A class of strongly stable approximation for unbounded operators Текст научной статьи по специальности «Математика»

ISSN 1810-0198. Вестник Тамбовского университета. Серия: естественные и технические науки

Том 24, № 126 2019

© A. Khellaf, S. Benarab, H. Guebbai, W. Merchela, 2019 DOI 10.20310/1810-0198-2019-24-126-218-234 УДК 517.984

A class of strongly stable approximation for unbounded operators

Ammar KHELLAF1 , Sarra BENARAB2 , Hamza GUEBBAI1 , Wassim MERCHELA2

Universite 8 Mai 1945 B.P. 401, Guelma 24000, Algeria ORCID: https://orcid.org/0000-0001-5282-7593, e-mail: amarlasix@gmail.com; khellaf.ammar@univ-guelma.dz

ORCID: https://orcid.org/0000-0001-8119-2881, e-mail: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz

2) Derzhavin Tambov State University 33 Internatsionalnaya St., Tambov 392000, Russian Federation ORCID: https://orcid.org/0000-0002-8849-8848, e-mail: benarab.sarraa@gmail.com ORCID: https://orcid.org/0000-0002-3702-0932, e-mail: merchela.wassim@gmail.com

Класс сильно устойчивой аппроксимации неограниченных операторов

Аммар ХЕЛЛАФ1 , Сарра БЕНАРАБ2 , ХамзаГЕББАЙ1 , Вассим МЕРЧЕЛА2

1)1 Университет 8 Мая 1945 24000, Алжир, г. Гельма, п.я. 401 ORCID: https://orcid.org/0000-0001-5282-7593, e-mail: amarlasix@gmail.com; khellaf.ammar@univ-guelma.dz

ORCID: https://orcid.org/0000-0001-8119-2881, e-mail: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz 2) ФГБОУ ВО «Тамбовский государственный университет им. Г.Р. Державина»

392000, Российская Федерация, г. Тамбов, ул. Интернациональная, 33 ORCID: https://orcid.org/0000-0002-8849-8848, e-mail: benarab.sarraa@gmail.com ORCID: https://orcid.org/0000-0002-3702-0932, e-mail: merchela.wassim@gmail.com.

Abstract. We derive new sufficient conditions to solve the spectral pollution problem by using the generalized spectrum method. This problem arises in the spectral approximation when the approximate matrix may possess eigenvalues which are unrelated to any spectral properties of the original unbounded operator. We develop the theoretical background of the generalized spectrum method as well as illustrate its effectiveness with the spectral pollution. As a numerical application, we will treat the Schrodinger's operator where the discretization process based upon the Kantorovich's projection.

Keywords: eigenvalue approximation; spectral pollution; generalized spectrum approximation, Schrodinger operator

For citation: Khellaf A., Benarab S., Guebbai H., Merchela W. A class of strongly stable approximation for unbounded operators // Vestnik Tambovskogo universiteta. Seriya estestvennye i tekhnicheskie nauki - Tambov University Reports. Series: Natural and Technical Sciences, 2019, vol. 24, no. 126, pp. 218-234. DOI: 10.20310/1810-0198-2019-24-126218-234.

Аннотация. С использованием метода обобщенного спектра получены новые достаточные условия решения проблемы спектрального загрязнения. Эта проблема, возникающая в спектральном приближении, вызвана тем, что приближенная матрица может иметь собственные значения, которые не связаны с какими-либо спектральными свойствами исходного неограниченного оператора. Мы разрабатываем теоретические основы метода обобщенного спектра, а также иллюстрируем его эффективность при наличии спектрального загрязнения. В качестве численного приложения рассматривается оператор Шрёдингера, а процесс дискретизации этого оператора основывается на проекции Канторовича.

Ключевые слова: приближение собственных значений; спектральное загрязнение; аппроксимация обобщенного спектра; оператор Шрёдингера

Для цитирования: Хеллаф А., Бенараб С., Геббай Х., Мерчела В. Класс сильно устойчивой аппроксимации неограниченных операторов // Вестник Тамбовского университета. Серия: естественные и технические науки. Тамбов, 2019. Т. 24. № 126. С. 218-234. DOI 10.20310/1810-0198-2019-24-126-218-234. (In Engl., Abstr. in Russian)

1. Introduction

Let (H,D(H)) be a self-adjoint unbounded operator on a Hilbert space H. With the purpose of finding the spectrum set sp(H) of the operator H by using numerical approach, the conventional methods used are the projection methods (see e.g. [1] and [2]). Precisely, let (Pk)ken be a sequence of orthogonal projections Pk : H ^ Lk , where the closed set Lk is a subspace of D(H). In the theory of spectral approximation, we seek whether or not lim sp(Pk H Pk) = sp(H). Generally, the result is negative, where for k large enough, the

k^x

set sp(Hk) may contain points that do not belong to the set sp(H).

