Exact bounds for the continuous spectrum of certain differential eigenvalue problems

Exact bounds for the continuous spectrum of certain differential eigenvalue problems

A~I~ KI~L~

University of Uppsala, Sweden and IIaile Selassie I University, Ethiopia

O. Introduction

Let t9 be an unbounded domain in the real n-dimensional cartesian spac.

R" and let a and k be reM-valued and Lebesgue measm'able functions on De The function k is not required to have a constant sign. We shall consider a Hilbert space realization of the spectral problem

( - ~ a2lax~ + a(x))u = ~(x)~ in x), (0.1)

j=l

u = 0 on the boundary, ~ ( 0 . 2 )

where i is the eigenvalue parameter. Under certain conditions (Sections 1 and 2) we shall deduce exact bounds for the positive and for the negative continuous spectrum of this problem. The case when k(x) = 1 for all x in [2 was treated by Arne Persson in [7].

The author wishes to express his gratitude to A. Pleijel and Arne Persson for their generous interest.

1. Conditions, the spectral problem

Let C~~ be the set of all infinitely differentiable real-valued functions with compact support in f2 a n d write

(u, v) = f (grad u grad v -t- auv)dx (1.1)

~2

when u and v are in C~~ I t is assumed that (u, u) is positive definite on C~(12). Furthermore there shall exist a constant C such that

198 AI~KIV ~61~ MATESIATIK. VO1. 9 No. 2

f ]k]u2dz < C(u, u) ^(1.2)

on C~(9). Completion of C ~ ( ~ ) w i t h respect to l.I, lul = ( u , u ) 1/2, gives a Hilberr space H on which (u, v) and, because of (1.2), also a form corresponding to

K(u, v) = f kuv dx on C~(9) (1.3)

~2

can be defined as limits on C a u c h y sequences in C~(D). The form K(u, v) becomes s y m m e t r i c a n d bounded. H e n c e

K(u, v) ~-- (Ku, v) (1.4)

defines a selfadjoint operator K. U n d e r suitable conditions t h e spectral problem for K, i.e. for t h e e q u a t i o n K u = / ~ u , is equivalent to the eigenvalue problem (0.1), (0.2) if A ~ #-1, see [8], [9]. Our aim is n o t to discuss such conditions b u t to o b t a i n e x a c t b o u n d s for t h e continuous spectrum of K as defined b y (1.4).

The eigenspace corresponding t o a real interval I is d e n o t e d b y H(I). F o r reference we state t h e following consequences of t h e spectral theorem. (1) I f u # 0 is an element

of

H(I), then (Ku, u)/(u, u) i.e. K(u, u)/(u, u) belongs to I. (2) The spectrum is discrete in I i f H(Io) is finite dimensional for every closed interval i o contained in the open kernel of I.

2. Further conditions

Precompactness of a q u a d r a t i c form Q on a linear space w i t h scalar p r o d u c t (. , .) can be characterized b y a compactness inequality. To every s > 0 there shall exist a finite n u m b e r of (. , .)-bounded linear forms L1, L 2 .. . . , LN such t h a t

N

IQ(u, u)l ~ z(u, u) + ~ iLj(u)#. (2.1)

j = l

P r e c o m p a c t is c o m p a c t if t h e space is complete.

L e t Sr be t h e intersection of f2 a n d t h e sphere { x : l x I < r}. Define b y a-(x) = 89 ~ a(x)). We assume that for every finite r the integrals

a -

f u=dx, f a-(x)u dx, f l (x)lu2dx

Sr S r Sr

(2.2)

are precompact on C~ (#2) with respect to the scalar product in (1.1). I)recompactness implies boundedness. Thus for e v e r y r and e v e r y u in C~(D)

E X A C T B O U N D S : F O R T H E C O I ' ~ T I N U O U S S P E C T B , U1VI 199

f u ~ dx < Cl(r)(u,

U)~ (2.3)

S r

where the constant

C~(r)

depends on r. Similar inequalities are valid for the other integrals in (2.2).

It is furthermore assumed that on C~(D)

f grad

u dx < Q(r)(u,

^U)~

(2.4)

Sr

where the constant C2(r ) depends upon r.

The inequality (2.3) shows t h a t every element u in H (Cauchy sequence) determines a function also denoted by u which is square integrable over St.

Since r is arbitrary the definition of u can be extended to the whole of f2.

According to (2.4) the function u has generalized locally square integrable derivatives of the first order. For u in H, values can be a t t r i b u t e d to the left h a n d sides of (2.3) and (2.4) either as limits on Cauchy sequences or as integrals in which the corresponding function and its generalized derivatives are introduced.

The inequalities (2.3), (2.4) remain valid in H. On account of their boundedness the other two integrals in (2.2) can be similarly defined as limits or as integrals.

The integrals (2.2) are compact on H.

