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NEW Y07.:< UNIVCRSITY
INSTITUTE Or MATHi-.-AATiCAL SCIENCES

AFOSR276 ^ Washington Place, New York 3, N. Y.



^E ET PR/f i^gyy^ YORK UNIVERSITY



w

kf^ ^"^ ,j> Division of Electromagnetic Research



^ 1^1 lY ^ Institute of Mathematical Sciences



RESEARCH REPORT No. BR-36



Asymptotic Estimates for the
Sturm-Liouville Spectrum



HARRY HOCHSTADT



Air Force Office of Scientific Research
Contract No. AF 49(638)-229
Project No. 47500

FEBRUARY, 1 961



KEW YORK UWrVERSITY

Institute of Mathematical Sciences
Division of Electromagnetic Research.



Research Report No. BR- 56



ASYMPTOTIC ESTIMATES FOR THE STURM-LI OUVTLLE SPECTRUM



Haxry Hochstadt




H^ Virginia.
Department of Defense contractors must be established for ASTIA services,
or have their 'need to know' certified by the cognizant military agency
of their project or contract.



- i '



Abstract



It Is shown that the differential equation

y'» + [X + $(x)]y =

can, -under suitable conditions, be solved by ass\milng a solution
of the form

y = A(x) sin cp(x)



where



q)'(x) = \/rT"$(^+i^^^^|^ sin 2cp(x)



^'W = -^WsTT^-^'^W



Use of the first equation leads, when boundajy conditions are
applied, to asymptotic estimates of the eigenvalues.

In partictilar, in the case of Hill's equation, it is shown
that the Instability intervals vanish faster than any inverse
power of k, k being the order of the corresponding eigenvalues,
when $(x) is an analytic function.



- 11



Table of Contents

Page

Introduction 1

Hie method 2

The Sturm-Li ouvi He problem 3

Hill's equation 1^

The stability intervals for Hill's equation l8

A result due to Erdelyi 21

A special case 25

Bibliography 26



- 1 -

Introduction
In the classical SttLrm-Liouvllle problem one Is concerned
with, the differential equation



^p(x) g+ (Xp(x) + g(x))y =



subject to the ■bo\xndary conditions

ay(0) + py'(O) =

7jM + 5y'(rt) = .

The finite interval under consideration can be taken to be the
interval (O^it) , X is a parameter and p(x) , p(x)^ g(x) are real
functions, and furthermore p(x) and p(x) are positive, a, p, 7, 5
are constants such that |a| + |3| ^ ^^^ l/l + |5| 4 0* "-^^ ^^
then interested in determ in ing those values of X for which non-trivial
solutions of the differential equation can be found, which satisfy
the boundary conditions.

If p(x) and p(x) are doubly differentiable, one can by means
of the Llouville transformation reduce the equation to the form



,2

^+ (X + $(x))y = .
dx

Borg'- -J discussed this equation and the asymptotic form of
the characteristic values of X. His chief interest was a discussion



- 2 -

of the inverse Stiirm-Llouvllle problem, where one is interested in
reconstructing the differential equation from a knowledge of the
Sturm-Llouville spectrum.

In this publication a method is developed, which leads to
Borg's results in a very efficient fashion, and has the additional
advantage of yielding improved error estimates. Furthermore under
fairly general conditions one can easily obtain asymptotic series for
large eigenvalues and the corresponding eigenfunctions. The method
leads to particularly precise information in the case of Hill's
equation.

As will be observed subsequently the method is intimately
related to the W-K-B method, with the additional advantage of
providing a rigorous foiondation for the latter method.

The method

If we consider the differential equation



y" + Q^(x)y =



where Q(x) Is a dlfferentlable non-vanishing function, it seems
reasonable to assume a solution of the form



A(x) sin cp(x)



If in addition one postulates that



y' = A(x) Q(x) cos cp(x) ,



- 5 -

one can by means of the original equation show by a direct
calculation that



9



'(x) = Q(x) + 1^^ sin 2cp(x



A'(x) = -A(x) 1^ cos^ cp(x) .

SimllsLTly, if one assumes that

y = A(x) cos 9(x)

y' = -A(x) Q(x) sin rp(x)
one obtains

9'(x) = Q(x) - 11^ sin 2 cp(x)
A'(x) = -A(x) l^ sin^rpCx) .

Evidently once 'p(x) has been found from the first of these equations,
A(x} can be obtained by an integration from the second equation, and
it also follows that A(x)/A(0) is a positive fimction, since it is
an exponential.

The Sturm-Li ouvi lie problem

We now turn to the equation

y" + [X + $(x)]y =

subject to the boundaiy conditions



- 1^ -

ay(0) + Py'(O) =
7yM + 5y'(3t) = ,
and subdivide the problem into foiir distinct subcases:

la. a = 7 = 1

p = 5 =
Ila. a = 5 = 1

P =

7 arbltraiy

lb. 3=5 = 1

a, 7 arbitrary
lib. p = 7 = 1

5 =

a arbitrary

Problem la.



We now consider



y" + [X + $(x)]y =



where y(o) = y(jt) = ,

and (x) is a differentiable function of mean value zero^ i.e.



$(x)dx =
o



- 5 -
Under the assumption that

y = A(x) sin cf(x)



y' = A(x) \/X + $(x) cos cp(x)



we obtain



1 $'(x)



cp'(x) = v^X~r¥(^ + ^ ^^^^^1^^ sj,n'2rp(x



where

cp(0) =
9(rt) = fcrt

and k is any sufficiently large integer. Here sufficiently large
means so large that for the corresponding value of X

\ + $(x) > .