The weakness of projection method is well known in numerical analysis as the spectral pollution problem, this is an important problem in several areas in the field of applied mathematics (see e.g. [3], [4] and [5]).

In this paper, we use an alternative method, the generalized spectral method, which has been introduced in [6]. This new method is based on the concept of the generalized spectrum, (see [7] and [8]).

Let T and S be two bounded linear operators defined on a Banach space X , we define the generalized resolvent,

re(T, S) = {z E C : (T — zS) : X ^ X is beijective }.

The complementary set of the generalized resolvent set is the generalized spectrum, denoted sp(T, S). We say that Л is a generalized eigenvalue of the couple (T, S) if there exists u EX \ {0} such that

Tu = ЛSu.

The subspace Ker(T — AS) is called the generalized spectral subspace corresponding to A.

The space of all bounded linear operator defined on the Banach space X is denoted by BL(X). We consider now an unbounded operator (A,D(A)) defined on X , we recall that the resolvent set of A is given by

re (A) = {z e C : (A — zl) : D(A) ^ X, is beijective and (A — zl )-1 e BL(X)}, and the spectrum set of A is sp(A) = C \ re (A).

In this work, under the assumption re(A) = 0 , we prove that each spectral problem associated to A has an equivalent generalized spectral problem which means that there exist two bounded operators T and S defined on X , satisfying sp(T, S) = sp(A). Furthermore, if A is an eigenvalue of A, then A is a generalized eigenvalue of the couple (T, S) and

Ker(A — AI ) = Ker(T — AS). (1.1)

Through the numerical approximation of the bounded operators T and S by sequences of bounded operators (Tk)keN and (Sk)keN defined on X, where they converge in an appropriate sense to T and S, we prove that

lim sp(Tfc, Sk) = sp(T, S).

The limit here is understood as a combination of the following Property U and Property L, where they are naturally extended from the classical case with S = I (see [9]).

Property U: if Ak e sp(Tk, Sk) and Ak ^ A , then A e sp(T, S).

Property L: if A e sp(T, S), then there exists (Ak)keN such that Ak e sp(Tk,Sk) and Ak ^ A.

We organize this paper as follows: throughout section 2, we construct the theoretical foundations of the generalized spectral method. This theory is a generalization of the classical case when S = I (see [9]). In section 3, we prove that the Property U and Property L hold under appropriate convergence of (Tk)keN and (Sk)keN to T and S respectively. Finally, a numerical application is given for the case of Schrodinger's operator, where our numerical results show the coherence and the effectiveness of the generalized spectrum method (see [11]).

2. Generalized spectrum

Let (X, || ■ ||) be a Banach space. The space BL(X) is the set of all bounded linear operators on X equipped with the subordinated operator norm,

||A|| = sup{||Ax|| : x e X, ||x|| = 1}, A e BL(X).

Let T and S be two operators in BL(X), for z e re(T, S), we set

R(z,T, S) = (T — zS)-1

as the generalized resolvent operator. Let A E sp(T, S) be a generalized eigenvalue, we say that A has a finite algebraic multiplicity if

dim Ker(T — AS) < x>.

We remark that, if the operator S is invertible, then

sp(T, S) = sp(S-1T),

but if S-1 does not exist, the generalized spectrum set can be a bounded set, or the whole C, or an empty set.

The next three results are a generalization of a classical case when S = I. The proofs are provided in [6].

Theorem 2.1. Let A E re(T,S) and p E C, where |A — < \\R(A,T,S)S||-1 . Then p E re(T, S) .

Corollary 2.1. The set sp(T,S) is closed in C .

Theorem 2.2. The function R(^,T,S) : re(T,S) ^ BL(X) is analytic, and its derivative is given by R(^,T,S)SR(,T, S) .

We consider now an unbounded operator A with domain D(A) C X. The following theorem shows that every unbounded operator allows a pair of two bounded operators in BL(X) which expresses it in the terms of the generalized spectrum.

Theorem 2.3. If re(A) = 0, then there exist T,S E BL(X) such that

sp(A) = sp(T, S).

In particulary, A is an eigenvalue for A if and only if A is a generalized eigenvalue for the couple (T,S) . In addition, the equality (1.1) is satisfied.

P r o o f. Let a E re(A). We define S,T : X ^ D(A) as follows:

S = (A — al)-1, T = A (A — al)-1.

