MARUO, KENJI

Maruo, Kenji

Osaka University Knowledge Archive

Asymptotic distribution of eigenvalues of non-symmetric operators associated with strongly elliptic sesquilinear forms

Osaka City University Repository

Maruo, K.
Osaka J. Math.
9 (1972), 547-560
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF
NON-SYMMETRIC OPERATORS ASSOCIATED W I T H
STRONGLY ELLIPTIC SESQUILINEAR FORMS
KENJI MARUO
(Received January 31, 1972)
1. Introduction
The main object of this paper is to extend the result of K. Maruo and H.
Tanabe [4] on the eigenvalue distribution of symmetric elliptic operators to a
non symmetric case. Some amelioration of the result of [4] on the remainder
estimates in WeyPs formula as well as the formula under less restrictive smooth-
ness assumptions is also obtained.
Let Ω be a bounded domain in Rn having the restricted cone property. We
use the same notations as those of [4] to denote various norms and functional
spaces. In this paper it is assumed that 2m>n as in the previous paper [4].
Let ΰ b e a sesquilinear form denned in H
m
(Ω) X H
m
(Ω) satisfying
Re£[w, u]>80\\u\\2m for any u^V a-( 1 )
o
where V is a closed subspace of H
m
{Cί) containing H
m
{Ω) and δ0 is some positive
constant independent of u. We assume that B has the following form
B[u,v]=B0[u,v]+Bλ[u,v] (1.1)
where B
o
 which is the principal part of B is a symmetric integro-differential
sesquilinear form of order m with bounded coefficients
B0[u, v] = I Σ a*fkx)T>*uO* vdx
and 5 j is a not necessarily symmetric sesquilinear form satisfying
UMIm-i + IML-ilML) «-(2)
for any uy v^V i.e. Bλ is the lower order part of B, Let A be the operator
associated with the form B: an element uoίV belongs to D(A) and Au=f^L2(Ω)
if B[u, v]~(f> v) holds for any v^V. A is a not necessarily symmetric
operator in L2(Ω) and all rays arg λ = θ different from the positive real axis are
rays of minimal growth of the resolvent of A, By N{i) we denote the number
548 K. MARUO
of eigenvalues of A whose real part does not exceed t. The main conclusion of
this paper is that the following asymptotic formula holds:
N(t) = C0tn'2m + 0(tn'2m) as t->oo , (1.2)
if the coefficients of B
o
 are Riemann integrable,
and
N(t) = C0tn'2m + 0(f»-<»'2m) as *->oo (1.3)
for any θ<hj{h + 2) if B
o
 has uniformly Hoelder continuous coefficients of order
h and for any θ<(h+ l)/(A + 3) if the coefficients of B
o
 belong to the class C1+h
in some domain containing Ω. The formula (1.3) is an improvement of the
corresponding result obtained for symmetric operators in [4] where (1.3) was
established only for #</z/(/i + 3) and θ<(h + l)l(h + 4) respectively making some
more restrictive assumptions and in order to prove (1. 3) for (/z+l)/(A + 4)<
#<l/2 still more hypotheses were required.
The author wishes to thank Professor H. Tanabe and Mr. M. Nagase for
suggesting this problem and helpful advices.
2. Main theorem
As was stated in the introduction let Ω be a bounded domain in Rn having
the restricted cone property (p. 11 of S. Agmon [1]) and it is assumed that
2m>n. For ^ e Ω w e write δ(#) = min {1, dist (x, 9Ω)}. Suppose that
ί S(x)-pdx<oo
JΩ
for some positive number p < 1 which will be specified later.
Since all coefficients of of B
o
 are bounded it follows from a — (2) that for
any u, v^V
\B[u,v]\<K\\u\\
for some constant K.
We state various smoothness assumptions on the coefficients of B
o
:
they are Riemann integrable, i.e. continuous almost everywhere in Ω:
* - ( 0 )
they are uniformly Hoelder continuous of order /zinΩ: s — ( 1 )
they belong to C1+Λ(ΩX) where Ωx is some domain containing Ω and C1+Λ(Ωj)
is the subclass of functions in C^Ω^ with derivatives Hoelder continuous of
order h in Ωlβ s — ( 2 )
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 549
Main Theorem. The following asymptotic formulas for N(t) hold as f—* oo:
N(t) = C
a
tnlim + o (tn/2m) under s-(0)
N(t) = C
o
tn<*m + Q (fn-<»'2m)
for any θ satisfying
0<θ<h/(h + 2) under s-( 1)
0<θ<(h+l)/(h + 3) under s-(2)
where
c
 sin (nβm)(
 C{χ)dχ
nβm JQ
REMARK. AS was mentioned in the Introduction the remainder estimates
described in the main theorem is an improvement of those established in [4].
