Trace Formulae for Matrix Integro-Differential Operators

Trace Formulae for Matrix Integro-Differential Operators

Chuan-Fu Yang

Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094, Jiangsu, People’s Republic of China

Reprint requests to C.-F. Y.; E-mail:chuanfuyang@tom.com

Z. Naturforsch.67a,180 – 184 (2012) / DOI: 10.5560/ZNA.2011-0068 Received July 4, 2011 / revised October 15, 2011

In this paper, we consider the eigenvalue problems for matrix integro-differential operators with separated boundary conditions on the finite interval and find new trace formulae for the matrix integro-differential operators.

Key words:Matrix Integro-Differential Operator; Eigenvalue; Trace formula.

Mathematics Subject Classification 2000:34A55, 34B25, 47E05

1. Introduction

In this paper, we will find trace formulae for the following matrix integro-differential operators L(Q,M;h,H):

−Y⁰⁰(x) +Q(x)Y(x) + Z x

0

M(x,t)Y(t)dt=λY(x), x∈(0,π),

(1)

with boundary conditions

Y⁰(0)−hY(0) =0 (2)

and

Y⁰(π) +HY(π) =0, (3)

whereλ is a spectral parameter,Y(x) = [y_k(x)]_k=1,dis a column vector,Q(x)andM(x,t)ared×dreal sym- metric matrix-valued functions, andhandHared×d real symmetric constant matrices. M(x,t) is an inte- grable function on the setD₀^def={(x,t): 0≤t≤x≤π, x,t∈R},Q∈C¹[0,π], where C¹[0,π] denotes a set whose element is a continuously differentiable func- tion on [0,π]. In particular, h=∞ in (2) means the Dirichlet boundary conditionY(0) =0, andH=∞in (3) means the Dirichlet boundary conditionY(π) =0.

For the matrix Sturm–Liouville equation (when M=0 in (1)) properties of spectral characteristics were provided in [1–4], and asymptotics of eigenvalues for the integro-differential operator withd=1 in (1) were given in [5–9].

Gelfand and Levitan [10] discussed the Sturm–

Liouville problem

−y⁰⁰(x) +q(x)y(x) =λy(x), x∈(0,π), (4) with the Neumann boundary conditions

y⁰(0) =y⁰(π) =0 (5)

and obtained the remarkable formula for the regular- ized trace as follows:

∞ n=0

∑

λn−n²−1 π

Z π

0

q(x)dx

= 1

4[q(0) +q(π)]− 1 2π

Z π

0

q(x)dx,

where q ∈C¹[0,π] and λ_n (n=0,1,2, . . .) are the eigenvalues of the Sturm–Liouville problem (4) and (5). For the Sturm–Liouville problem (4) with Dirich- let boundary conditions and the eigenvaluesλn (n= 1,2, . . .), they got the following formula:

∞ n=1

∑

λ_n−n²−1 π

Z _π

0

q(x)dx

=−1

4[q(0) +q(π)] + 1 2π

Z π

0

q(x)dx.

Since then, this kind of trace formulae for dif- ferential operators was found by a number of au- thors (see references). A trace formula of a differen- tial operator has many applications in the inverse prob- lem, in the numerical calculation of eigenvalues, in

c

2012 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

the theory of integrable systems, etc. Sadovnichii and Podol’skii [11] stated several sharp methods to trace formulae of second-order operators, high-order opera- tors as well as partial differential operators.

Using Rouch´e’s theorem for operator-valued func- tions in [12], we can suitably locate the eigenvalues of L(Q,M;h,H)and find a precise description for the for- mula of the square root of the large eigenvalues up to theo(¹_n)-term, which are similar to the results in [1,2]:

(i) Let λn^(j)(j=1,d; n=0,1,2, . . .) be eigenval- ues of the operatorL(Q,M;h,H), thenλn^(j)satisfy the asymptotic formula

ρ_n^(j)def= q

λ_n^(j)=n+ω₁^(j) nπ +κ_1,n^(j)

n ,

where ω₁^(j) are the characteristic values of the d×d real symmetric matrixω1=h+H+¹₂^R₀^πQ(x)dxand

∑n|κ_1,n^(j)|²<∞.

(ii) Letλn^(j)(j=1,d;n=1,2, . . .) be eigenvalues of the operatorL(Q,M;h,∞), thenλn^(j)satisfy the asymp- totic formula

ρn^(j)def

= q

λn^(j)=n−1

2+ ω₂^(j) n−¹₂

π +κ_2,n^(j)

n ,

where ω₂^(j) are the characteristic values of the d×d real symmetric matrix ω2 = h+ ¹₂^R₀^πQ(x)dx and

∑_n|κ_2,n^(j)|²<∞.

(iii) Let λn^(j)(j=1,d;n=1,2, . . .)be eigenvalues of the operator L(Q,M;∞,H), then λn^(j) satisfy the asymptotic formula

ρ_n^(j)def= q

λ_n^(j)=n−1

2+ ω₃^(j) n−¹₂

π +κ_3,n^(j)

n ,

where ω₃^(j) are the characteristic values of the d×d real symmetric matrix ω₃ = H+¹₂^R₀^πQ(x)dx and

∑n|κ_3,n^(j)|²<∞.