It is clear that T,S E BL(X). To show that sp(A) = sp(T,S), we prove that re(A) = re(T,S). Let A E re(A), i. e. there exists operator (A — AI)-1 E BL(X). Then

(A — AI)(A — al)-1 = I + (a — A)(A — al)-1 E BL(X).

So as

[(A — AI)(A — aI)-1]-1 = (A — aI)(A — AI)-1 = I + (A — a)(A — AI)-1 E BL(X),

we get

A(A — a/)-1 — A(A — a/)-1 e BL(X) ^ (T — AS)-1 e BL(X).

Thus, it is proved that A e re(T, S).

Inversely, let A e re(T, S). To show that (A — A/)-1 e BL(X), we prove that (A — A/) is bijective. Firstly, check the injectivity. Let u e D(A), using the fact that A commutes with (A — a/)-1 we have

(A — a/)-1 Au = A(A — a/)-1u = u + a(A — a/)-1u. (2.2)

Taking into consideration the equality (2.2), we find

(A — A/)u = 0 ^ (A — a/)-1(A — A/)u = 0 ^ (A — A/)(A — a/)-1u = 0

^ [A(A — a/)-1 — A(A — a/)-1]u = 0 ^ (T — AS)u = 0 ^ u = 0.

Secondly, prove the surjectivity. For all y e X we show that (A — A/)x = y has a solution x e D(A). Put x = (A — a/)-1(T — AS)-1y; it is clear that x e D(A) (the fact that (A — a/)-1 : X ^ D(A)), moreover we have

(A — A/)(A — a/)-1(T — AS )-1y = [A(A — a/)-1 — A(A — a/)-1](T — AS)-1y

= (T — AS )(T — AS)-1 y = y.

Furthermore, we can see, upon the choice of the vector x, that

||x||<||(A — a/)-11| ||(T — AS)-1|| ||y||,

so

|| (A — A/)-1|| < ||(A — a/)-1|| ||(T — AS)-1||,

which implies that (A — A/)-1 e BL(X) and therefore A e re(A).

Now, we show that the equality (1.1) holds. Let A be a generalized eigenvalue of the couple (T, S) , then there exists u e X\{0} such that Tu = ASu , thus

Tu = ASu ^ A(A — a/)-1u = A(A — a/)-1u

^ u = (A — a)(A — a/)-1 ^ u e D(A).

By applying (A — a/) on Tu = ASu , we find that Au = Au . Inversely, let A be an eigenvalue of A, then Au = Au. So, by applying (A — a/)-1 on Au = Au and using the fact that (A — a/)-1Au = A(A — a/)-1u for all u e D(A), we find that Tu — ASu = 0 . □

We note that the choice of the couple (T, S) as a function of the resolvent operator of A is not unique (see the numerical application below).

The next results represent the theoretical background of the generalized spectrum approach.

Theorem 2.4. Let T, S E BL(X), and let A be a generalized eigenvalue with finite algebraic multiplicity, isolated in sp(T, S) . We denote by r the Cauchy contour separating A from sp(T, S) . Then the operator

P = — f(T — zS)-1S dz (2.3) 2in Jr

defines a projection from X to X , and we have

P X = Ker(T — AS). (2.4)

P r o o f. To show that the operator P given by (2.3) is a projection form X to X , see the book [8, p. 50]. Now to prove the equality (2.4), firstly we fix a E re(T,S), where for any Cauchy contour r associated with A we assume that a E r. For p E r, we have

pS — T = (aS — T )[(a — p)-1I — (aS — T )-1S](a — p)

which gives

(pS — T)-1 = [(a — p)-1I — (aS — T)-1S]-1(a — p)-1 (aS — T)-1.

Thus, we can see that (a — A)-1 is an eigenvalue of the operator (aS — T)-1S. Indeed

u E Ker(T — AS) ^ (T — AS)u = 0 ^ (aS — T)-1(aS — T + T — AS)u = u

^ (aS — T)-1Su = (a — A)-1u ^ u E Ker((aS — T)-1S — (a — A)-1I).

We reverse the last process and get

Ker(T — AS) = Ker((aS — T )-1S — (a — A)-11).

Now, under the choice of a , we can see that for all Cauchy contour r , n(r) is also a Cauchy contour of the eigenvalue (a — A)-1 where n(p) = (a — p)-1.