F r o m the inequality (1.2) it follows t h a t for u and v in H the value of

K(u, v)

equals the integral in (1.3) in which the corresponding functions in

L~o~(dx)

are inserted. These functions belong

to L2( I]c Idx).

The inequality (1.2) is valid also in H.

The possibility t h a t a non-zero element of H m a y determine a function u which is 0 almost everywhere is not excluded. However, an eigenspace of K be- longing to an interval I of positive distance from the origin is in one-to-one cor- respondence with its functions in

L~or ).

This follows from statement (1), end of Section 1.

3. Sulficient conditions

As in Courant & Hilbert [1], pp. 515--521, it can be proved t h a t for evory e > 0 a Friedrich's inequality

f

^{j = l}

S r Sr S r

holds on C~(D). The functions Wl, w2 . . . . , w N are bounded and integrable with compact supports. The proof is simplified since, when Sr is not contained in ~2, the functions in

C~(D)

vanish identically in neighbourhoods of the common b o u n d a r y of Sr and D.

200 AI~KIV 1~51~ MATEMATII~. VO1. 9 No. 2

I f a-(x) and k(x) are bounded in every St, and i f (2.3) and (2.4) hold, then the compactness of the integrals (2.2) is a consequence of (3.1). For according to (2.3) the linear forms in (3.1) are (. , .)-bounded and b y the use of (2.4) one can introduce (u, u) in the s-term. Thus the first integral in (2.2) is precompact on C~(Y2), compact on H. Because of the boundedness of a-(x) and ]c(x) in Sr the last two integrals in (2.2) are majorized b y the first one, hence compact.

_Remark. Under the conditions of Sections 1 and 2 we shall obtain bounds for the continuous spectrum of K. A n y set of conditions which imply ours will evidently do for the same purpose. Such a set (not assuming boundedness of a - in S~) is due to Arne Persson when ]c(x) is identically 1. His assumption t h a t a(x) be bounded from below for large values of Ixl is, however, no~ necessary. I n Section 9 we shall indicate an example in which a(x) is not bounded from below but all conditions of Sections 1 and 2 are fulfilled.

4. Result to be obtained

L e t

L + = sup (K(u, u)/(u, u)), L;- = inf (K(u, u)/(u, u))

when u varies in C~~ S J, a n d consider the non-increasing a n d non-decreasing limits

L + = l i m L +, L - - - ~ l i m L ~ - .

r --r o o r - + o o

An easy consequence of the compactness of the third integral in (2.2) is t h a t L + > 0 and L - < 0. To see this take R > r. For Q(u, u) = [[k]u~dx, the precompactness

SR

relation (2.1) holds with certain L1, L2 . . . . , LN. The space C~'(S R -- St) is of infinite dimension and thus contains functions u :# 0 for which Lj(u) = 0, j = 1, 2 , . . . , N . For such a function the precompactness relation gives

f lklu2dx <

S(U~

u)

so t h a t SR

IK(u, u) l/(u, u) < s.

Thus L + > - - s and L7 ~ s for a n y e > 0 which shows t h a t L + ~ 0 a n d L - < 0. We shall prove the following

THEOREm1. The spectrum of K outside L - ~ # ~ L + is discrete and this interval contains no smaller interval with this property.

E X A C T B O U N D S ]FOR T H E C O N T I N U O U S SPECTRU1V~ 2 0 1

5. A basic inequality

For 0 < ~ < r let ~0 be an infinitely differentiable function which is identically 0 for ]x I ~ ~, identically 1 for Ix[ > r and has all its values in the closed interval [0, 1]. With u, also ~u belongs to C~~ and (~u, ? u ) ~ 0. E a s y computations give

(~u, q~u) = (u, u) -- f ( 1 - - cf2)grad2udx -- f (1 -

Sr Sr

+ 2 f u~ grad u grad ~ dx § f u~ grad2 q dx.

S r S r

q~2)au2dx -+-

According to the Cauchy--Schwarz inequality and b y the help of (2.4) it follows t h a t for all u # 0 in C~~

O <__ (q~u, qvu)/(u,u) < l

+ f a-u dx/(u,u) + f +

S r S r

C~ f u2dx/(u, u).

-4- (5.1)

S r

Here the constants C 1 and C 2 depend upon the choice of ~ iuctuding the choice of ~ and r.

A first consequence of (5.1) is t h a t for any element u = {un}~ in H, also qu = {~u,}~ belongs to H. For (5.1) shows, on account of the boundedness of the integrals (2.2), that with u, also {~u,}~ is a Cauchy sequence (with elements in C ~ ( ~ 9 - - S o ) ). A transition to the limit proves (5.1)fox" u in H.

6. Upper bound for the positive continuous spectrum

/ .

