proceedings of the Royal Society of Edinburgh, 103A,215-228, 1986
Limit-point and limit-circle criteria for Sturm-Liouville
equations with intermittently negative principal
coefficients
W. N. Everitt
Department of Mathematics, University of Birmingham, Birmingham B15 2TT,
England
I. W. Knowles
Department of Mathematics, University of Alabama at Birmingham,
Birmingham, Alabama 35294, U.S. A.
and
T. T. Read
Department of Mathematics, Western Washington University, Bellingham,
Washington 98225, U.S.A.
(MS received 15 January 1985. Revised MS received 9 January 1986)
Synopsis
Limit-point and limit-circle criteria are given for the generalised Sturm-Liouville
expression
@u')'
w-'(-
+qu)
differential
on [a, b)
where
(i) p, 7, and w are real-valued on [a, b),
(ii) p - , q, w are locally Lebesgue integrable on [a, b),
(iii) w > 0 almost everywhere on [a, b)
and the principal coefficient p is allowed to assume both positive and negative values.
1. Introduction
We consider differential expressions of the type
lu
1
= -{ - (p(x)ul(x))'
4x1
where - ~ < a < b S w , w > 0 , and
p - l , q,
and
+ q(x)u(x)),
w E L:,,[a,
b).
x E [a, b)
(1.1)
(1.2)
It is known [lo] that condition (1.2) is both necessary and sufficient for the
eigenvalue problem lu = ilu to have Caratheodory solutions on [a, b) for any
A€@.
Following Weyl [24] (cf. also [19]) we define 1 to be of limit-point type at b if
not all solutions to ly = hy lie in Lk[a, b), and of limit-circle type otherwise.
Here, as usual, Lt[a, b) denotes the set of (equivalence classes of) functions u
216
W. N. Everitt, I. W. Knowles and T. T. Read
such that wfu is square-integrable in the usual sense. We also set
A = {UE L$[a, b): u, pu' ~Ac,,,[a, b), lu E L$[a, b)),
(1.3)
where AC,,,[a, b) denotes the set of functions that are locally absolutely
continuous on [a, b), and define the maximal operator L, for 1 in L$[a, b) to be
the restriction of I to the domain A. It is known [I91 that the number and type of
boundary conditions that one must apply to functions in A to obtain self-adjoint
restrictions of L is precisely determined by a knowledge of the limit-pointllimitcircle dichotomy for 1.
There is already a significant body of results on this dichotomy that enable one
to classify most equations of interest; the independent results of Evans [a] and
Read [20,21] are known to cover most of the previous limit-point criteria, and
that of Kuptsov [17] includes essentially all of the known limit-circle criteria. In
addition, a recent theorem of Atkinson [I] provides a concise and elegant
condition for the limit-point condition that is both necessary and sufficient.
Most of the known conditions assume that in addition to (1.2), p > 0 on [a, b).
Our aim here is to show that this condition may be removed. In addition the main
result (Theorem 1) combines the features of the results in [8] and [20,21]. We
also show that the limit-circle result of Kuptsov extends to the general case as
well.
We note that Sturm-Liouville equations with principal coefficients taking both
positive and negative values arise naturally when one separates variables in
certain partial differential equations that exhibit both hyperbolic and elliptic
behaviour. Such equations occur for example in the study of wave motion in a
rotating stratified fluid (cf. [Ill).
2. Limit-point criteria
The main result of this section may be stated as follows:
ASSUMPTION.
There exists a sequence of numbers {y,);=, c [a, b) such that
denote a
THEOREM1. If {y,) is the sequence defined above, let
sequence of disjoint intervals in [a, b) with each I,,, c (y,, Y,+~)for some n.