We put B = (aS — T)-1S and z = (a — p)-1 for any p E r. Following this notation we have

(pS — T )-1S = z[—I + z (zI — B)-1 ]. Thus, integrating over the r, we get

^ i(pS — T)-1Sdp = 2- [ z[—I + z(zI — B)-1] -2 2nt Jr 2nt Jn(r) z2

= ^ i \—z-1I + (zI — B)-1] dz

Jn(r)

If 1 , T 1

-— - dz I +— (zI - B)-1 dz = -P{(a-X)-1}, Jn(r) z Jn(r) }

where P{(a-A)-i} is the spectral projection associated with the operator (aS—T) 1S around (a — A)-1. Hence, according to the spectral decomposition theory,

P X = P{(a-A)-i}X = Ker((aS — T )-1S — (a — A)-1 /) = Ker(T — AS).

□

Now, we show some results obtained in the qualitative aspect for the generalized spectrum theory.

We denote by B(0, k) C C the ball with center 0 and radius k > 0.

Theorem 2.5. Let T, S e BL(X), then there exists k > 0 such that sp(T, S) C B(0, k) if and only if 0 e sp(S) .

Proof. We assume that sp(T, S) C B(0, k), then for a e re(T, S), we get

AS — T = (aS — T)[(aS — T)-1S — (a — A)-1](A — a). (2.5)

As a e re(T, S), we have that

A e sp(T, S) ^ (a — A)-1 e sp((aS — T)-1 S).

So, the inclusion sp(T, S) C B(0,k) implies the relation 0 e sp((aS — T)-1S); otherwise,

0 e sp((aS — T)-1S) implies ro e sp(T, S). Thus 0 e sp((aS — T)-1 S) gives 0 e sp(S).

□

We denote by spp(T, S) the set of all generalized eigenvalues. It is clear that when X is a finite-dimensional space, the generalized spectrum consists only of the generalized eigenvalues, except

Theorem 2.6. Let T, S e BL(X), if S is compact, then

sp(T, S) = spp(T, S) U

Proof. We use the expression (2.5). Since the operator (aS — T)-1 S is compact, sp(T, S) is a set of isolated points. Let A e sp(T, S), then there is y e sp((aS — T)-1S), where 7 = (a — A)-1. Hence there exists u e X \ {0} such that

(aS — T)-1Su = ^u ^ (aS — T)-1(aS — AS)u = u

^ u + (aS — T)-1(T — AS)u = u ^ Tu = ASu.

□

3. Generalized spectrum approximation

Let T, S E BL(X), where re(T, S) = 0 , and let (Tk)keN and (Sk)keN be two sequences in BL(X). We will use the following conditions: (H1) S is a compact operator in BL(X) , (H2) \\(Tk - T )x|| — 0, || (Sk - S )x|| — 0 for all x EX , (H3) W(Tk - T)T|| - 0 , (H4) W(Sk - S)T||- 0 .

In the sequel, we write ■ -— ■ to express the pointwise convergence, while the norm convergence is denoted by ■ ■ .

Proposition 3.1. (see [6]) Let T, T, S, S E BL(X) . For all z E re(T, S),

if HR(z,T,S) (T - T) - z(S - S) H < 1, then z E re(T,S), and the next inequality is satisfied

HR(z,fM< WR(z,T,S)W

1 — \\R(z,T,S) (T — T) + z(S — S)

Remark 3.1. According to our assumptions in (H1) — (H4) we can easily conclude that

[(Tk — T) — A(Sk — S)] (T — zS) 4 0,

for all z E re(T, S).

Proposition 3.2. Let A, B and C be three bounded operators such that 0Esp(B) and AB 4 C, then B-1A 4 b-1CB-1.

P r o o f. We note that \\B-1A — B-1 CB-1|| < l\B-1\l \\AB — C|| l\B-1l\. □

Theorem 3.7. Property U. For k E N, under (H1) — (H4), if Ak E sp(Tk,Sk) and Ak 4 A, then A E sp(T, S).

Proof. We assume that A E re(T, S). Since the set re(T, S) is open in C, as stated in Corollary 2.1, there exists r > 0 such that

E := {p E C : \p — A\ <r}c re(T, S).

On the other side, for all z E E and for all k E N, we find that

Tk — zSk = (T — zS) [I + R(z, T, S)[(T — Tk) — z(S — Sk)]].

Using Remark 3.1 and Proposition 3.2 with

A =[(T — Tk) — A(S — Sk)], B = (T — AS),

so, there exists k0 E N such that

\\R(z,T,S) [(T — Tk) — A(S — Sk)] \\< 2,

for all k > k0 • Then, by Proposition 3.1, we find z G re(Tk, Sk) such that

||R(z,Tk,Sk)||< 2||R(z,T,S)||, Vk > ko,

but Ak A A, thus there exists ki G N such that Ak G E C re(Tk, Sk) for k > ki, which form a contradiction. □

In numerical test, we calculate the quantity

sup {sp(T, S)) : ^ G sp(Tk, Sk)}, its convergence to 0 implies the Property U. We mention that

sp(T, S)) = inf — y|.

y€sp(T,S)

Lemma 3.1. (see [10]) Let Pi and P2 be two projections on X such that

||(Pi — P2)Pl|| < 1,

then dim PiX < dim P2X .