For u in C~(Y2) it is clear t h a t

K(u, u) = K(cfu, c/u) - ~ / ( 1

^{- -}

q92)ku2dx, a

relation which remains valid for u in H. With u in H this formula is divided b y (u, u). I f ~u = 0 the quotient

K(q~u, ?u)/(u, u)

vanishes. I f ~u # 0 the quotient equals

(K(~u, q~u)/(~u, cfu)) 9 ((q~u, ~u)/(u, u)).

Here

K(~u, ~u)/(~u, q~u)

< L~-. For u in H the quotient (~u,

cfu)/(u, u)

can be estimated according to (5.1). In both cases

202 ARKIV t'6g MATEMATIK. Vol. 9 No. 2

( \1/2

f f

S r S r

S r S r

where t h e c o n s t a n t s d e p e n d u p o n the choice o f ~v including t h e choice o f q a n d r , q < r .

Consider H ( L + 4- 8 < # < co) for b > 0. B e c a u s e o f (1.2) t h e e n t i r e s p e c t r u m o f K lies in t h e closed i n t e r v a l [ - - C , C ] a n d H ( L + 4 - 8 <_tt < co) = H ( L + 4- 8 < # <_ C). T h e dimension o f this space is c l a i m e d t o be finite. A s s u m e t h e c o n t r a r y . F i r s t choose s in 0 < e < 8/2 a n d t a k e ~ so large t h a t L + < L + 4 - e . A choice o f ~0 (and r > ~) gives c e r t a i n values t o C~ a n d C s in (5.1). T a k e e 1 > 0 so small t h a t

(L + 4- e){1 4- e I 4- Ol#l 1/2 4- C2el} 4- e 1 < L + 4- 8 .

T h e integrals (2.2) a p p e a r i n g in (6.1) are c o m p a c t , t h u s can all be m a j o r i z e d b y an expression

N

ex(u, u) 4- ~, ]Lj(u)I s . j=l

I f H ( L + 4- 8 < # < co) is o f infinite dimension it contains n o n - z e r o e l e m e n t s u for w h i c h t h e linear f o r m s Lj(u) vanish, j = 1, 2 . . . N . F o r such an e l e m e n t it follows that

K(u, u)/(u, u) <_ (L+ +

e){ 1 + q + qq~/~ + c s q } + q < L+ 4- 8.

Since u belongs t o H ( L + 4- 8 _< # < oo) t h e last i n e q u a l i t y is v i o l a t e d b y s t a t e - m e n t (1) a t t h e e n d o f Section 1. T h u s , a c c o r d i n g t o s t a t e m e n t (2) in Section 1, t h e v a l u e o f L + is a n u p p e r b o u n d for t h e p o s i t i v e c o n t i n u o u s s p e c t r u m .

7. Best upper bound for the positive continuous spectrum

This section d e p e n d s o n l y on (1.2) a n d t h e positive definiteness o f (1.1); c o m p a r e [10], p. 360.

L e t M > 0 a n d assume t h a t H 1 = H ( M < # < co) is finite-dimensionM.

T h e n H 1 is s p a n n e d b y eigenfunetions vj, v2 . . . v N w h i c h can be t a k e n o r t h o - n o r m a l i z e d w i t h r e s p e c t to (. , .). T h e eigenspaces H 1 a n d H 2 = H ( - - co < # < M) are o r t h o g o n a l c o m p l e m e n t s , also o r t h o g o n a l w i t h r e s p e c t to K ( . , .). P u t u = u 1 + u s , w h e r e u 1 belongs t o H 1 a n d u s belongs t o I t s. T h e n ( u , u ) = (u v us) + (u s, u2) a n d K ( u , u) = K ( u v us) 4- K ( u s, %). According t o (1.2),

E X A C T B O U N D S FOI~ TX-IE C O N T I N U O U S SPECTRU?Cf 203

K(%, %) <_ C(%, %)

a n d because of statement (1) in Section 1,

K(%, %) <_

M(%, %).

Hence

K(u, u)/(u, u) < (0 - M ) ( u . u~)/(u, u) + M .

(7.1)

Write % =

ClV 1 -3 7 C2V 2 -~-

. . . - t - C N V N , where cj = (u 1,

vj) = (u, vj).

The eigen- values & corresponding ~o

vl,

j = 1, 2 . . . . , N, are positive and cj =

#ilK(u,

vj).

Let u be a function in

C ~ ( D - - S , ) .

On account of (1.2)

I f

~ 2 - - S r D - - S r

so t h a t

]cjl2(u,u) -1

tends to 0 when r tends to infinity, j = 1, 2 , . . . , N . I f

( U l , U l ) = C 1 -~- C 2 -~- . . . @ C 2 2 i s i n t r o d u c e d i n t o (7.1) t h e l i m i t property o f t h e

coefficients c i shows t h a t , according to the definition of L + in Section 4, L + = lim sup (K(u,

u)/(u, u)) < M.