Suppose that, in each interval I,, there exists a real function Q, E L1(Im), a
non-negative function a, EAC,,, supported in I,, a non-negative measureable
function k,, constants 6 > 0, K > 0, a ( 0 S a S 1)) and G,,, > 0 such that q =
q1 - q2 with qi E L:,, i = 1,2, and the following conditions hold:
~ ama$ql
(i) (1 + K,(cu, 6)) ( p (( 0 3 where in I,, where H, = Q,a;-,-IY and
S
+ (1 + 6)a2H2,/ Ipl
c u , = l if p L O on I,,
=-1 i f p S O on I,,
and Kl(a, 6 ) > (a+ 3)'/46;
S k,
almost every-
Limit-point criteria for Sturm-Liouville equations
(ii) f (- or,q2 + Q,) 2 -KG,~$,,,
inf { J p ( t ) lf :E Jrn);
(iii) C , @il
$1, am(amqlak+ k,)f
for
Ip I-'
217
all intervals J c Im, where p, =
2
=m
where
Then the operator L is of limit-point type at b.
The proof of this result will be delayed until Section 4. However, a few remarks
on the theorem and its proof are in order here. In the formulation and proof of
Theorem 1, we have followed that of Theorem 1 in the paper [8] of Evans. The
main modification of the result of Evans is the inclusion of the functions a, in
condition (i) by analogy with the other main limit-point condition, that of Read
([21, Theorem 31; see also [20, Theorem I]). In this way we are able to obtain a
limit-point condition that includes the essential features of all known limit-point
conditions, and more importantly in the present instance, one that is also valid for
the case in which the coefficient p may take both positive and negative values in
the manner described above.
It should be noted that Theorem 1 can easily be modified to include the more
complicated result of Evans mentioned above. In this form, L is of limit-point
type at b if in addition to condition (i) above, one has, for each m, a positive
function h, E AC,,, and real functions 4, E AC,,,, supported in I,, and such that
(ii)'
( - ( Y , ~ z + Qm)hm 2 -KGmpm6
for all intervals J c I,, where p, = inf { l p ( t ) (t: E I,);
(iii)' C;=, @
Jzm ~
il
, h k @ ~ ( ~ q+ k,);
~ a kI p ( - f = m,
where
SJ
Here the functions a, need not be supported on I,.
In another direction, for the most important choice of a in Theorem 1, cu = 1,
the theorem can be much simplified by omitting G,, Q,, and @, without
affecting its generality. On setting a = 1, G, = 0, and Q , = H k = amq2 in
Theorem 1, we have
THEOREM1'. L is limit-point at b if, in each interval I,, there exist a
non-negative function a, E AC,,, supported in I,, a non-negative measurable
function k,, a positive constant 7 , and functions qiE L:,,, i = 1,2, such that
4=41-92and
(i) ( 1 + C,) lpl a: - a r n a ~ q+l (1 + r ) a k ~ kIp / I S km almost everywhere in I,,
where H, = fLamqz,Cl is some number greater than 4q-', and a,,, is defined as
before, and
(ii)
Ern inf (wlk,)
$Im
arn(amq,ak+ kmIf I P
I-;
= m.
zm
To see that this includes Theorem 1 when a = 1, we shall need the following
extremely useful result of Atkinson ( [ 2 , Lemma 51).
218
W. N. Everitt, I. W. Knowles and T. T. Read
LEMMA
(Atkinson). Let f be real-valued and integrable on an interval I = [a, 61.
For any positive real number y, a necessary and sufficient condition for the
existence of an absolutely continuous function g with g' S f and IglS y on I is that
f o r a l l a S a S p S b , JCfZ-2y.
Now suppose that the hypotheses of Theorem 1 hold with a = 1. Then by
condition (i) of Theorem 1 with a = 1, there exist constants K , and 6 with
K , > 46-', and functions a,, k,, and Q , such that
almost everywhere on I,, where H, = S Q,.
Also, by condition (ii) of Theorem 1 and Atkinson's lemma, for each
m , amq, - Q , = gk - em where em L 0 and Ig, I S K G , ~ ~Thus
.
Choose q < 6 so that K , > 4q-' and define
Cl = K,. Define
E
> 0 by (1 + & ) ( I + q ) = 1 + 6 ; set
47 = 41 + amern, q,# = %(gm + Hm)',
and k z = k , + K1(l + q ) ( l + E - ~ ) G ~ u ; .