Lemma 3.2. Let z G re(T, S), under (H1) — (H4) there exists a positive integer k0 G N such that for k > k0 , z G re(Tk, Sk) and

R(z, Tfc,Sfc) R(z,T, S).

Proof. Let z G re(T, S) , we have

Tk — zSk = (T — zS) [I + R(z, T, S)[(T — Tk) — z(S — Sfc)]],

for all k G N. As stated above in the demonstration of Theorem 3.7, we find z G re(Tk, Sk) for all k > k0, and R(z,Tk, Sk) is uniformly bounded for all k G N.

On the other side, for z G re(T, S) fl re(Tk, Sk),

R(z, Tk, Sk) — R(z, T, S) = R(z, T, S) [(T — Tk) — z(S — Sk)] R(z, Tk, Sk).

Since R(z, T, S) [(T — Tk) — z(S — Sk)] A 0 (according Remark 3.1 and to Proposition 3.2) and R(z, Tk, Sk) is uniformly bounded for all k G N, we have that

R(z, T, S) [(T — Tk) — z(S — Sk)] (Tk — zSk)-i 0.

□

Theorem 3.8. Let A be a generalized eigenvalue of finite type, isolated in sp(T, S) . We denote by r the Cauchy contour separating A from sp(T, S) . Under (H1) — (H4), there exists ko e N such that for each k > ko, we have

dim P X = dim PkX,

where

P = — ¿/R(Z'T'S)Sdz, Pfc = — ¿/R(z>Tfc ,Sfc )-1Sfc dz. Proof. For z e r and k > k0, we see that

R(z,Tfc,Sfc)Sfc — R(z,T,S)S = [R(z,Tfc, Sfc) — R(z,T,S)] S

— [R(z, Tfc, Sfc) — R(z, T, S)] (S — Sfc) — R(z, T, S)(S — Sfc).

From (H1) — (H4) we easily find that (S — Sk)(T — zS) -— 0 , thus according to Proposition 3.2 we have R(z, T, S)(S — Sk) -A 0 . Now by using Lemma 3.2, we have

R(z, Tfc, Sfc)Sfc — R(z, T, S)S — 0.

Finally, we apply Lemma 3.1 and find that dim PX = dim PkX for k > k0 . □

Theorem 3.9. Property L. Let A be a generalized eigenvalue of finite type, isolated in sp(T, S) . Under (H1) — (H4) there exists a sequence Ak e sp(Tk, Sk) such that Ak — A .

Proof. Let r be the Cauchy contour separating A from sp(T, S). We set

Afc e int(r) n sp(Tfc, Sfc).

Since re(T, S) 9 z — R(z,T, S)S and re(Tk, Sk) 9 z — R(z,Tk, Sk)Sk are analytic functions, and Pk -— P , we find

(Afc)fceN = 0 ^^ int(r) n sp(T, S) = 0.

We fix e > 0 such that the sequence (Ak)keN belongs to B, where

B = {z e C : |z — A| < e}.

On the other hand, it is enough to show that every convergent subsequence of (Ak)keN converges to A itself. Indeed, let a subsequence (Ak/)k'eN converge to A where A = A. By Property U proved in Theorem 3.7, we see that A e sp(T, S), but A e B and sp(T, S) n B = {A} , hence A = A , thus Ak — A . □

The last theorem shows that for every generalized eigenvalue A of finite type isolated in sp(T, S) , there exists a sequence (Ak)keN converging to A such that Ak e sp(Tk, Sk). The next result shows that the generalized eigenvectors associated to Ak converge to the generalized eigenvector associated with A.

We define the notion of gap between two closed subspaces Z and Y of X as

gap(Z,Y) = max {y(Z,Y),j(Y, Z)},

where

Y(Z, Y) = sup {dist(z, Y) : z E Z, \\z\\ = ^. Theorem 3.10. Let M = PX and Mk = PkX for k E N. Then gap(M, Mk) 4 0. Proof. Let u E M = PX such that \\u\\ = 1. For k E N large enough we have dist(u,Mk) < \\u — Pku\\ = \\Pu — Pku\\ < \\P — Pk\\. Let u E Mk = PkX such that \\u\\ = 1. For k E N large enough

dist(u,M) < \\u — Pu\\ = \\Pku — Pu\\ < \\P — Pk\\, which implies gap(M, Mk) < \\Pk — P\\. □

4. Numerical application

As an example for which the numerical results are available by other approaches, we consider the following problem from [11]; it is also studied in [13].