Thus L + is the least upper bound of the positive continuous spectrum.

8. Bound for the negative continuous spectrum

A change of sign of

k(x)

shows t h a t L - is the greatest lower bound for the negative continuous spectrum. Thus the spectrum of K outside L - < / x < L + is discrete and this interval is the smallest one with this property. For a differential problem of type (0.1), (0.2) this means t h a t the spectrum in

1/L- < ,~ < 1/L +

is discrete and t h a t no larger interval has this property.

9. Example

We shall show t h a t our conditions can be fulfilled even with a function

a(x)

which is not bounded from below for large values of Ix I.

The domain ,Q is taken as the entire space R n. L e t p and P be functions on R", p real-valued and P vector-valued with values in R". I f u belongs to C~(R"), partial integrations give

f (grad

R n

= f

( g r a d u +

uP)2dx + f (lo -- P2)u2dx,

,2

R n R n

(9.1)

204 ARKIV t~SR MATEMATIK. Vol. 9 No. 2

w h e r e a = p - - d i v P a n d p2 is t h e scalar p r o d u c t o f P b y itself. T h e i n t e g r a t e d p a r t s v a n i s h since u has c o m p a c t s u p p o r t . T h u s t h e left h a n d side o f (9.1), i.e.

(u, u) is positive definite o n C~~ ") p r o v i d e d Io > p2 for all x.

T a k e p = c o n s t a n t a n d P ( x ) -~ g r a d f ( r ) w i t h r = Ixl a n d

f'(r)

= A + sin (#), w h e r e A is a c o n s t a n t . T h e n

a ( x ) = p - - 2r cos (#) - - (n - - 1)r-l(A + sin (r~))

w h i c h is n o t b o u n d e d f r o m below. I f A < 0 t h e f u n c t i o n a - ( x ) is locally b o u n d e d . W e h a v e p2 = (A + sin (re)) 2 a n d it is easily seen t h a t we c a n t a k e 10 > (IA[ + 1) 2 such t h a t (2.3) a n d (2.4) h o l d w i t h t h e integrals e x t e n d e d over t h e entire space R"

(the c o n s t a n t s o f course i n d e p e n d e n t o f r). This is still t r u e if a positive locally b o u n d e d f u n c t i o n is a d d e d t o a(x). T h e n e g a t i v e p a r t o f t h e n e w f u n c t i o n a ( x ) is still locally b o u n d e d . This gives a simple w a y o f c o n s t r u c t i n g ~ couple of locally b o u n d e d f u n c t i o n s a ( x ) a n d k ( x ) for w h i c h (1.2) and, on a c c o u n t o f S e c t i o n 3, also t h e o t h e r c o n d i t i o n s o f Sections 1 a n d 2 are satisfied. T h e n e w f u n c t i o n a ( x ) c a n be chosen u n b o u n d e d f r o m below.

References

1. Cov~,~T, R. & HILBERT, D., Methoden der mathematischen P h y s i k I I . Berlin 1937, 2. EASTmAn, M. S. P., On the limit points of the spectrum. J . London Math. Soc. 43 (1968),

253--260.

3. Gr.AZMAN, I. M., On the character of the spectrum of many dimensionM singular boundary problems. Dokl. Alcad. N a u k . S S S R 87 (1952), 171--174 (Russian).

4. --~)-- On the application of the method of decomposition to many dimensional singular boundary problems. ~lat. Sb. ~V. S. 35 (77) (1954), 231--246 (l~ussian).

5. MOLCA~OV, A. M., Criterion for discrete spectrum of semi-bounded Sturm--Liouville operators. Trudy Moskov. Mat. Obg~. 2 (1953), 169--199 (Russian).

6. NILSSO~, N., Essential self-adjointness and the spectral resolution of Hamiltonian operators.

Kungl. Fysiogr. Sdllslc. i L u n d F6rh. 29 (1959), 1--19.

7. PERSSO~, A., Bounds for the discrete part of the spectrum of the semibounded Schr6dinger operator, iVIath. Scan& 8 (1960), 143--153.

8. Pr.~IjEL, ~., Le probl~me spectral de certaines 6quations aux d6riv6es partielles. A r k . .Mat. Astr. Fys. 30 A, n:o 21 (1944), 1--47.

9. --))-- Sur les op6rateurs differentiels de type @lliptique. Arlz. ~Vlat. Astr. tZys. 32 A, n:o 14 (1945), 1--4.

10. I~IESZ, F. & SZ.-~AGY, B., Lemons d'analyse fonctionnelle. Fourth edition. Paris, Budapest 1965.

Received December 12, 1970 Abiy I~ifIe

Department of Mathematics I~Iaile Selassie I IJniversity

Box 1176 Addis Abeba Ethiopia