Furthermore applying the theorem to the sesquilinear form (Auy Av) where A is
the elliptic operator satisfying the conditions of R. Beals [3] we may prove
Theorem C of [3] with 0<θ<h/(h + 2) instead of 0<θ<h/(h + 3) if the order of
A is greater than τz/2.
Following the method of S. Agmon [5] or Dunford-Schwartz [6] it is possi-
ble to show that the generalized eigenfunctions of A are complete in L2(Ω) under
our assumptions.
3. Some lemmas
As in the previous paper [4] we extend the operator A to a mapping on V
to F * where F* is the antidual of F. This extended operator which is again
denoted by A is defined by
B[u, v] = (Au, v) for any v^V
where the bracket on the right stands for the duality between F* and F in this
case.
Identifying L2(Ω) with its antidual we may consider F c L 2 ( Ω ) c F * alge-
braically and topologically, and as is easily seen V is a dense subspace of F *
under this convention. The resolvent of A thus extended is a bounded linear
operator on F* to F. We denote by p(A) the resolvent set of A and d(X) the
distance from the point λ to the positive real axis for a complex number λ.
Lemma 3.1. The resolvent set p(A) of A in either sense contains the set
{λ: rf(X)^C|X|1"1/2m, | λ | ^ C } for some constant C. The eigenvalues {λj}J=0
of A have finite multiplicity and eigenvalues of A can have only oo as a limite point.
550 K. MARUO
Proof. We put (A — λ) u =f for any u e D(A). We see that
B[u,u]-X(u,u) = (/,«) (3.1)
From (3. 1), (1. 1), α-(2) and Im50[M, w]=0, we get:
I Im λ I IMI?<||/lϋMI0+2iqMUI«IL,-1 (3.2)
Applying to the last term HWIUIMIL-X Young's inequality and then using the
interpolation inequality, for any positive constant St and δ2 < 1 we find that
SJIulli+er^lNll+Sr^r^lNIS} (3.3)
From (3. 1) and a—(1) we get
δ | |« | | i^ |λ | | | « | |3 + | |u | |
β
| | / | |
β
. (3.4)
Putting δ, = 8ψ = I λ I "1/2m and combining (3. 2), (3. 3) and (3. 4) we find that
(iimλi-^iλi-πNi^α+^iλi-'mi/iuNio (3.5)
If I Im λ I > CI λ 11"1/2m for large C, we know that
N | 0 < K 3 / | I m λ | | | / | | 0 . (3.6)
If Re λ < 0 we get
|Reλ | |M| 0 2 <: | | /H 0 N| 0 (3.7)
from (3. 1).
Combining (3. 6) and (3. 7) we find that there is a constant Kt independent
of λ such that
||M||0<ί:4/4λ)||/||0 (3.8)
On the other hand for an adjoint operator A* we find the same estimate (3. 8).
Thus the null space of the operator (A* — X)consists only of zero and we know
{λ:rf(λ)>C|λ|1-1/2" ί, \\\>C)ap(A).
Next we put (A — X)u = f for any M G F .
From (1. 1), a — {\) and a — (2) it follows that
^ } . (3.9)
For any number δ3 such that 0 < δ 3 ^ l we know
M^.^KάhML+zr^MW*}. (3.10)
From the inequality
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 551
|λ||(«,c)|^||/|μiNL+^||«IUHL for any Ϊ E F
it follows that
IMII«llv£ll/llv+tf,IML (3.11)
Combining α - ( l ) , (3.9), (3. 10) and (3. 11) and putting δ 3 = |λ |" 1 / 2 m we get
the following estimate:
£\\f\\v \\«\L+κt\\\id(\){{i+\\\-^\\u\\m\\f\\y.