We consider the unbounded operator A defined on L2(0, by the differential equation

Au := —u" + x2u, x E [0, +<x), u(0) = 0. This is the harmonic oscillator problem with domain

D(A) = H2(0, +<x) n{u E L2(0, <x): i x2 \u\2dx < ro}.

First, according to the theory of pseudo spectrum for self-adjoint operators (see [6], [11] and [14]) we can find

sp(A) = U sp(Aa), (4.6)

a )

a>0

where Aa is the Schrodinger operator which has the same formula as A in L2(0,a), but with the Dirichlet condition at the point a. The domain of Aa is given by

D(Aa) = H2(0,a) n Hl(0,a).

Let a > 0 , we denote by La the Laplacien operator defined on L2(0, a) by

Lau = -u", D(L) = H2(0,a) n H^(0,a).

Proposition 4.3. (see [12]) La is invertible and its inverse is the bounded operator Sa defined by

S„u(x) = / G{0,a}(x,y)u(y)dy, u G L2(0, a), Jo

where

j x(a-y) 0 < x < y < a,

G{0,a}(x,y) = S y(a-x) nz / /

I y( a ) 0 < y < x < a. Let Ta be a bounded operator defined on L2(0, a) to itself by

Tau(x) = u(x) + / G{0,a}(x,y)y2u(y)dy. 0

Theorem 4.11. sp(A) = y sp(Ta,Sa).

a>0

Proof. According to (4.6), we need only to show that sp(Aa) = sp(Ta,Sa) for all a > 0.

Let A be an eigenvalue of Aa with the eigenvector u G D(Aa) \ {0}. By applying Sa to Aau = Au, we get Tau = ASau, which implies that A is a generalized eigenvalue of the couple (Ta, Sa) with the eigenvector u G L2(0, a) \ {0} .

Inversely, let A be a generalized eigenvalue of the couple (Ta,Sa) with the eigenvector u G L2(0, a) \ {0} , i. e. Tau = ASau , so

u = ASau — Sa(vu) ^ u = Sa(Au — vu),

where v(x) = x2 . Since Au + vu G L2(0, a) , we have u G D(La) = D(Aa), then

u + Sa(v u) = ASau ^ Lau + v u = Au.

□

Now, for a > 0 we use Kantorovich's projection method to approach the operators Ta

and Sa. We define a subdivision of [0, a] for n > 2 by

a

=-, Xj = (i — 1)hn, 1 < i < n.

n — 1

Let Ta,n and Sa,n be the approximation operators of Ta and Sa by means of Kantorovich's projection methods (see [9]), given for all x G [0,a] by

Ta,„u„(x) w u„(x) + G{0,a}(xi,y)y2u„(y)d^ei(x),

¿=1

Sa,„u„(x) w G{0,a}(xj,y)u„(y)dy)ej(x),

0

where, for 2 < i < n — 1,

\x xi \ ei(x) = {1 h

0, otherwise

1 — —7-, xi-1 < x < xi+1

x2 - x

eAx) = < ~h , x1 < x < x2

0, otherwise, x xn_1

. . . — , xn-1 < x < xn

e n ( x ) = hn

0, otherwise.

By applying Kantorovich's projection method [9] to the equation Tau = ASau, we get the approximate equation

un(x) + y2( / G{0,a}(xi,y)y2un(v)dy)ei(x) i=i Jo 7

= An G{0ia}(xi,y)un(y)dy^jei(x), x E [0,a].

i=1

Denote by 31 and 32 the two vectors

aa

fa(i)= G{0,a}(xj,y)y2un(y)dy, fc(i)= G{0,a}(xj,y)un(y)dy, 1 < i < n, Jo Jo

then we can rewrite the previous approximate equation as

nn

un(x) + p1(i)ei(x) = p2 (i)ei(x). (4.7)

i=1 i=1

Multiplying first equation (4.7) by G{0,a}(xj,x)x2 for 1 < j < n and integrating over [0, a] we obtain

A^y^ 32 (i)[ G{o,a}(xj ,x)x2ei(x)dx) = / G{o,a}(xj ,x)x2un(x)dx i=1 o o

+ 1P1(i){i G{o ,a} (xj, x^x ei (x^dx 1.

i=1 Jo

The latter equation is equivalent to the matrix equation

P1 + A/31 = AnAP2, (4.8)

where A is a matrix defined by

A(i,j)= G{o,a}(xj,x)x2ei(x)dx, 1 < i,j < n.

o

In the same way, multiplying equation (4.7) by G{0,a}(xj, x) for 1 < j < n and integrating over [0, a] , we also obtain

Aray^ (i) ( / G{o,tt}(xj, x)ej(x)dxj = / G{o,tt}(xj,x)ura(x)dx i=1 Vo 'J0

+ V]&(i)( / G{0,a}(xj,x)ei(x)dx i=i V70

the latter equation is equivalent to the matrix equation

& + Bft = AraB^2,

where B is a matrix defined by

B(i, j) = / G{o,tt}(xj,x)ei(x)dx, 1 < i,j < n.