The differential equation can be rewritten as an Integral equation

X x

9(x) = J \/\ + $(x)" dx + i J ^ I '^^^^ sin 2cp(x) dx .
o o

From this equation we see that



^i)(x) = / ^X + ii^ dx + O(-) ,



- 6 -
and by an iteration

X X ^ X

cp(x) = J VTTTT^ dx + I J Y^ sin I2J \/X + (x) outside its interval
of definition (0,n) as an even function of period 2jt and mean value
zero. Then



$(x) = ^ a cos nx ,



n



and we find that



V^ = k + ^ + -^ f^^i^) dx + 0{-\)



8itk'

o



Now that an asymptotic estimate of \ is known one can find asymptotic
estimates of cp(x) and A(x) and therefore y(x) . Then one finds that



y(x) =



X

A(0) sin [2kx + ^ / f (x) dx]
o

[k + $(x)]^/^



[x.o(i)]



It is interesting to compare the eigenvalues with those of
the corresponding problem for $(x) = 0. Then

y" + Xy =



- 8



y(0) = y(rt) =



and



\A - V^ = A + o(-^)



for ^'(x) dlfferentlable. When '(x) is not differentlable



^^ 8rtk^ -* k^

o



Problem lb.



For the equation with constant coefficients we have in this



case



y " + y.y =
ay(0) + y'(0) =

7jM + y'(rt) =

so that

(7-a) V^
tan VX « =



X + 7a



- 9 -
One can show from, this transcendental equation that





c


\


2




^


^




7-


a\




1


7 -<


X-


7-a


\ Jt


-)


+


3


Jt




irk






1.5









V5C . ^ . ^ . ^"-^ ^ ' . o(-^)

When we t\im to the more general problem.

y" + [x + ^(x)Jy =
ay(0) + y'(0) =
/(it) + y(it) =

we assume a solution of the foim

y = A(x) sin (^(x) + A)
y' = A(x) Vx + $(x) cos (cp(x) + A) .

If



_^ Vx + (x). Therefore one can apply Integration

by parts to the integral

jt/2 . .■ It/2

r (;(x) + •••

and for X. we use

X = hk^ + en + ....



22 -



Retaining only € terms we have immediately



^^ l6k



Use of the boundary conditions



i(0) = ^(f) =



and the Fourier series



0(x) =


>;

n=l


a cos 2mc
n


shows that






M- =


^2k
2




so that







. v2 '^2k
X = ilk + — 7:— -



Similarly for case 1. ) we find



) v2 '^2k
X = 4k - — TT-



so that



23



l\+l -^1 = l^^2kl



It is evident from the previous development that Erdelyi's
result is not correct in general, hut Magnus has shown'- -' that in
an average sense it is correct. If



7^ < /g < 7^ <



denote the \ corresponding to even solutions of period jt and



^ < "2 < ^3 <



correspond to odd solutions of period «. Magnus shoved that



/ , ^ n n / , 2n

n=l n=l



A similar result holds for cases 3- ) a-nd k. ) .

A Special Case

The previous results were all derived tmder the assumption
that '^(x) was differentiable. One can show that the asymptotic
forms developed here do not hold when (x) is not differentiable.
Such an example was treated in detail in reference [5] . The equation
considered there is

y" + X^Q(x)y =



21^



Q(x) = 1



X < 1



1 < X < L



and it Is shown there that



^2n



2n
1 + a(L-l)



2 ^ ^ 2



2n-l



2n ~[
1 + a(L-l)J



- + 9 ^
2 ^ ^ 2



"■k



2n-l ~ ]
+ a(L-l)J



1 + a(L-l) i 2 ^ ^ 2



^2n-l



2n-l '] £ £
1 + a(L-l) 2 "^ ^ 2



[x] denotes the greatest integer less than or equal to x and the
0's must satisfy the inequality



< < 1 .



Here evidently



^1



k = 1 + a(L-l) * °(^)



and the instability intervals are 0(l),



25



ACKNOWLEDGMENT

The author wishes to thank Prof. W. Magnus for
arousing his interest in this area of research^ and
for helpful suggestions made in the course of numerous
discussions.



- 26 -



Blbxlography



[l] Borg, Goran

[2] Magnus W. and
Shenltzer^ A.

[3] Erdelyi, A.



[V] Magnus, W.



[5] Hochstadt, Earry



Elne Umkehrung der Sturm-Llouvlllesciien
Elgenwertaufgabe; Acta Math., 78, I-96 (1946)

Hill's Equation, Part I. General Theory;
N.Y.U. , Inst. Math. Sci., Div. EM Res.,
Research Report No. BR-22, 195?.

Ueber die rechnerische Ermlttlung von
Schwingungsvorgangen etc . } Archiv der
Elektrotechnik, 29, k73-h89 (1955).

The discriminant of Hill's equation j
N.Y.U. , Inst. Math. Sci., Div. EM Res.,
Research Report No. BR-28, 1959.

A special Hill's equation with discontinuous
coefficients} N.Y.U. , Inst. Math. Sci., Div.
EM Res., Research Report No. BR- 52, I960.



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Ar^ptotic_^sUmates^or_bhe
'sturm-Io,ouv3^^^__spectrura

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Online LibraryHarry HochstadtAsymptotic estimates for the Sturm-Liouville spectrum → online text (page 1 of 1)