss
a
+ max
s,tT
E7`it
I1
+
I2
I-nS F(g,t)f(s,)dg
maxa
F(g, t)f(s, )dg +
11
I
+
1
F(g, t)f(s, g)dg
0
I3,
respectively.
I
1
- max S
s,t ET
t)f(s, 6)d0 < max Ill (
0
12 = max I F(s, t)f(s, s)
s,tET
max I F(s,
I F(t, t)1(q)
s,t E T
0
Fcg, ts(s,g)clt1
max S s Fcg, 01 di max
I maxi (f(s, t) I
0
I
2
< max If I (maxi F ) + max I fs(s, t) I (maxi-4
13 < max Si s Ft(g, t) I f(s,
I
I dg
I
max I f(s, s)
0
Ft(u,
t) I dg)
Odu 0
fg
(s, t)(t(u, Odu)dg
0
max If' max I 55F
(u, Odul
t
0
+ max
s,
max$
0
F (u, Odu
I
0
.
87
I1 + 12 + 13 < maxi f 1 (maxi
S srt(u, t)dul + max F(s,
+ max Ifs(s, t)1 (max
t)1cq)
u, t)duld
s, t)1 max
+ max
o
Let
0
0
max
s
+
12
0
+
13
u, t)dui4
max Sb
d
0
1
1 F( ,t)1 4
U, t)dul + maxi F(s, t)1 + max
M = max { f max
0
_< M(max) f 1+ maxi fs(s, t)1+ maxi fti )
11Af II
C
1 <1 M
=>
if M < 1, we have that
11 K(x,
t)-Kn(x,
< Mn
t)11
CT
1-M
11K1 (x, t)-K0(x, t)11 ci
,
t)
axn(x,
conwhich implies that Kn converges uniformly to K,
at
ax
(x,
t)
converges uniformly to
and
verges uniformly to aK(x,
'
ax
at
aK(x, t)
ax
88
APPENDIX B
Kantorovich Method [5]
A brief synopsis of the results of Kantorovich [5] upon which
Chapters II and III are based:
Let
X be a complete subspace of the normed space
Let
P be a linear operation such that
r-.)
1--)(x) = X;
P
(B.1)
In space
X:
Kx
x - Hx = y
(B. 2)
In space
X:
1") IV
A)
Kx
X,
XCX
P
/0 r.)
-"""
x - Hx
= Py
H and g are linear operations in X and
respectively.
X,
The following results are based upon the lemma:
Let
B-space
V be a linear operation from the B-space
Y
and let there exist for every
yEY
X
into the
an x E X
such
that
v(x)-y11 <
where
q<1
has for every
114 < NI1y11
and N are constants. Then the equation
y E Y,
a solution x E X
ilx11 <
satisfying
V(x) = y
89
Conditions.
For every
n)
x E X,
II PHX-HCc II <
For every
X E X,
there exists an
5E
exists such that
rt)
xEX
such that
<
ji
An element
where
Ti,
may depend on
< r1211Y11,
II
y.
Theorem Bl. If Conditions I and II are satisfied and K has a linear
inverse, then if
q= 101(1+1 X 11/
(B.3) the equation
Also
<
"1"(-X1
11
3,) II,
IITIC 11 111 K1 II < 1,
= 3jr.
has a solution x
where
for any niy E X.
'1/4)
N=(1+1X11 )11K
II.
Theorem B2. If Conditions I, II are satisfied, the linear operation
-1
and the Equation (B..1) has a solution
x,
then
II x.23)c* II <B II x.3 II,
where
x
is a solution of the Equation B.2) and
fr"+(ill 1X1+11211KH)(1+IIK1
PK!).
=
Theorem B3. If
K has a linear inverse
K
satisfies Condition A (for each
n = I, 2, ... )
90
the Conditions I, II and III are
n = 1, 2, ...,
3) for each
satisfied, and
liM
= 0,
nco
=0, and lim 12011
II
1
n--oo
then the approximate equations are soluble for sufficiently
large
and the sequence of approximate solutions con-
n
verges to the exact solution:
urn
II
n__.0o
and there exists constants
Qo' Q1' and Q2
such that
N*
II X.