(4.9)

So, by using this process, we have transformed the equation (4.7) into the system of two matrix equations (4.8) and (4.9), namely

ßi + Aß! = A„Aß2, ß2 + Bßi = A„Bß2.

We also can write this system as

+ A)^i + O raxra & = A ft + AraAft,

+ ft = An°raxraft + AraBft,

where /raxra is the identity matrix with dimension n x n and Onxn is the null matrix with dimension n x n. This leads to the matrix generalized eigenvalue problem

A + ^raxra Oraxra

^^ ^n x n

ßi

ß2

An

Oraxra A Oraxra B

ßi

ß2

Finally, we use the command "eig" in Matlab to calculate the generalized eigenvalue of

A +1 B

On

nxn ^raxra ^raxra

A

nxn °raxra B

We mention that Kantorovich's projection method gives the norm convergence (see [9]) which satisfies our assumption in (H1) — (H4).

We fix n = 200 to approach the eigenvalues in our example.

The following table 1 shows that the Kantorovich's method converges perfectly compared with the exact eigenvalue.

0

Table 1: The numerical results for a=5

Exact eigenvalue Kantorovich's method

3 3.0001972

7 7.0009887

11 11.0026039

15 15.0103317

19 19.0806050

5. Conclusion

Our study shows the efficiency of the generalized spectrum method, theoretically and numerically. This technique appears to be a computationally attractive tool for resolving the spectral pollution. We resolved this spectral pollution by treating the analytical question: to find the bounded operators T and S representing the spectrum proprieties of an unbounded operator A in the theory of generalized spectrum.

References

[1] M. Levitin, E. Shargorodsky, "Spectral pollution and second-order relative spectra for self-adjoint operators", IMA J. Numer. Anal., 24:3 (2004), 393-416.

[2] E. B. Davies, "Spectral enclosures and complex resonances for general self-adjoint operators", LMSJ. Comput. Math., 1 (1998), 42-74.

[3] E.B. Davies, M. Plum, Spectral Pollution, 2002, arXiv: math/0302145v1.

[4] S. Bogli, "Convergence of sequences of linear operators and their spectra", Integral Equations Operator Theory, 88 (2017), 559-599.

[5] J. Hinchcliffe, M. Strauss, S.E. Zhukovskiy, "Spectral enclosure and superconvergence for eigenvalues in gaps", Integral Equations Operator Theory, 84:1 (2016), 1-32.

[6] H. Guebbai, "Generalized spectrum approximation and numerical computation of eigenvalues for Schrodinger's operators", Lobachevskii Journal of Mathematics, 34 (2013), 45-60.

[7] A.J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, California, 2005.

[8] I. Gohberg, I. S. Goldberg, M. A. Kaashoek, Classes of Linear Operators. V. I, Springer Basel AG, 1990.

[9] M. Ahues, A. Largillier, B. V. Limaye, Spectral Computations for Bounded Operators, Chapman and Hall/CRC, New York, 2001.

[10] M.T. Nair, "On strongly stable approximation", J. Austral. Math. Soc, 52:2 (1992), 251-260.

[11] A. Khellaf, H. Guebbai, S. Lemita, M. Z. Aissaoui, "Eigenvalues computation by the generalized spectrum method of Schrodinger's operator", Computational and Applied Mathematics, 37:5 (2018), 5965-5980.

[12] G.F. Roach, Green's Functions, Cambridge University Press, New York, 1982.

[13] A. Aslanyan, E.B. Davies, "Spectral instability for some Schrodinger operators", E. Numer. Math., 85:4 (2000), 525-552.

[14] H. Guebbai, A. Largillier, "Spectra and pseudospectra of convection-diffusion operator", Lobachevskii Journal of Mathematics, 33:3 (2012), 274-283.

Список литературы

[1] M. Levitin, E. Shargorodsky, "Spectral pollution and second-order relative spectra for self-adjoint operators", IMA J. Numer. Anal., 24:3 (2004), 393-416.