(iv) Let λn^(j)(j=1,d; n=1,2, . . .)be eigenvalues of the operator L(Q,M;∞,∞), then λn^(j) satisfy the asymptotic formula

ρn^(j) def=

q

λn^(j)=n+ω₄^(j) nπ +κ_4,n^(j)

n ,

where ω₄^(j) are the characteristic values of the d×d real symmetric matrix ω4= ¹₂^R₀^πQ(x)dx and

∑_n|κ_4,n^(j)|²<∞.

However, some trace formulae for the matrix integro-differential operatorL(Q,M;h,H) have never been considered before. In this paper, we shall discuss the eigenvalue problem for the operatorL(Q,M;h,H) and find new trace formulae.

2. Result

For simplicityA_{i j} denotes the entry of matrixAat theith row and jth column and trAdenotes the trace of the matrixA;I_dis ad×didentity matrix and 0_dis ad×dzero matrix.

Theorem 1.

(i) For the operator L(Q,M;h,H): let λn^(j)(j = 1,d; n=0,1,2, . . .) be eigenvalues of the operator L(Q,M;h,H), then we have the trace formula

∞ n=0

∑

" _d

∑

λn^(j)−n² −2

πtrω1

#

=1

4tr(Q(0) +Q(π))

−1

πtrω1+1 2

Z _π

0

trM(t,t)dt−1

2tr(h²+H²).

(6)

(ii) For the operator L(Q,M;h,∞): let λn^(j)(j = 1,d; n = 1,2, . . .) be eigenvalues of the operator L(Q,M;h,∞), then we have the trace formula

∞

∑

n=1

"

d

∑

j=1

λ_n^(j)−

n−1 2

2!

−2 πtrω₂

#

=1

4tr(Q(0)−Q(π)) +1 2

Z π

0

trM(t,t)dt−1 2trh².

(7)

(iii) For the operator L(Q,M;∞,h): let λn^(j)(j= 1,d; n = 1,2, . . .) be eigenvalues of the operator L(Q,M;∞,h), then we have the trace formula

∞ n=1

∑

"

d

∑

λn^(j)−

n−1 2

2!

−2 πtrω3

#

=1

4tr(Q(π)−Q(0)) +1 2

Z π

0

trM(t,t)dt−1 2trH².

(8)

(iv) For the operator L(Q,M;∞,∞): let λn^(j)(j= 1,d; n = 1,2, . . .) be eigenvalues of the operator

L(Q,M;∞,∞), then we have the trace formula

∞ n=1

∑

"

d j=1

∑

λn^(j)−n²

−2 πtrω4

#

=−1

4tr(Q(0) +Q(π)) +1

πtrω₄+1 2

Z π

0

trM(t,t)dt.

(9)

3. Proof

We only give the proof for (6) in Theorem1. Anal- ogously, we can also prove that (7) – (9) in Theorem1 hold.

LetΦ(x,λ)be a solution of (1) satisfyingΦ(0,λ) = I_d,Φ⁰(0,λ) =h, then we have

Φ(x,λ) =cos(ρx)I_d+h ρsin(ρx) +

Z x 0

sinρ(x−t) ρ Q(t) +

Z x t

M(ξ,t)sinρ(x−ξ)

ρ dξ

Φ(t,λ)dt. (10)

Using integration by parts and the iterative method, we can compute

Φ(x,λ) =cos(ρx)I_d+

h+1 2

Z _x 0

Q(t)dt

sin(ρx) ρ +

Q(x)−Q(0)

4 −1

2 Z x

0

[Q(t)h+M(t,t)]dt

−1 8

Z _x 0

Q(t)dt 2)

cos(ρx) ρ² +o

e^τx

|ρ|²

,

(11)

whereλ=ρ²,τ=|Imρ|.From (11), we obtain Φ⁰(x,λ) =−ρsin(ρx)I_d

+

h+1 2

Z _x 0

Q(t)dt

cos(ρx) +

Q(x) +Q(0) 4 +1

2 Z _x

0

[Q(t)h+M(t,t)]dt

+1 8

_{Z x}

0

Q(t)dt 2)

sin(ρx) ρ +o

e^τx

|ρ|

.

(12)

In virtue of (11) and (12), we get the character- istic matrix w(λ) for the boundary-value problem

(1) – (3):

w(λ) =Φ⁰(π,λ) +HΦ(π,λ)

=−ρsin(ρ π)I_d+ω1cos(ρ π) +ω₅sin(ρ π)

ρ +o e^{τ π}

|ρ|

,

(13)

where

ω1=h+H+1 2

Z π

0

Q(x)dx and

ω₅=Q(0) +Q(π) 4 +1

2 Z _π

0

[Q(t)h+HQ(t) +M(t,t)]dt+Hh

+1 8

_{Z π}

0

Q(t)dt 2

.