If <ί(λ)^C|λ|1-1 /*'" with | λ | sufficiently large there is a constant K10 in-
dependent of λ such that
ll«IL<^o|λ|μ(λ)||/|μ (3.12)
On the other hand we put (A* — X)u = f for any u^V. Then we find the same
estimate (3. 12) for A*. Thus we see that
The last part of the lemma is a simple consequence of Rellich's theorem.
Q.E.D.
For a bounded operator S on F * to V we use the notations
K*^L2 etc, to denote the norms of *S considered as an operator on V* to F, F*
to L2(Ω), etc.
Lemma 3. 2. There exists a constant C
x
 such that
i) WiA-xΓW^^KCjdix) ϋ)
iϋ) WiA-xyWy^vKCAWIdix) iv)
if d(X) > CI λ 11~1/2m, I λ I ^  C where C is the constant in the statement of Lemma 3.1.
Proof. The statement i) is clear from (3. 8).
If M = (i4-λ)-1/for any/<EΞL2(Ω) we get;
from a — (1) and i).
The statement iii) is clear from (3. 12). Finally with the aid of (3. 12) and
the following inequality
552 K. MARUO
we can easily show iv). Q.E.D.
Lemma 3. 3. Let S be a bounded operator on V* to V. Then S has a
kernel M in the following sense:
Sf{x) = \
a
M(x, y)f{y)dy for
M(x, y) is continuous in Ω x Ω and there exists a constant C2 such that for any
\M(x,y)\
Proof, see [4]. Q.E.D.
Lemma 3. 4. There are positive constants Cz and C4 such that
B0[uyu]>C2\\u\\2m-CA\\u\\l for any u<=V.
Proof. From a — (1) and the interpolation inequality, we can easily show
the statement. Q.E.D.
4. Estimates of the resolvent kernel
We shall estimate the difference between the resolvent kernel of A and that
of the operator A
o
 associated with B
o
 + C49 thus BQ\uy υ] + CA(u, v) = (Aouy v) for
any u, υ^V. Obviously for the operator A
o
 the analogues of Lemma 3. 2 hold.
Let£
λ
 be the operator defined by
SJ =(A- λ ) " 1 / - (A - λ)" 1 / for any / e V* .
Lemma 4.1. There is a constant C7 such that for d(X) > C \ λ 11"1/IM, | λ | > C,
i) \\s
λ
\\
v
*+
v
<c5\x\id(\)
....
 i l Q I (
iv) l |S
λ
| | ^ 2 <C 5 /</(λ) ( I λ I -
Proof. Let (A - λ ) " 1 / - (A
o
 - λ)~7= S
λ
f= u. Now we know that
(^_ λ)-i_μ o_ λ )-i = (A0-X)-\A0-A) (A-\)->.
On the other hand, since the operator A
o
 is self-adjoint we know
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 553
(SJ, φ) = ((A0-A)(A-\y>f, (A0-\Γφ)
C4((^-λ)-1/, (Λ-^)-'Φ)
γ, (^0-X)-» (4.1)
for any φ e V*.
Combining (4. 1), Lemma 3. 2 and the interpolation inequality we find that
there are constants K1 and K2 such that
+ \\(A-\y1f\\
m
_1\\(A0-xTφ\\m}
<κ2( i x i /φ,))21 λ r I / 2 Ί i/i ivi IΦI w
Then we get
||S
λ
|l™,*<C51λ Ild(\)( I X I '-
The remaining inequalities can be proved in a similar manner. Q.E.D.
Since m>n/2 there exist the resolvent kernels K
λ
(x, y) and K%(x, y) of the
operator A and ^ 40 such that
(A-XΓf(x) =
(Λ-λΓ/(*) = \QKΪ(X, y)f(y)dy for any /eL2(Ω).
Theorem 4. 2. .For #;ry ^^// positive numbers py S and any non-negative
integer j , the following inequality holds:
I Kk(x, x)- C(x)(-xy^>» I <CJίI λ I ^2mld(X){Ύh+i I λ I /d(λ)
+ (T11 λ 11"1/2W/d(\)Y + I λ I ^^-/^(λ) + (I λ I ''^Six)d(X)Y}] (4. 2)
/or </(λ)>|λΓ 1 / 4 w -{-£, γ > 0 , T ' Ί λ l 1 " 1 ' 1 1 " / ^ ) ^ ! , α/zrf | λ | sufficiently large,
where ί = 0 wwrfβr ί—(1) ami i = l z/w&r 5 —(2). C 6 is a constant depending on
p, S,j but not onX, y or x, and C(x) is the function defined in the main theorem.