H z = am(gm+ H,),
We now show that conditions (i) and (ii) of Theorem 1' hold with q, C,, qp,
q f , HZ, and k: replacing 6 , K , , q l , q2, H,, and k , respectively. Observe that
(1 + CI) I P I (a&)' - arnakq? + (1 + q)ak(Hz)21I P I
S (1 + K1) lpl ( d J 2
- amaiq, + (1 + q ) ( a k l Ipl){(l+ E)H%+ ( 1 + ~ - ' ) g % )
S k , + K1(l + q ) ( l + ~ - ~ ) G g =
a kz.
Finally,
L const.
O-'
m
im
o m ( a m q l+ok,)+
~
lp
1-1
by condition (iii) of Theorem 1, as required.
As a third remark, we note that if p 5 0 on [a, b ) then clearly, to modify a
given limit-point criterion to cover this case, one need merely replace q by -q in
the statement. The presence of the term a;, in Theorem 1 indicates that this is
precisely what happens in the more general situation considered above.
Next, notice that conditions (i) can be replaced by the condition
- amakql + (1 + K,(a, S ) } U ~ H(PI ~5/ k , almost every(i)' ( 1 + 6 ) lpl
where on I, for each m, where K l ( a , 6 ) > ( a + 3)2/46. This follows by an
obvious modification of the proof in Section 4. It was a feature of condition (i) of
[21,Theorem 31 that it is enough to have
Limit-point criteria for Sturm-Liouville equations
219
rather than having the two terms on the left being bounded separately (such a
situation could occur for example in the derivation of Corollary 6 below). In this
sense, Theorem 1 allows even more latitude as one could conceivably have (i) (or
(i)') being true without necessarily having the term 02H:/lpl
bounded as is
required, for example, in [21, Theorem 31. (See also [22, Remark 2 following
Theorem 1.)
Finally, we note that the method may also be applied to produce like results for
higher order differential expressions (cf. [9]), and to related questions involving
appropriate partial differential operators (cf. [7, 14, 221).
As one might expect from the generality of the statement, the theorem has a
wide variety of corollaries, and we now undertake a discussion of some of the
more important of these results. We shall use the simpler form Theorem 1' in
their derivation. In the remainder of this section we shall continue to assume that
each interval I, c (y,, y,,,) for some value of n. We begin with
COROLLARY
1 (cf. [8, Corollary 1; 14, Corollary 2.41). Let {I,)~=, denote a
sequence of disjoint intervals in [a, b). Suppose that
pm = inf {lp(t)l: t E I,) > 0
w, = inf {w(t): t E I,) > 0
for all m. Suppose also that for some K > 0, all intervals J c I,, and all m,
(a) sr cu,q 2 - ~ p t , ( l I Ipl-4-I
~
and
(b) CE=, wm(Sc IPI-4)' = W Then L is of limit-point type at b.
Proof. Set I, = [a,, b,],
and define c, by
and omby
By (i) and Atkinson's lemma, amq = am(ql- qz) where cu,qlL 0 and
Then it is clear that (i)' is satisfied if we choose k, = k for all m, for a suitably
W. N. Everitt, I. W. Knowles and T. T. Read
220
large constant k. Condition (ii)' also holds since
i nw
m
m
J
I",
omk!, 1pl-l
+ const. C w , ~ i , =
m.
m
We also have
COROLLARY
2. (cf. [20, Theorem 3; 61). Let {Im)z=, denote a sequence of
disjoint intervals in [a, b). Suppose that there exists a sequence of numbers
{v,)~,, such that for each m
(a) v m J $ l K > O
where Jmis defined by (2.1),
(b) Cz=l vilwm =
2 3 t
(c) J I ( ~ 9 ) 5- CvmJdm~
where (amq)-(x) = max (-LY,~(x),0) denotes the negative part of the function
wmq. Then L is of limit-point type at b.
