Trace for Differential Pencils on a Star-Type Graph

Trace for Differential Pencils on a Star-Type Graph

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@njust.edu.cn Z. Naturforsch.68a,421 – 426 (2013) / DOI: 10.5560/ZNA.2013-0020

Received October 19, 2012 / revised January 22, 2013 / published online May 1, 2013

In this work, we consider the spectral problem for differential pencils on a star-type graph with a Kirchhoff-type condition in the internal vertex. The regularized trace formula of this operator is established with the contour integration method in complex analysis.

Key words:Differential Pencils; Star-Type Graph; Trace Formula.

PACS numbers:03.65.Nk; 73.63.-b; 85.35.-p 1. Introduction

We consider the differential equations on a graph

−y⁰⁰_i(x) + [2ρp_i(x) +q_i(x)]y_i(x) =ρ²y_i(x), i=1,r; r≥2, r∈N, (1) whereρ is the spectral parameter, while the functions y_i(x)andy⁰_i(x)fori=1,rare absolutely continuous on [0,π]and satisfy the matching conditions

y_i(π) =y_j(π), i,j=1,r;

r i=1

∑

y⁰_i(π) +βy1(π) =0 (2) at the central vertex v0. We assume that real-valued functions p_i(x)∈W₂²[0,π] with ^R₀^πp_i(t)dt =0, i= 1,r, q_i(x)∈W₂¹[0,π]; β is a real constant given by (2). The real-valued functionsp(x) ={p_i(x)}_i=1,rand q(x) ={q_i(x)}_i=1,r on a graph are called potentials.

The matching conditions (2) are called the continuity condition together with a Kirchhoff-type condition.

We consider on a graph the boundary value problem for (1) with the matching conditions (2) and the fol- lowing boundary conditions at the boundary vertices v1, . . . ,vr:

y⁰_i(0)−h_iy_i(0) =0, i=1,r, (3) whereh_iare real numbers. Denote the problem (1), (2), and (3) byL=L(p,q,h,β), whereh={h_i}_i=1,r. Then (1) describes the wave propagation on a graph, where ρ is the wave number (momentum) andρ²the energy;

p(x)and q(x)describe the joint effect of absorption and generation of energy and the regeneration of the force density, respectively. Because of important ap- plications in quantum mechanics, it is interesting to investigate the spectral characteristics of the operator pencilL, and if at least one of the colliding particles is a fermion, it is relevant to solve the inverse problem in the presence of central and spin–orbital potentials.

The theory of regularized traces of ordinary dif- ferential operators has a long history. First, the trace formulas for the Sturm–Liouville operator with the Dirichlet boundary conditions and sufficiently smooth potential were established in [1]. Afterwards, these investigations were continued in many directions (see, [2–7], etc.). The trace formulas can be used for approximate calculation of the first eigenvalues of an operator, and in order to establish necessary and suffi- cient conditions for a set of complex numbers to be the spectrum of an operator.

Differential operators on graphs (networks and trees) often arise in natural sciences and technology.

The spectral problems of quantum graphs have been studied in [8–18], etc. Some results on trace formu- las for Laplacians on metric graphs have appeared in [10,14] and other articles.

Recently, the author considered the spectral problem for the Sturm–Liouville differential operator onr-star- type graph [16], and later found that by using a refined estimate for a fundamental pair of solutions to equa- tion (1) in [19], we can establish the trace formula for the operatorL, which extends the result of the previous work of the author [16].

© 2013 Verlag der Zeitschrift f¨ur Naturforschung, T¨ubingen·http://znaturforsch.com

2. Result

In [15], the author considered the spectral problem for the differential pencilLand gave asymptotics of the eigenvalues for L. The eigenvalue set ^Sr_j=1{ρ_n^(j)}^∞_−∞

for the differential pencilLbehaves asymptotically as follows (see Theorem 2.3 in [15]):

ρn⁽¹⁾=n+ω+β rn +O

1 n²

and

ρn^(j)=n+1 2+ c_j

n+¹₂+O 1

n²

, j=2,r, wherec_j, j=2,r, are the solutions (all real but not necessarily different) of the equation forx

P(x)^def= ∑^r

i=1

Πl6=i(x−ω_l) =0, where

ω_i(x) =1 2

Z x 0

[p²_i(t) +q_i(t)]dt+h_i, i=1,r, and

ω0= ∑^r

i=1

(2−r)p_i(π)+r p_i(0)

2 , ω= ∑^r

i=1

ωi(π).