Proof. Combining Lemma 4. 2, 6. 2, 7.2 and 7.3 of [4] we get
\Kl(x,
 x
)-C{x){-X)-^^\ <K3[\XΓ'»»ld(X){Ύ»+ηd(X)
+ (7-1\X\1-1/2m/d(X)y + (\X\1-1'2ml8(x)d(X)Y) + I λ l ^ " 1 ^ - 1 ] (4. 3)
where i = 0 or 1 according as we assume s — (1) or s — (2).
Formally we replaced d(X) by some power of | λ | at this point (Theorem
7. 1 of [4]); however, in this paper we postpone this replacement for a little while
to obtain better remainder estimates as was stated in the introduction.
On the other hand applying Lemma 3. 3 and Lemma 4. 1 to S
λ
 we get
554 K. MARUO
I Kk(x, y)-K°(x, y) \ <K4( | λ | /</(λ))21 λ | c«-iv*«-i (4. 4)
Combining (4. 3) and (4. 4) the desired estimate (4. 2) is obtained. Q.E.D.
Next we shall consider the case of the assumption s — (0). We denote P
aβ
the set of points where a
aβ
 is continuous and put P= Π P
aβ
. We fix a
l * l = l/3l = m
point x
o
^P and set
B*'[u> »] = ( Σ a
Λβ
(x0)DauD^vdx for «, v
Lemma 4. 3. There exist positive constants C7 and C^independent of u and
x0 such that
Proof. There is a constant K
s
 such that
o
for any ξ^Rn. That the desired inequality holds for any u^H
m
(Ω) is a well
known fact. Q.E.D.
We put B2[u, v]=B2'[u, v\ + C2(u, v) for u, υ^Hm(Ω). We know that
B2[u, u]>K6\\u\\i for ueHm(ίl) (4.5)
from Lemma 4. 3.
We denote by A2 the operator associated with B2 under the Dirichlet
boundary condition. By definition for any u, v^H
m
(Ω) we have
Bt[u, υ] = (A2u, v)
where the bracket on the right denotes the pairing between the antidual H_
m
(Ω)
o o
of H
m
(Ω) and H
m
{Ω) this case. Obviously for the operator A2 the analogues of
Lemma 3. 1 and Lemma 3. 2 hold.
We denote by ξ(x) a function in Co(Rn) the support of which is contained
in the set {x^Rn: \x\ <1} and which takes the valued 1 at the origin. We
write ξs(x) — ξ((x — #o)/δ) where δ is any positive number <δ(#0).
Let 5
λ δ
 be the operator defined by
S
λ8f=ξδ{(A-\Γf-(A2-X)-\rf)} for / E P
o
where rf is the restriction o f / G P to H
m
(Ω).
o
Obviously S
λζ
 is a bounded operator on F* to H
m
(Ω) and hence a fortiori
to V. Since a
aβ
 is continuous at x0 for any a and β with | a \ = \ β \ = m there
is a positive number θ8 such that
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 555
O as δ^O and
\a
aβ
(x)-a
aβ
(x0)\<θs for \x-xo\<8 (4.6)
Lemma 4.4. IfXis real < 0 and S'ι\\\ ~1/Zm < 1 we get
ϋ) \\SJ\y^<Ct{θB+δ"11 λ I ~^m} I λ I
iϋ) \\Su\\L^v<ZC.{et + SrlI λ I -^m} I λ I "
iv) ||Sχ,|L«
Proof. Let M=(^-λ)" 1 /-(^ 2 -λ)" 1 (r/)andf = |δίί=S' λ S/. Noting that
o
we have
- B2[Ό, V]-BJU, ξBΌ]+BJu9 fa»]-λ(M, fβ»)
λ)-1/, ξ
δ
^] . (4. 7)
In view of (4. 5) we get
\B2[v,v]-X(v,v)\>K,{\\v\\m+\\n\v\\0γ. (4.8)
Next from (4. 7)
\BJV,V]-\(Ό,V)\
< \Bt[v, v]-B2[u, ξ,v]\ + \(B2-B)[(A-\yιf, ξBv]\
*>y\y
ί Σ M
jΩ|«|=-|β|=ι»» γ
Σ {a
aβ
(x)-a
aβ
(x
| β | " «
= 7 1 + / 2 + / 3 + / 4 . (4.9)
Noting that ||r/||_OT<||/||v» we get, by Lemma 3. 2
for /
e
p (4.10)
for / E L ! ( Ω ) (4.11)
if 0<l<m.