Proof. We first note that, as in the proof of [20, Theorem 31, we can assume
without loss of generality that V , J ~ = 1. It is then enough to show that condition
(a) of Corollary 1 is satisfied. But
for all J c 1,. The result now follows.
An alternative form of Corollary 1 with (a) more restrictive and (b) less
restrictive may be given by using
(a)' - cu,q S K ( W , ~ ~ ) ~ W
all ~ J~c, I,,
all m,
(b)' Cz=l w; = w,
where Wm= SIm (w/ lp))f, instead of (a) and (b) as given in the corollary. This
may be derived from Theorem 1' by choosing cm so that
I,
defining amby
and taking km = w.
Similarly, the hypotheses of Corollary 2 could be altered to
(a)' vmWi, 2 K,
(b)' C;=, v,' = 03,
(c)' Si, ( ~ 9 ) - Cvm~L(~mpm)'.
These results become particularly simple when Ip[ = 1. As an example we have
COROLLARY
3 (cf. [4]). Let Ip I = 1 in I,, and let I, have length at least v > 0 for
Limit-point criteria for Sturm-Liouville equations
each m. Then L is of limit-point type at co if
(a) S j q L - K ( i f p = 1 on I,)
s J q S K ( i f p = -1 on I,)
for all intervals J c I,, and
(b)
w, =
The proof of this statement follows immediately from Corollary 1. These
conditions are satisfied for example if w is bounded away from zero on [0, m), and
either q is bounded below on an infinite collection of intervals with length at least
v > 0 on which p = 1, or q is bounded above on a similar infinite collection of
intervals on which p = -1. Another immediate consequence of Corollary 3 is
4. Let Ip( = 1 on I, and assume that each I, has length at least
COROLLARY
v > 0. Then L is of limit-point type at if q E Lr[a,co) for some r, 1 $ r < w, and
The next result is an analogue of [3, Theorem 51, which in turn extends the
well-known criterion of Levinson [IS](see also Sears [23]).
COROLLARY
5. Let I, = [a,, b,] be disjoint intervals in [a, b ) , and let
[c,, em]c (a,, b,). Suppose that there exists a function a and positive constants
K,, K2, 6 , and y such that in each interval I, the function a is positive and
absolutely continuous, and
(a) (1 + 6 ) I P ) (0')' - mrnqo25 K I W ,
( b ) J j J P J - ~ W T UL y
where J denotes either component of I,\[c,,
em],
(c)
W ~ Ip(-f
O
= m.
Then L is of limit-point type at b.
Proof. Define
= +)(I
= 0,
- r,
p
a-lwt lP\-:), x E [a,, c,)
elsewhere
where r, and s, are constants, chosen so that am(am)= a,(b,) = 0. We will use
Theorem 1' with a, as defined above, 9 , = q, q2 = 0, C, = 612, and q > 4ICl. On
the intervals [a,, c,) and [em,b,) where a, has the form a,(x) = a ( x ) f ( x )with
0 S f ( x )5 1, we have
Thus we may take k, = K6w, where K, > K,
+ K,.
Consequently, noting that
W. N. Everitt, I. W . Knowles and T. T. Read
222
cumqla2,+ k , > ( K , - K1 - K5)w, we have
inf (wik,)
m
] ~ , ( o ; ~+~k,)iu ~lp-4
I,
Im
a w f Ipl-l
= 00.
This completes the proof.
Note that this result is quite close to Theorem 3 of [21],except for the presence
of condition (b). This condition, which may be traced back to our use of the
compact support functions a,, seems to be necessary when p is not necessarily of
one sign in a neighbourhood of b.
The final result that we derive here is an analogue of Theorem 2 of [21].
COROLLARY
6. Suppose that there is a sequence {I,) of pairwise disjoint
intervals on which cumq is non-negative. If there are constants c < 1 and y > 0 such
that
m=l
where J, is a sub-interval of I, and each component J of ImVmsatisfies
then L is of limit-point type at b.