00 + Q1/11 li
-Xn11
+ Q29211 Pit
Condition A. The existence of a solution of Equation (B.3) for every
implies its uniqueness.
yEX
As discussed in Chapter II, similar results can be derived between the space
and a space
X
In particular: Let
r.)
onto
cp
X,
co
to the entirety of
0
P=
0
x
x;
0
1-19
0
x = cp
= x - XHx
Let
Equation (B.2) transforms to
1 .....
A.)
^-1
X.
be a linear extension of the operation
yc
X. P:x; 90(2) =
-
x
9,
which is isomorphic to
be a linear operation defining a 1:1 mapping
0
Let
X.
on
X,
0
=
=
,
(PY
ill 9-01
0
= 91C90
91
For every V
Condition Ia.
E
3e,
11
171907- 911X II <
Theorem Bla. If Ia and II are satisfied, and
.rT
II
exists, then if
1C1
1X1[5(1+1K111)119-0111 + 1 11(P-01(PK11)11K-111
*
(B.4) the equation
handside
y e X,
11;*1
has a solution x
< 1.
for every right-
with
<= (1- I
I 1 )11
1111901
1)K-1
II
Theorem B2a. If Ia, II, and III are satisfied, linear operator
exists and the Equation (B. 2) has solution x
,
then
<
where
111
0
K
+ c( 1+
11
0
K
cpKII ),
12IIKIL
< 11IXI
Theorem B3a. If the following conditions are satisfied
K has a linear inverse
"R
satisfies Condition A
Conditions Ia, II and III are satisfied for every n,
n = 1, 2, ...,
lirn 711 90111
n-00
where
-1
=
Urn 1111i
n-00
(P0
= HI/1 11 2 119-16°11
0
n--00
0,
1
92
then the approximate Equation (B.4) is soluble for sufficiently large
and the sequence of approximate solutions con-
n
verges to the exact solution. Also
1
II x
where
stants.
< 6-1111 (P 0-1 II
xn
=
_1*
xn,
+ -5 1
and
II g9-
0
-5
,
(1211
-52,
+ -5 2
112 II q)-1
0
and
3
are con-
93
APPENDIX C
Synopsis of Basic Theory [6]
Let
(C.1)
y"-q(x)y =
with
00) - hy(0) = 0
(C. 2)
0 <x < 00, q(x)
where
number.
Q(x, X)
is a real integrable function; h is a real
denotes the solution of (C.1) with the initial condi-
tions (C. 2).
Theorem (Gelfand and Levitan)C1.
(C.1)
Suppose
Q(x, )'.)
is the solution of
satisfying the initial conditions (C. 2) and that
q(x)
locally integrable derivatives. Then there exist functions
and
H(x, t),
each have
each of the variables, such that
K(x, t) cos Nradt
cp(x, X) = cos Nirx
0
+S' H(x, t)cp(t, X)dt
0
,+ 1xq(t)dt,
0
K(x, t)
integrable derivatives with respect to
m+1
cos N/Xx = co(x, X)
has m
aK(x, t)
= 0,
t=0
at
94
8H(x, t)
at
- hH(x, t))
t=0
= K(x,
H(x, t) = 0
K(x, t) = 0,
Further, if
M > 1,
then
0,
xx - q(x)K
K
H(x, -t) = 0 for t > 0;
t > x.
for
Ktt
and
Hxx = Htt - q(t)H.
Theorem (Gelfand and Levitan). Suppose
q(x)
has m locally
integrable derivatives and that
X< 0
p()),
o(X) =
P(X) -
then the integral
tion
where
,
cos NX x do())
(x) = X( , 0)
I.(x)
2
has
Nrx--,
X>0
converges boundedly to a func-
in any finite range of values of x,
m+1
as N
00,
integrable derivatives.
Theorem (Gelfand and Levitan) C2. The kernel
K(x, t)
satisfies the
integral equation
K(x, s)F(s, t)ds = 0,
F(x, t) + K(x, t)
0 < t < x,
0
where
p(X),
F(x, t) =
X<0
N cos N/Xx cos NITtdo-(X.), r(X) =
n
oo
_oo
2
p(X)TrNg,
X > 0.
Theorem (Gelfand and Levitan) C3. The integral equation
95
K(x, s)F(s, t)ds = 0
F(x, t) + K(x, t)
0
has a unique solution K(x, t) for every fixed x.