We have
(4.12)
556 K. MARUO
From (4. 10) and (4. 12) it follows that
\\v\\
m
 for any / e F * (4.13)
and
. (4.14)
for a n y / e F * .
From (4. 6) it follows that
I/, g
<κ1A\\f\\v*\Mm+ iλi^iHi.). (4.15)
From a — (2), (4. 12) and the interpolation we know
<K
u
\xm\f\\v<ML+\M1/2\Mo). (4.16)
Combining (4. 8), (4. 13), (4. 14), (4. 15) and (4. 16) we find that
where K17 is a positive constant independent of X and δ.
Thus the statements i) and ii) are clear. The inequalities iii) and iv) can be
proved similarly. Q.E.D.
Lemma 4. 5. For any x^P we have
lim(-\γ-n'2mK
λ
(xy x) = C(x).
Proof. From Lemma 3. 3 and Lemma 4. 4, it follows that if λ < 0 and
I K
λ
(x0, x0)-K°(x0, x0) I < i £ 1 8 ( 0 δ + δ " 1 ^ ) I λ I -1+»<2m (4. 17)
where Kl(x, y) is the kernel of the operator (A2 — λ)"1.
On the other hand, from Agmon [2], we get
+ I λ I -1 + c w~^ / 2 > w/δ^0)) (4. 18)
where p is the any positive constant.
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 557
In view of (4. 17) and (4. 18) with p = 1/2 we find
| * x ( * . , * , ) - ( - λ ) - 1 + 1 ^ G ( *
β
) |
<κ20(θs+δ"11 λ i -1/2m+s(χoy1/21 x i - 1 / t m ) i x | -ι+nμm.
Thus we know
lim ( - xy-^mK
λ
(x0 x0)) = C(x0) Q.E.D.
5. Proof of the main theorem
First we shall consider the relation between the resolvent kernel and eigen-
values.
Lemma 5.1. We get the following equality and estimates:
under s — (0) tfί λ-» — co.
iii) //
+ 0 [ I λ I «+i
] a; I λ I
ί = 0 or 1 M«i/er ί = ( l ) o r s "~(2) respectively p is the any positive number such
that 0 < £ < 1 and C l o = ί C(Λ)rf^.
JΩ
Proof. For the statement i) see § 13 of Agmon [1].
From Lemma 3. 2 and Lemma 3. 3 we see that
λ l ^ - 1 . (5.1)
Since a
aβ
(x) are Riemann-integrable functions we find that the measure of
(Ω — P) is zero. Using Lemma 4. 5, (5. 1) and Lebesgue theorem we know that
lim ( (-\γ-n/2fnK
λ
(xy x)dx = [ lim (-\Y'n/2mKJx, x)dx .
Thus ii) is proved.
Putting 7 = 1711~1/2fn+zjd{\) in (4. 2) and integrating both sides over Ω we
get the desired estimate since the second term is smaller than the first if j is
558 K. MARUO
sufficiently large and the third term is dominated by the integral of the last.
Q.E.D.
Lemma 5. 2. Under s—(0) it follows that
N(t) = C0tnί2m+S(tnf2m).
Proof. Using Lemma 5. 1 (ii) and arguing as in § 14 of Agmon [1] we get
the desired statement. Q.E.D.
Lemma 5. 3. There is a constant C
n
 such that
Re λy > C
n
y2m/H for large j .
Proof. From j<N(Rt λy) and Lemma 5. 2 we can easily show the estimate.
Q.E.D.
L e m m a 5. 4. If d(\)>C\\\1~ί/2m+ζ and | λ | is sufficiently large then we
have the following estimate
Proof. We have the following equality
Σ (λy-λ)" 1 - ! ] (Re λy-λ)"1 = - Σ Im λy(λy-λ)"1 (Re λy-λ)"1
= - Σ - Σ =Λ+Λ.