Proof. Following [21],we may assume (extending each J, slightly if necessary)
that, for some y > 0, each Zm = [a,, b,] may be decomposed into a union
of intervals with common endpoints, with a, = c,, b, = c4, Jm = [c,, c3],SO that
Q (wi lPl)+
c1
cwi 1 ~ 1 ) ; =
=
c3
I:'lq/plf =
lo
(WI
$1);
Im
and
Define
J
1(1
c2
lqlp~+.
c1
om(x)
=
( w /1
~ 1 ) ~ . X E [CO,cl],
= om(c1)exp
[c
lqlp14], x
E
[c,, c,],
= u r n ( c l ) e x p [ c r l q / p l 4 ] , xE[c2,c31,
=r
( w 1 IPI ) ; , * E [ ~ 3~, 4 ) ,
= 0,
elsewhere.
Limit-point criteria for Sturm-Liouville equations
223
We now use Theorem 1' with omdefined as above, q, = q, q, = 0, Cl = c-2 - 1,
and km = cW2w.Clearly, as ( p(
= w on [c,, cl] U [c,, c4], condition (i)' is
satisfied there. Also, on [cl, c3], we have cV21pl(ok)2- amqoL=O; thus
condition (i)' is satisfied.
For (ii)' we have, since om(cl) = y SIm (wl Ip1)4,
5
m=l
inf (wlk,)
1
om(amq,oL
+ k A i \PI-'
Im
As is noted in [21], this result extends the well-known criterion of Ismagilov
[13]. We obtain a useful special case of Corollary 6 when w = lp) = 1 and
gmq2 qm> 0 on each interval I,. In this case it follows that L is of limit-point
type at b if
where dm is the length of I,. In particular, consider the case a = 1, b = m, w = 1,
lpl = 1 on intervals I, with length at least v > 0, and q(x) = x" sin (nxP), where
cu,P >O. Then it follows by the method outlined in [21,p. 2721 that L is of
limit-point type at a~provided a > 2P - 2.
3. Limit-circle criteria
It is known (see [IS]) that, in the case p > 0, virtually all known conditions for
the limit-circle case are included in the result [17, Theorem 11 of Kuptsov. This
criterion is a direct consequence of an estimate for solutions y to Ly = 0 obtained
by a consideration of the quadratic form
where y and @ > 0 are arbitrary functions in AC,,,.
Furthermore, an inspection of the derivation of this estimate (see [12,16] for
simpler versions) shows that no essential use is made of the assumption that p be
positive. The only point that need be checked is whether or not the form E is
positive definite in this new situation; but this follows easily when one notes that
for any solution y to Ly = 0, y and the quasi-derivative (see [19, para. 171) py '
cannot have a common zero. Thus we have
THEOREM
2. Let p and q satisfy the basic conditions (1.1), (1.2), and suppose
W. N. Everitt, I. W. Knowles and T. T. Read
224
that there exist functions y and $I > 0 in AC,,, such that
Then L is of limit-circle type at b.
Of the many possible Corollaries that one could formally derive from Theorem
2, we isolate two.
COROLLARY
7. Let p and q satisfy the basic conditions (1.1), (1.2), and suppose
that there exist functions y E AC,,, and q, such that pq, E AC,,, is positive and
(P40l1- 4Y
(a) -2-,
P40
P
Then L is of limit-circle type at b.
Proof. Set $2 =pq, in Theorem 2. Then, using (a) and (b), we have that the
left hand side of (3.1) is majorised by
1
b
SK
w@qo)-$ exp
[(I
( ~ 4 0 ) ' 2YI+ 1 IY Z2pqo
P
G
P
yl l ) ]
The result now follows immediately.
It should be stressed here that in the above theory, p need not be continuous.
In particular, consider the following
EXAMPLE.
For fixed
E
> 0, define p, q, and w on [I, m) as follows:
p(x) = -1, x E (n, n
= 1, otherwise;
q(x) = x2+&,x
E
+ n - " ) , n = 1, 2, . . . ,
(n, n
+
n = 1, 2, . . . ,
- - X 2 + ~ , otherwise,
and set w = 1. Then it is clear that p(x)q(x) = -x2+" on [I, a). In Corollary 7
replace q by -q and set 2y = (1 + ~12)t-'for t 2 1. Then conditions (a) and (b)
above are easily seen to be satisfied. Also, for all x 2 1,
Limit-point criteria for Sturm-Liouville equations
225
Thus
and (c) is satisfied also. Hence, by the corollary, the corresponding operator L is
of limit-circle type at m.