The trace formula is interesting and of significance for the inverse spectral problem. How is it possible to obtain the second-order trace formula for eigenvalues of the problemL? It was noted that the formula of the second-order trace comprises lots of information on the operator spectrum, the boundary condition param- eters, and the potential of a graph.

Before giving the main result, we need the following notations. Fori=1,r, denote

b1,i =1 2

Z π

0

[p²_i(t) +q_i(t)]dt, a_1,i =1

2[p_i(π) +p_i(0)], c_1,i =1

2[p_i(0)−p_i(π)], d_2,i =1

8[5p²_i(π)−2p_i(0)p_i(π)−3p²_i(0) +2q_i(π)−2q_i(0)]−1

2b²_1,i, d1,i =−1

4[p⁰_i(π) +p⁰_i(0)] +1

2b1,i[p_i(π) (4)

−p_i(0)] +1 2

Z π

0

p_i(t)[p²_i(t) +q_i(t)]dt,

e_1,i =1

4[p⁰_i(π)−p⁰_i(0)]−1

2b_1,i[p_i(π) +p_i(0)]

+1 2

Z π

0

p_i(t)[p²_i(t) +q_i(t)]dt, e2,i =1

8[3p²_i(π)−2p_i(0)p_i(π) +3p²_i(0) +2q_i(π) +2q_i(0)] +1

2b²_1,i,

A₁={±0,±1,±2, . . .}, A₂={0,±1,±2, . . .}, and

ρ₋₀⁽⁰¹⁾=ρ₀⁽⁰¹⁾=0 ; ρn⁽⁰¹⁾=n, n∈Z\ {0}; ρ_n^(0j)=n+1

2, n∈Z, j=2,r.

(5) The main result of this work reads as follows.

Theorem 1. For the eigenvalue set^Sr_j=1{ρn^(j)}^∞_−∞for the differential pencil L, we have the trace formula

n∈A

∑

₁

h ρn⁽¹⁾

2

− ρn⁽⁰¹⁾

2

− 2 πr

r

∑

i=1

(b_1,i+h_i)−2β πr i

+

∑

n∈A₂

( _r

∑

i=2

h ρ_n⁽ⁱ⁾2

− ρ⁽⁰ⁱ

n )²i

−2(r−1) πr

r

∑

i=1

(b_1,i+h_i) )

=− 2 πr

r

∑

i=1

(b_1,i+h_i)−2β πr−

r i=1

∑

(2d_2,i+b²_1,i) (6) +2

r

r

∑

i=1

(d_2,i+e_2,i)−

r

∑

i=1

h²_i−β² r² +ω3, where

ω₃= 1 r²

h ^r

i=1

∑

(a_1,i−c1,i)i2

+2 r

r i=1

∑

a1,ic1,i+r−2 r

r i=1

∑

c²_1,i.

(7)

3. Proof of Theorem1

Denote byϕ_i(ρ,x),i=1,r, the solutions of (1) sat- isfying the conditions

ϕi(ρ,0) =0, ϕ_i⁰(ρ,0) =1, (8) and byψi(ρ,x), i=1,r, the solutions of (1) satisfying the conditions

ψi(ρ,0) =1, ψ_i⁰(ρ,0) =0. (9)

Thus, for the solutionssi(ρ,x), i=1,r, of (1) satisfy- ing the conditions

s_i(ρ,0) =1, s⁰_i(ρ,0) =h_i, we have

si(ρ,x) =ψi(ρ,x) +hiϕi(ρ,x),

s⁰_i(ρ,x) =ψ_i⁰(ρ,x) +h_iϕ_i⁰(ρ,x). (10) Then the solutions of (1) which satisfy the conditions (3) are

y_i(ρ,x) =A_i(ρ)s_i(ρ,x), (11) whereAi(ρ)are constants. Substituting (11) into (2), we obtain the following equation for the eigenvalues of the problemL:

ϕ(ρ)^def=φ(ρ) +β χ(ρ) =0, where

φ(ρ)^def=

r

∑

i=1

s⁰_i(ρ,π)×

∏

j6=i=1,2,...,r

s_j(ρ,π) (12) and

χ(ρ)^def=

r

∏

i=1

s_i(ρ,π). (13)

Using a refined estimate for a fundamental pair of solutions of the equation (1) in [19], we get

si(ρ,π) =ψi(ρ,π) +hiϕi(ρ,π)

=cos(ρ π)−c1,i

cos(ρ π)

ρ + (b_1,i+h_i)sin(ρ π) ρ

+ (d_2,i−h_ib_1,i)cos(ρ π)

ρ² + (d_1,i+h_ia_1,i)sin(ρ π) ρ²

+o e^{τ π}

ρ²

(14)

and

s⁰_i(ρ,π) =ψ_i⁰(ρ,π) +h_iϕ_i⁰(ρ,π)

=−ρsin(ρ π) + (b_1,i+h_i)cos(ρ π) +a_1,isin(ρ π) + (e_1,i+h_ic_1,i)cos(ρ π)

ρ + (e_2,i+h_ib_1,i)sin(ρ π) ρ

+o e^{τ π}

ρ

,

(15)

whereτ=|Imρ|.

Denote

ϕ0(ρ) =−rρsin(ρ π)cos^r−1(ρ π). (16) Here, we point out that ϕ0(ρ) is the characteris- tic equation for L(0,0,0) and its zeros, ρn^(0j),j = 1,2, . . . ,r, are defined by

ρ_n⁽⁰¹⁾=n,n∈A₁, and

ρn^(0j)=n+1

2,n∈A₂, j6=1.

Let the contourΓN, integerN=0,1,2, . . .→∞, de- note the following sequences of circular contours, tra- versed counterclockwise:

The contourΓ_N is the circle of radius(N+¹₄)²with its center at the origin.

Obviously,ρ_n^(0j)don’t lie on the contourΓ_N. Substi- tuting the expressions (14) and (15) into (12) and (13), we have on the contourΓ_N

φ(ρ) ϕ0(ρ)=1

r

r

∑

i=1

1−b1,i+h_i

ρ cot(ρ π)−a1,i

ρ

−(e_1,i+h_ic_1,i)cot(ρ π)

ρ² −e_2,i+h_ib_1,i ρ² +o

1 ρ²

·

∏

l6=i

h 1+T_l

ρ +V_l ρ²+o

1 ρ²

i ,

where

T_l=−c_1,l+ (h_l+b_1,l)tan(ρ π)

V_l=d2,l−h_lb1,l+ (d_1,l+h_la1,l)tan(ρ π). Moreover, by calculation, we obtain

φ(ρ) ϕ0(ρ)=1

r

r i=1

∑

1−b_1,i+h_i

ρ cot(ρ π)−a_1,i ρ

−(e_1,i+h_ic_1,i)cot(ρ π)

ρ² −(e_2,i+h_ib1,i) ρ² +o

1 ρ²

·

1+∑l6=iT_l ρ

+∑l6=iV_l+∑i₁<i₂6=iTi₁Ti₂

ρ²

+o 1

ρ²

=1−∑^ri=1(b_1,i+h_i)

r ×cot(ρ π) ρ +r−1

r

r

∑

i=1

T_i ρ

(17)

−1 r

r i=1

∑

a_1,i ρ +r−1

r

r i=1

∑

Vi

ρ²+r−2

r

∑

i₁<i₂

T_i1T_i2 ρ²

−∑^r_i=1(b_1,i+h_i)∑^r_i=1T_i−∑^r_i=1(b_1,i+h_i)T_i

r ·cot(ρ π)

ρ²

−∑^ri=1a_1,i∑^ri=1T_i−∑^ri=1a_1,iT_i

r · 1

ρ²−∑^ri=1(e_1,i+h_ic_1,i) r

·cot(ρ π)

