Spectral problems on compact graphs

• 1.    Perturbed Operators on Compact Graphs. Recently, the methods of mathematical modelling began to play an important role in the study of the frequency-resonance characteristics of various technical devices described by linear dynamical systems and computer diagnostics of technical systems based on frequencies of natural oscillations. In this case, usually, mathematical model is direct or inverse spectral problem for Sturm -Liouville’s operators on geometric graphs. The methods for finding eigenvalues and eigenfunctions of abstract discrete semibounded operators defined on compact graphs are developed in the article.

• 2.    Double-Edge Graph. As an example, consider a compact graph G, consisting of two edges E 1 aiid E ₂ on Figure.

  

  Graph G

  
  

Let G = G ( V , E ) be a finite associated oriented compact graph. Here V = {V i } =1 - set of vertices. E = {E j } j =1 - set of edges. Supf>ose. that each edge E j has the length of l j >  0 and cross-sectional area, d j >  0. On the edges E of the graph G we consider the operator

T + P = ( T 1 + P 1 ,T 2 + P 2 , ..,T о + P j o)

acting in the Hilbert space

H = L2(g) = {g = (g 1 ,g2,...,gjо), gj E L2(0,lj), j = 1,jо} with the scalar product [1]

l j0         j

(g,h) = 52 dj / gj hj dx, g, h E H.

j =1     ₀

Here T j are discrete semibounded operators, and P j are bounded operators, defined in L ₂(0 , l j ) ( j = 1 , j 0 ). We consider the boundary value problem

( T j + P j ) U j = TU j , U j = U j ( X j ) , X j e (0 ,l j ) , j = 1 ,j 0 ,                 (1) E   d k dT k     - E   d m du m     =0 ,            (2) EkEEa ( Vs ) d x k xk = 0    EmEE- ( Vs )    d x m xm = l m U i (0) = U k (0) = U m ( l m ) = U h ( l h ) , E i ,E k e E a ( V s ) , E m ,E h e E ш ( V s ) , ^s e N .

Let E ^a ( V s ) denote a set of edges witli origin at the vertex V_s. ai id E ^ш ( V s ) - set of edges with an end at the vertex Vs. Conditions (2) mean that the flow through each vertex must be equal to zero, and (3) means that the solution u = ( u 1 , u ₂ , ...,U j 0) at each vertex must be continuous. Also consider the boundary value problem:

Tjvj = Xvj, vj = vj (xj), xj e (0,lj), j = 1 ,j0,(4)

E  dk Ek"   - E  dm Tm    =0,(5)

EkEEa(Vs) dxk xk =0 E E (Vs) dxm xm=lm vi (0) = Uk (0) = Um (lm) = uh (lh ),

Ei, Ek e Ea(Vs), Em, Eh e Eш(Vs), ^s e N.

We denote by {X _k } k =1 eigenvalues of the problem (4) - (6), numbered in the order of nondecreasing of their magnitudes, and denote by { v _k = ( v 1 _k ,v _{2 k} ,...,V j 0 ,_k ) } / E1 - eigenvector-functions, corresponding to these eigenvalues X_k. Approximate solution of the problem (l)-(3) can be found in the form

n

u ( n ) = ^a k v k , k =1

where a_k, with undetermined coefficients and vector-functions v _k = ( v 1 _k , v _{2 k} ,..., v j 0 ,_k ), form a countable basis in the space L ₂( G ) with energy norm Ц v _k||T ₊ p, induced by the energy scalar product

( v k , v m ) T + P = ( T + P ) v k

l j0

vm = j=1

( T j + P j ) V jk V jm dx.

The space L ₂(G) with enei‘gy norm | |v _k | _{T + P} is dene)ted by H T ₊ P.

Theorem 1. If T + P is a semibounded from below operator, acting in the Hilbert space L ₂ (G). then solutions of the. problem (4)-( 6) form a basis lit the. energy space. HT ₊ P.

Доказательство. The system { v _k}k =1 is a basis in the Hilbert space HT ₊ P in that case, if it is closed in this space. Considering (4) and boundedness of operator P , we get

I I v k I I T + P = ( v k , v k ) T + P = (( T + P ) v k , v k ) = ( T v k , v k ) + ( P v k , v k ) =

= ( X k v k , v k ) + ( P v k , v k ) = X k I I v k I I² + ( P v k , v k ) < X k I I v k I I² + |P v k I I • I I v k I I <      (⁸)

< X k I v k 1 ² + IP !•! v k 1 ² = ( X k + IP I ) •! v k 1 ² .

The operator T is positive defined in space L 2 ( G ). Denote by H_T energy space, which is a replenishment of space L ₂ ( G ) by the norm | | v _k||_T , induced by the energy scalar product, defined by the relation[2]:

( v k , V m ) T =

( T V k , V m

l j0        j

) = Edd

•j ^-1    0

T j v jk v jm dx^.

Considering (4), we get, that | |v _k|| T = ( Tv_k, v_k ) = X_k| |v _k| |², from whence

| | v k | |² = ^ T λ k

Let c be the lower bound of operator T + P. Then

.

II ^v k ||T ₊ p =

(( T + P)v k , vk) > c(v k , v k ) = c ( Tv k , v k ) = c-| | v k || T . λ k              λ k

Vector-functions vk are eigenfuncticins of operator T and form a, Liasis in L2(G) [1]. Therefore, the system of these functions is closed in L 2(G), and he nee, in HT. Considering (8) - (10), we get c ||vk ||T < ||vk ||T+p < (1 + P) • ||vk ||T.