The eigenvaluesλ_n^(j) of the operatorL(Q,M;h,H) can be located by determining whether the matrix- valued functionw(λ)is singular or not. We can rewrite w(λ) =w0(λ) +ε(λ),

wherew₀(λ) = (−ρsin(ρ π))I_dandε(λ)is a remain- der. We shall see thatw0(λ)has a quite neat and sim- ple form from which we can determine those values ˆλ making w₀(λˆ) be singular. By the extension theo- rem of Rouch´e’s theorem on operator-valued functions in [12], we are getting close to locate the eigenvalues ofL(Q,M;h,H).

Let∆₁(λ) =o(e^{τ π}/|ρ|²), then w⁻¹₀ (λ)w(λ) =I_d−cot(ρ π)

ρ ω1− 1 ρ²ω₅ + ∆1(λ)

sin(ρ π).

(14)

DenoteΓN₀def

={ρ:|ρ|=N0+¹₂}andG_δ ={ρ:|ρ−k|

≥δ, k=0,±1,±2, . . .}, where N₀ is a sufficiently large integer, andδ >0 is sufficiently small. Accord- ing to [13, p. 7], we obtain

|sin(ρ π)| ≥C_δe^{τ π}, λ∈G_δ, |ρ| ≥ρ^∗, (15) whereρ^∗=ρ^∗(δ)sufficiently large, andC_δ is a con- stant with respect to δ. From (15) and ∆₁(λ) = o(e^{τ π}/|ρ|²), we get

∆1(λ) sin(ρ π)

=o 1

|ρ|²

onΓ_N0. (16)

Thus we have

∆(λ)^def=det[w⁻¹₀ (λ)w(λ)]

=det

I_d−cot(ρ π) ρ ω₁− 1

ρ²ω₅+o 1

ρ²

.

Using the Laplace expansion of determinants, we ob- tain that

∆(λ) =

d

∏

1−cot(ρ π)

ρ ω1,ii− 1

ρ²ω5,ii+o 1

ρ²

+acot²(ρ π) ρ² +o

1 ρ²

,

where

a=−

d−1 i=1

∑ ∑

i<j

ω1,i jω1,ji.

Moreover, we have

∆(λ) =1−cot(ρ π) ρ

d i=1

∑

ω1,ii− 1 ρ²

d i=1

∑

ω5,ii

+cot²(ρ π) ρ²

d−1

∑

i=1

∑

i<j

ω1,iiω1,j j+acot²(ρ π) ρ² +o

1 ρ²

.

Expanding ln∆(λ)by the Maclaurin formula, we ob- tain that onΓ_N0

ln∆(λ) =−cot(ρ π) ρ

d

∑

i=1

ω1,ii− 1 ρ²

d

∑

i=1

ω_5,ii

+cot²(ρ π) ρ²

d−1 i=1

∑ ∑

i<j

ω1,iiω1,j j−cot²ρ π 2ρ²

d i=1

∑

ω1,ii

!2

+acot²(ρ π) ρ² +o

1 ρ²

.

Let λn^(j)(j =1,d; n =0,1,2, . . .) be the zeros of the function detw(λ)andµn=n(n=0,±1,±2, . . .) be the zeros of the function detw₀(λ). From (14), for sufficiently large N0, we see that the numbers λn^(j)

(n≤N₀)are insideΓN0 and the numbersλn^(j)(n>N₀) are outside Γ_N0. Obviously, µ_n=n do not lie on the contourΓN₀. We have the following identity:

N₀ n=0

∑

d j=1

∑

(2λ_n−2n²) =− 1 2πi

I

ΓN0

2ρln∆(λ)dρ. (17)

By calculating residues, we get for sufficiently largeN0 I

ΓN0

cot(ρ π)dρ=2πi

N₀ n=−N

∑

1 π, I

Γ_N0

cot²(ρ π)

ρ dρ=−2πi+o(1), I

Γ_N0

o 1

|ρ|

dρ=o(1).

(18)

Substituting (18) into (17), we have

− 1 2πi

I

ΓN0

2ρln w(λ) w0(λ)dρ=1

2tr(Q(0) +Q(π)) +2(2N₀+1)

π trω₁+ Z π

0

trM(t,t)dt

−tr(h²+H²) +o(1).

(19)

From (17) and (19), we obtain

N0

∑

n=0

"

d

∑

j=1

(λ_n−n²)−2 πtrω1

#

=1

4tr(Q(0) +Q(π))

−1

πtrω₁+1 2

Z π

0

trM(t,t)dt

−1

2tr(h²+H²) +o(1).

(20)

LettingN₀→+∞in (20) yields

∞ n=0

∑

"

d

∑

(λ_n−n²)−2 πtrω1

#

=1

4tr(Q(0) +Q(π))

−1

πtrω₁+1 2

Z π

0

trM(t,t)dt−1

2tr(h²+H²).

This completes the proof of Theorem1.

Acknowledgements

The author would like to thank the referees for valu- able comments. The author is grateful to Professor V.

A. Yurko for many useful discussions related to some topics of spectral analysis of differential operators.

This work was supported by the National Natural Sci- ence Foundation of China (11171152/A010602), Nat- ural Science Foundation of Jiangsu Province of China (BK 2010489), the Outstanding Plan-Zijin Star Foun- dation of NUST (AB 41366), and NUST Research Funding (No. AE88787).

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