[2] E.B. Davies, "Spectral enclosures and complex resonances for general self-adjoint operators", LMSJ. Comput. Math., 1 (1998), 42-74.

[3] E.B. Davies, M. Plum, Spectral Pollution, 2002, arXiv: math/0302145v1.

[4] S. Bogli, "Convergence of sequences of linear operators and their spectra", Integral Equations Operator Theory, 88 (2017), 559-599.

[5] J. Hinchcliffe, M. Strauss, S.E. Zhukovskiy, "Spectral enclosure and superconvergence for eigenvalues in gaps", Integral Equations Operator Theory, 84:1 (2016), 1-32.

[6] H. Guebbai, "Generalized spectrum approximation and numerical computation of eigenvalues for Schrodinger's operators", Lobachevskii Journal of Mathematics, 34 (2013), 45-60.

[7] A.J. Laub, Matrix Analysis for Scientists and Engineers, SIAM, California, 2005.

[8] I. Gohberg, I. S. Goldberg, M. A. Kaashoek, Classes of Linear Operators. V. I, Springer Basel AG, 1990.

[9] M. Ahues, A. Largillier, B. V. Limaye, Spectral Computations for Bounded Operators, Chapman and Hall/CRC, New York., 2001.

[10] M.T. Nair, "On strongly stable approximation", J. Austral. Math. Soc, 52:2 (1992), 251-260.

[11] A. Khellaf, H. Guebbai, S. Lemita, M. Z. Aissaoui, "Eigenvalues computation by the generalized spectrum method of Schrodinger's operator", Computational and Applied Mathematics, 37:5 (2018), 5965-5980.

[12] G.F. Roach, Green's Functions, Cambridge University Press, New York., 1982.

[13] A. Aslanyan, E.B. Davies, "Spectral instability for some Schrodinger operators", E. Numer. Math., 85:4 (2000), 525-552.

[14] H. Guebbai, A. Largillier, "Spectra and pseudospectra of convection-diffusion operator", Lobachevskii Journal of Mathematics, 33:3 (2012), 274-283.

Information about the authors

Ammar Khellaf, Post-Graduate Student, Mathematics Department. University 8 Mai 1945, Guelma, Algeria. E-mail: amarlasix@gmail.com; khellaf.ammar@univ-guelma.dz ORCID: https://orcid.org/0000-0001-5282-7593

Sarra Benarab, Post-Graduate Student, Functional Analysis Department. Derzhavin Tambov State University, Tambov, the Russian Federation. E-mail: benarab.sarraa@gmail.com ORCID: https://orcid.org/0000-0002-8849-8848

Hamza Guebbai, Associate Professor of the Mathematics Department. University 8 Mai 1945, Guelma, Algeria. E-mail: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz ORCID: https://orcid.org/0000-0001-8119-2881

Информация об авторах

Хеллаф Аммар, аспирант, кафедра математики. Университет 8 Мая 1945, г. Гель-ма, Алжир. E-mail: amarlasix@gmail.com; khellaf.ammar@univ-guelma.dz ORCID: https://orcid.org/0000-0001-5282-7593

Бенараб Сарра, аспирант, кафедра функционального анализа. Тамбовский государственный университет им. Г.Р. Державина, г. Тамбов, Российская Федерация. E-mail: benarab.sarra@gmail.com ORCID: https://orcid.org/0000-0002-8849-8848

Геббай Хамза, доцент кафедры математики. Университет 8 Мая 1945, г. Гель-ма, Алжир. E-mail: guebaihamza@yahoo.fr; guebbai.hamza@univ-guelma.dz ORCID: https://orcid.org/0000-0001-8119-2881

Wassim Merchela, Post-Graduate Student, Functional Analysis Department. Derzhavin Tambov State University, Tambov, the Russian Federation. E-mail: merchela.wassim@gmail.com ORCID: https://orcid.org/0000-0002-3702-0932

There is no conflict of interests.

Corresponding author:

Wassim Merchela

E-mail: merchela.wassim@gmail.com

Received 15 February 2019

Reviewed 08 April 2019

Accepted for press 20 May 2019

Мерчела Вассим, аспирант, кафедра функционального анализа. Тамбовский государственный университет им. Г.Р. Державина, г. Тамбов, Российская Федерация. E-mail: merchela.wassim@gmail.com ORCID: https://orcid.org/0000-0002-3702-0932

Конфликт интересов отсутствует.

Для контактов:

Мерчела Вассим

E-mail: merchela.wassim@gmail.com

Поступила в редакцию 15.02.2019 г. Поступила после рецензирования 08.04.2019 г. Принята к публикации 20.05.2019 г.