λk                             λk

Therefore, H_{T + P} consists of the same element, that H_T, so that, the system of functions { v _k }_k— ₁ is dosed in H_T + p-                                                              D

Corollary 1. Further, by the theorem 1, when solving the problem (1) - (3) in the form (7). we will use the. first n solutions of the problem (4) - (6) as coordinate functions v _k. ( k = 1 , to ). If necessary, the. eh'merits of the system { v _k}^ =1 should be. normalized.

In the articles of S.I. Kadchenko [3, 4] linear formulas for eigenvalues of perturbed discrete operators were obtained. Analogously to these papers, we can prove the theorem.

Theorem 2. If T = ( T 1 , T 2 ,...,Tj ₀) is a discrete semibounded operator, and P = ( P 1 ,P ₂ ,...,P j ₀) - bounded operator, acting in a separable Hilbert space. L ₂( G ). and the. system of vector-functions { v _k = ( v 1 _k,v _{2 k},...,Vj ₀ ,_k ) }k -1 forms an orthonormal basis in L 2( G ), then eigenvalues ц_т of operator T + P can be found from formulas;

µm

l j0 j

= X_m + ^2 d j / P j V jm V jm dx + 5 ( m ) .

j -1    ₀

(П)

m — 1

Here 5(m) =    [bk(m — 1) — bk(m)]. a,nd mk(m) mn-th Galerkin's approximation of k-th k-1

eigenvalues.

Following the Galerkin method, coefficients a_k (k = 1 , n ), included in (7), are found from the solution of a system of linear homogeneous equations n

52 a k [( v k , v m ) T + p — b ( v k , v m )] = 0 .                        (12)

k -1

Using the theorem 2, by the formulas (11) we find n eigenvalues цк k = 1, n) of operator T + P. We substitute some pm into the system (12) instead of parameter p. Then the determinant of this system is equal to zero, and the system (12) will have nontrivial solutions. We denote the coefficients ak, included in (7) and corresponding to this solutions, via, a km) k = 1, n). We use the normalization condition (u m (n), u m (n)) = 1. Convert it. taking into account the system orthonormality {vkInLp n n                         n          2

(u m ⁽ n ) , u m ⁽ n )} = ^^ a m ) a ^{m )(} v k , v i ) = ^ a m^m )) = ¹ .        (из)

k =1 1 =1                         1 =1

Having supplemented the system of equations (12) by equation (13), we find the coefficients a km ) k = 1 M

On each edge E j ( j = 1 , 2) we introduce the real parameter X j. varying from 0 ti 11 l j. On graph G define a vector-function u = ( u 1 , u ₂). wliicli ec mipoiient uj is a, function of parameter X j E [0 , l j ]. i.e. corresponds to an edge E j ( j = 1 , 2). On each edge E j of the graph G introduce an equation of the form:

—u j + q j ( X j ) U j = pu j .

We will assume, that the components of vector-function u are interconnected by standard gluing conditions, including the condition (2), analogous to the Kirchhoff’s condition, and continuity condition (3). The continuity condition means, that, since the vertex V > is an incident to the edges E 1 and E ₂, then the values of the components of the vector-function u on these edges in the ends, corresponding to the vertex V >. are coincide:

u 1 ( l 1 ) = U 2 (0) .

Condition (2) means that the sum of normal derivatives of the components of the vector function u in the л"ertices V ( j = 1 , 3) is equal to zero. i.e. if V corresporiding to X j = 0. then the derivative of the component u j in the point, corresponds to the vertex V j, is taken with a sign " +". and witli a, sign "  — ". if V j corresponds to X j = l j:

du 1 dX ₁

x 1 = 1 1

+ du ₂ dX ₂

= 0 .

x 2 =0

In the boundary vertices V1 and V3 the conditions (2) transform to the Neumann’s conditions:

du 1

dX 1

= 0 , x 1 =0

du ₂

dx ₂

= 0 .

x 1 = l 2

We use the system of coordinate functions { v k } ^ =1 to construct the solution of the problem (7), (14) - (18) and while finding the eigenvalues {^ k } ^ =1 by the formulas (11). To find the system we solve the boundary value problem:

-v j = ^Xvj , ^j = ¹ , ² , v 1 ( 1 1 ) = v 2 (0) ,

dv 1 dx 1

x 1 = l 1 dv 1

dx 1 dv 2

dx 2

+ dv < 2

dx 2

x 1 =0

x 2 =0

= 0 ,

= 0 .

x 1 = l 2

= 0 ,

It can be shown that eigenvalues of the spectral problem (19) are

Xk=(ипс) ’ and eigenfunctions are v 1 k (X1) = Ck cos VXx 1, v2k (x2) = Ck cos VX 11 cos VXx2 — sin yX 11 sin yXx2

The constants C _k are determined from the normalization condition.

Through v _k = ( v 1 _k ,v _{2 k} ) denote the vector-functions corresponding to the eigenvalues X _k. To find eigenvalues p _k and vector-functions u _k of the boundary problem (14) - (18) a computational experiment was conducted. Verification of the spectral characteristics was performed by substituting them into the ecpiation (14). The Table shows the norms of the left and right side of the ecpiation (14) and the difference between them.

Conclusion. New algorithm for finding eigenvalues of abstract discrete semibounded operators on geometric graph is developed. Numerous computational experiments have shown high computational efficiency of the algorithm.