Our final corollary is an analogue of [S,Theorem 20, p. 14091.
COROLLARY
8. Let p and q satisfy the basic conditions (1.1), (1.2), and suppose
that pq E AC,,, is negative on [a, b ) , and that p l(pq):lr E AC,,,. Suppose also that
Then L is of limit-circle type at b if
[w l p q l - i<
m.
Proof. This result follows easily from Theorem 2 if we set $ = ( v p q ) 4
and
y = -p(qrr]) where r] = $-i.
4. Proof of Theorem 1
Without loss of generality, it is enough to consider only real elements of A.
Consequently, let u E A be real. From an integration by parts we have (setting
- ( P U ~ ) ~ qu = t u ) ,
+
so that
im Im im
=
Thus, for any
> 0,
o:u(ru) -
qo;u2 - z
d
pomo,uru.
226
W. N. Everitt, I. W. Knowles and T. T. Read
Next, by another integration by parts,
(4.2)
for all E ~e3
, > 0. Finally, as in [8], note that condition (ii) of the theorem implies,
via Atkinson's lemma, that there exists on I, an absolutely continuous function
gm with
Thus
for all s4,e5> 0. Returning now to (4.1), we write
and use (4.2) and (4.3) to estimate the second term on the right hand side. This
gives
where Cl= e4 + E;'
so that
+ eyl,
+
and C2 = e3(l + c ~ ) ~ / 4E;',
NOWwe choose el to
~s
E=E1+&2+E5<1,
c1=[1+
K,(a, 6)1(1- 711,
G = (1 - 71)(1+ 61,
(4.4)
Limit-point criteria for Sturm-Liouville equations
227
where 0 < r] < 1. Since it is not obvious that such numbers exist, we digress briefly
for an explanation. Clearly, since E~ and E , can be chosen arbitrarily small, it is
enough to find positive values of E , , E,, and E , for which
+ E~ = 1 and
where p = (1 + c ~ ) ~ /Now
4 . if
difficultto show that E;'
E;' = f ( e l ) where
+
P E +~ E;'
=1
+ 6 , then
it is not
and that f ( e l ) > 0 for 0 < E , < 6 ( 1 + 6)-l, attaining its minimum value of
1+ ( a + 3)2(46)-1when E , = 6 { 6 + (3 a)/2}-' < 1. We now assume that E , has
this value. Since Kl(a, 6 ) > ( a + 3)2(46)-1,(4.7) must now hold if we define E~
and E~ by
E~ = 1 and p~~ + E;' = 1 + 8. Henceforth we regard the value of e2
as fixed. By decreasing E , slightly so as not to invalidate (4.7), we can obtain
(4.5). Finally, by decreasing the value of E , slightly, again without invalidating
(4.7), we can obtain (4.6), as required.
Thus, by (4.4) and condition (i),
+
+
where K2 depends only on
E,,
. . . , E~ and not on m. Thus
{Ipl U : ( U ' ) +
~ (amqlo&+ km)u2}s K,
I
{o; lu(ru)l+ @,wu2}.
(4.8)
G
Suppose now that there are two real linearly independent solutions u and v to
Lu = 0 in ~ t [ ab,) with p ( u f v - u v ' ) = 1. Then we have almost everywhere on I,,,
o m ( a m q l a+~km)f
= om(PI1 (cu,qlo%+ k,)f
J u ' v- uv' 1
S IpI u ~ ( u '+)( ~
v ' ) ~+} (amqloi+ km)(u2+ v2).
Hence, from (4.8),
Thus, summing over m , we obtain
which contradicts condition (iii), and completes the proof.
W . N . Everitt, I. W . Knowles and T. T. Read
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