ρ² −∑^r_i=1(e_2,i+hib1,i)

r · 1

ρ² +o

1 ρ²

, where

r i=1

∑

T_i=−

r i=1

∑

c1,i+

r i=1

∑

(b_1,i+h_i)tan(ρ π),

r i=1

∑

V_i=

r i=1

∑

(d_2,i−h_ib_1,i) +

r i=1

∑

(d_1,i+h_ia_1,i)tan(ρ π),

i₁

∑

<i₂

Ti₁Ti₂ =

∑

i₁<i₂

[(b_1,i1+hi₁)(b_1,i2+hi₂)tan²(ρ π)

−(b_1,i1+h_i1)c_1,i2tan(ρ π)−(b_1,i2+h_i2)c_1,i1tan(ρ π) +c_1,i1c_1,i2],

r i=1

∑

(b_1,i+h_i)T_i=

r i=1

∑

[(b_1,i+h_i)²tan(ρ π)

−(b_1,i+h_i)c_1,i], and

r

∑

i=1

a_1,iT_i=

r

∑

i=1

[a_1,i(b_1,i+h_i)tan(ρ π)−a_1,ic_1,i]. Also, on the contourΓ_N, it yields

β χ(ρ) ϕ0(ρ) =−β

rρ

cot(ρ π)−c_1,1cot(ρ π)

ρ +b_1,1+h₁ ρ

+O 1

ρ²

·

r

∏

i=2

1−c1,i

ρ + (b_1,i+h_i)tan(ρ π) ρ

+O 1

ρ²

=−β

rρcot(ρ π) +β r

r

∑

i=1

c1,i

cot(ρ π) ρ²

−β r

r i=1

∑

b_1,1+h₁ ρ² +O

1 ρ³

.

(18)

Therefore, on the contour ΓN, the following equation holds:

ϕ(ρ)

ϕ0(ρ)=1+ω1

ρ +ω2

ρ²+o 1

ρ²

,

where

ω1=−∑^r_i=1(b1,i+h_i)

r cot(ρ π) +r−1 r

r

∑

i=1

T_i

−1 r

r

∑

i=1

a1,i−β

r cot(ρ π)

and ω2=r−1

r

r i=1

∑

V_i+r−2

r

∑

i₁<i₂

T_i1T_i2

−∑^r_i=1(b_1,i+h_i)∑^r_i=1T_i

r cot(ρ π)

+∑^ri=1(b_1,i+h_i)T_i

r cot(ρ π)

−∑^r_i=1a1,i∑^r_i=1Ti−∑^r_i=1a1,iTi

r

−∑^r_i=1(e_1,i+h_ic1,i)

r cot(ρ π)−∑^r_i=1(e_2,i+h_ib1,i) r

+β r

r

∑

i=1

c_1,icot(ρ π)−β r

r

∑

i=1

(b_1,i+h_i).

Next, the power series expansion implies lnϕ(ρ)

ϕ0(ρ)=ω₁

ρ +ω2−¹₂ω₁² ρ² +o

1 ρ²

, (19)

where

−1

2ω₁²=−(∑^r_i=1(b1,i+h_i))²

2r² cot²(ρ π)

−β r²

r

∑

i=1

(b_1,i+h_i)cot²(ρ π)−β²

2r²cot²(ρ π) +r−1

r²

r i=1

∑

(b_1,i+hi)

r i=1

∑

Ticot(ρ π) +β(r−1)

r²

r i=1

∑

T_icot(ρ π)−β r²

r

∑

i=1

a_1,icot(ρ π)

− 1 r²

r

∑

i=1

a1,r r

∑

i=1

(b_1,i+h_i)cot(ρ π)−(r−1)² 2r²

r

∑

i=1

T_i²

−(r−1)² r²

∑

i₁<i₂

T_i1T_i2−(∑^ri=1a_1,i)² 2r² +r−1

r²

r

∑

i=1

T_i

r

∑

i=1

a_1,i.