Rβλ^.^2|λ| Rβλ >2|\|
If Re λy<21 λ I there is a constant K2 such that
λ l 1 " 1 ^ (5.2)
from Lemma 3. 1.
On the other hand, if rf(λ)>C|λ11"1/2m+e and | λ | is sufficiently large,
then an elementary geometrical observation shows that there is a positive constant
K3 such that
(5.3)
for any j .
In view of Lemma 5.2, (5.2) and (5. 3) we get
IΛI< Σ | Imλy | I λ y - λ ΓM Re λ y - λ Γ 1
Rλ^2|λ|
Next from Lemma 5. 3 and Re λy >21 λ | we see
ASYMPTOTIC DISTRIBUTION OF EIGENVALUES 559
Thus we find
Σ I λ y - λ I '^K9 I λ I -
1+
«/2~+* f j |"Ci+ >
Reλ.>2|\| i=»0
< ί Γ 7 | λ Γ 1 + " / 2 w + β . (5.4)
On the other hand, from Lemma 3. 1 and Re Xj>2 \ X |, we get
I Im λ y 11 Re λ,—λ | ^ K, | X | "1 / 2 m . (5. 5)
From (5. 4) and (5. 5) we know that
Σ | I m λ , | Iλy-λΓMReλ,—λ, - λ Γ 1
λ/>2|λlRβ |
Q.E.D.
Now we follow the method of Agmon [2]. We put
λ - λ ) - 1 and /(*) = (ZwiV'f /(λ)<ίλ
where L(^) is an oriented curve in the complex plane from z to z = t + iτ not
intersecting [0, oo).
Thus for t>0, τ > 0
\I(z)-(τ/π) Re/(ar)-iV(ί) + iV(O)| < C 1 2 τ | I m / ( ^ ) | . (5. 6)
First we consider the asymptotic formula for N(t) under s — (1). If d(\)>
(5.7)
from Lemma 5. 1 and Lemma 5. 4.
We put z = t + if-h'2nκh+2^* and take
L(z) = {X =
U {λ; |
where t is a sufficiently large positive number.
From (5. 6), (5. 7) and JV(O) = O we find
\I(z)-N(t)\ <K12tn/2m-h/2m(h+2>+z. (5. 8)
On the other hand we know the following equality
f(X)d\ = (2τrO"1( {/(λ)-C 1 0(-λ)- 1 + n / 2 w} dX
LCzϊ J LCz)
-
1
 \ C10(-X)\-1+n'2mdX = / , + / , .
J LCzϊ
560 K. MARUO
In view of Lemma 5. 1 and Lemma 5. 4, putting 1 >p>hβ we get that
Iii I < ^ i 3 { ( I λ 11+*+ί»-w»+ηd(x)2+hl dx I
J L(z~)
I ^
_ι j l
Noting that
—.(
2πiJi nπβm
from (5. 8) and (5. 9) we obtain the desired estimate.
In case of s—(2) assuming that a — (3) holds for somep>(h-\-1)/2 if
and for any/><l if λ = l , we can prove the desired result in the same method as
above.
OSAKA UNIVERSITY
Bibliography
[1] S. Agmon: Lectures on Elliptic Boundary Value Problems, Van Nostrand Mathe-
matical Studies, Princeton, 1965.
[2] S. Agmon: Asymptotic formulas with remainder estimates for eigenvalues of elliptic
operators, Arch. Rational Mech. Anal. 28 (1968), 165-183.
[3] R. Beals: Asymptotic behavior of the Green's function and spectral function of an
elliptic operator, J. Functional Analysis 5 (1970), 484-503.
[4] K. Maruo and H. Tanabe: On the asymptotic distribution of eigenvalues of operators
associated with strongly elliptic sesquilinear forms, Osaka J. Math. 8 (1971), 323—345.
[5] S. Agmon: On the eigenfunctions and on the eigenvalues of general elliptic boundary
value problems, Comm. Pure Appl. Math. 15 (1962), 119-147.
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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF NON-SYMMETRIC OPERATORS ASSOCIATED WITH STRONGLY ELLIPTIC SESQUILINEAR FORMS

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ASYMPTOTIC DISTRIBUTION OF EIGENVALUES OF NON-SYMMETRIC OPERATORS ASSOCIATED WITH STRONGLY ELLIPTIC SESQUILINEAR FORMS

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