Theorem (Gelfand and Levitan) C4. Suppose the kernel
K(x, t)
satisfies the above integral equation. Then the function
Q(x, X) = cos \Fa
K(x, t) cos \TXtdt
0
satisfies the differential equation
conditions
cp" + {X-q(x)}9 = 0
and the initial
4'(0, X) = K(0, 0) = -F(0, 0) = h,
9(0, X) = 1,
q(x)
+
2dK(x, x)
dx
Theorem (Gelfand and Levitan) C5. The monotonically increasing
function
p().)
is the spectral function of a boundary value problem
of the type (C. 1)-(C. 2) with a function
q(x)
(having m integrable
derivatives) and a number h if and only if the following conditions
are satisfied:
a. If
f(x)
E(X)
is the cosine transform of an arbitrary function
of compact support in L2(0, 00)
oo
E2(X)dp(X) = 0,
_oo
and
96
then
f(x)
(a. e. ).
0
b. The limit
S
I,(x) = lim
N---00
cos NTXxd.o-(X),
-00
where
<O
p(X),
(k) =
p(X) -
2
X>0
exists boundedly in every finite range of values of
.
has
(x)
x
and
integrable derivatives with 1.(0) = -h.
m+1
Theorem. Basic Inverse Sturm-Liouville Theorem on [0, Tr] (Gelfand)
c6.
-y" + q(x)y = ky
y'(0) - hy(0) = 0,
The numbers
and
{Xn}
y'(Tr) + I-Iy(rr) = 0
{an}
are the spectral characteris-
tics of some boundary value problem (C.1)-(C. 3) for
function
q(x),
where
q(m) (x) E L(0, Tr)
[0, id
with a
iff the following asymptotic
estimates hold:
a
NTTn = n +
where
X.
k
for
0
1
+ 0( --n-),
= a+2 o(-)
n
n m and all the
1
Tr
an
> 0,
and if the function
97
cc
F(x, t) =
1
"K t cos \f-Kox cosT0
a0
1
cosNaxcosNrX
t
n
n
+
Tr
-
an
it
cosnxcosnti
1
has integrable derivatives of order
in the region
m+1
This implies that there exists a function K(x, t)
(0< x,t < TT).
such that
pX
K(x, s)F(s, t)ds = 0,
F(x, t) + K(x, t) +
where
0 et ex< IT for the kernel K; K(x, t) = 0 for t > x and
q(x) =
2
dK(x, x)
dx
Theorem (Gelfand and Levitan) C7. If all the an > 0,
\IT
a
+
,
n
=
2
al + c(
a0
0
2
3
1
and
),
)'
wherea0, al, b0 are constants, then there exists an absolutely
continuous function
corresponding to the given
q(x)
a
1
0
=
IT
r
Lh+H+
1
..c1
0
q(t)dt]
Xn
and
an.
98
Note: If
0(---) for
then
can be replaced by 0( n4 )'
a
n3
q(x)
has an absolutely continuous derivative.
The basis of Levitan's new result with respect to a variation of
boundary conditions is the formula
=FL
n n=0
h -h
00
1
(C.4)
n
n
/
Xk-Xn
-X
k
-X
n
which gives an expression for-the normalizing constants of a regular
Sturm-Liouville operator in terms of two of its spectra. In addition,
gives a conditional solution of the inverse problem in terms of
two spectra, for once we know the numbers
tkn}
and
{an},
we can
define the spectral function by the formula
1
an
n<X.
and then form the operator by the prescription given by the basic
Spectral Theory.
For the problem
-yu + q(x)y = Xy,
(C. 1)
with
y1(0)
h1y(0) = 0,
yl(71-) + Hy(Tr) = 0
there exists the set of eigenvalues
}
Similarly, for
99
-y" + q(x)y = Xy
(C.1)
with
y'(0) - h2y(0) = 0,
(C. 6)
yr(-tr) + Hy(ir) = 0
there exists the set of eigenvalues
{la
,}.
Theorem (Gas ymov and Levitan) C8. Suppose that we are given two
sequences of numbers
{Xn}
and
{p.n} (n = 0, 1, 2, .. )
satisfying
the following conditions:
The numbers
Xn
and
P.n
Xn
and
interlace,
satisfy the asymptotic estimates of Theorem
C7, and (a0 a'0) then there exist an absolutely continuous
function q(x) and numbers h1, h2, H such that {}
is the spectrum of the problem (C. 1)-(C. 5) and
spectrum of the problem (C. 1)-(C. 6); moreover
h2 - h1 = ir(ato - ao).
{-1.11}
the