(20)

The residue theorem in complex analysis tells us, on the contourΓ_N,

n∈A

∑

_N

h ρn⁽¹⁾2

− ρn⁽⁰¹⁾2i +

∑

n∈A_N0

r

∑

i=2

h ρn⁽ⁱ⁾2

− ρn⁽⁰ⁱ⁾2i

= 1 2πi

I

Γ_N

ρ² ϕ⁰(ρ)

ϕ(ρ) −ϕ₀⁰(ρ) ϕ0(ρ)

dρ

=− 1 2πi

I

Γ_N

2ρlnϕ(ρ) ϕ0(ρ)dρ,

(21)

where ρn⁽ⁱ⁾,ρn⁽⁰ⁱ⁾ are the zeros of the entire functions ϕ(ρ),ϕ₀(ρ) inside the contour ΓN listed with multi- plicity, respectively, and

AN={±0,±1,±2, . . . ,±N},

A_N⁰={0,1,2, . . . ,N−1,−1,−2, . . . ,−N}. Direct calculations yield that

1 2πi

I

Γ_N

tan(ρ π)

ρ dρ= 1 2πi

I

Γ_N

cot(ρ π)

ρ dρ=0 (22) and

1 2πi

I

ΓN

tan(ρ π)dρ=−2N₀ π , 1

2πi I

Γ_N

cot(ρ π)dρ=2N+1 π .

(23)

Using (22), (23), and (19), we have

− 1 2πi

I

Γ_N

2ρln ϕ(ρ) ϕ0(ρ)dρ

=− 1 2πi

I

ΓN

2ω₁+2ω₂−ω₁²

ρ +o

1 ρ

dρ

(24)

=4Nr+2 πr

r i=1

∑

(b_1,i+h_i) +4N₀+2 r β−

r i=1

∑

(2d_2,i+b²_1,i)

+2 r

r

∑

i=1

(d_2,i+e_2,i)−

r

∑

i=1

h²_i −β²

r² +ω3+o(1), where

ω3= 2 r²

∑

i1<i2

c_1,i1c_1,i2+2 r

r

∑

i=1

a_1,ic_1,i− 2 r²

r

∑

i=1

a_1,i

r

∑

i=1

c_1,i

+(r−1)² r²

r i=1

∑

c²_1,i+ 1 r²

r i=1

∑

a1,i

!2

= 1 r²

h ^r

∑

i=1

(a_1,i−c_1,i)i2

+2 r

r

∑

i=1

a_1,ic_1,i+r−2 r

r

∑

i=1

c²_1,i. (25)

Substituting (24) into (21) yields

n∈A

∑

_N

h ρn⁽¹⁾2

− ρn⁽⁰¹⁾2

− 2 πr

r i=1

∑

(b_1,i+h_i)−2β πr i

+

∑

n∈A_N0

( r

∑

i=2

h ρn⁽ⁱ⁾

2

− ρn⁽⁰ⁱ⁾

2i

−2(r−1) πr

r

∑

i=1

(b_1,i+h_i) )

=−2 πr

r i=1

∑

(b_1,i+hi)−2β πr−

r

∑

i=1

2d_2,i+b²_1,i +2

r

r

∑

i=1

(d_2,i+e_2,i)−

r

∑

i=1

h²_i−β²

r² +ω₃+o(1). (26) LettingN→∞in (26), we obtain

n∈A

∑

₁

h ρn⁽¹⁾2

− ρ⁽⁰¹

n )²− 2

πr

r i=1

∑

(b_1,i+h_i)−2β πr i

+

∑

n∈A₂

( r i=2

∑

h ρn⁽ⁱ⁾2

− ρn⁽⁰ⁱ⁾2i

−2(r−1) πr

r

∑

i=1

(b_1,i+h_i) )

=−2 πr

r i=1

∑

(b_1,i+h_i)−2β πr−

r

∑

i=1

2d_2,i+b²_1,i +2

r

r

∑

i=1

(d_2,i+e2,i)−

r

∑

i=1

h²_i−β² r² +ω₃. The proof of theorem is finished.

Acknowledgement

The author would like to thank the referees for valuable comments. This work was supported by the National Natural Science Foundation of China (11171152/A010602), Natural Science Foundation of Jiangsu Province of China (BK 2010489), and the Outstanding Plan-Zijin Star Foundation of NUST (AB 41366).

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