On spectra of a certain class of quadratic operator pencils with one-dimensional linear part
We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils a...
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| Дата: | 2007 |
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Institute of Mathematics, NAS of Ukraine
2007
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860509412974657536 |
|---|---|
| author | Pivovarchik, V. N. Пивоварчик, В. Н. |
| author_facet | Pivovarchik, V. N. Пивоварчик, В. Н. |
| author_sort | Pivovarchik, V. N. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2020-03-18T19:51:39Z |
| description | We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found. |
| first_indexed | 2026-03-24T02:40:42Z |
| format | Article |
| fulltext |
UDC 517.5 + 517.43
V. N. Pivovarchik (South-Ukr. Ped. Univ., Odessa)
ON SPECTRA OF A CERTAIN CLASS
OF QUADRATIC OPERATOR PENCILS
WITH ONE-DIMENSIONAL LINEAR PART∗
PRO SPEKTRY PEVNOHO KLASU
KVADRATYÇNYX OPERATORNYX V’QZOK
Z ODNOVYMIRNOG LINIJNOG ÇASTYNOG
We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a
pencil linear with respect to the spectral parameter describes the viscous friction in problems of small vibrations
of strings and beams. Patterns in location of eigenvalues of such pencils are established. If the viscous friction
(damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules
in the location of the purely imaginary eigenvalues are found.
Rozhlqnuto pevnyj klas kvadratyçnyx operatornyx v’qzok,wo vynykagt\ u bahat\ox zadaçax fizyky.
Linijna za spektral\nym parametrom çastyna v’qzky opysu[ v’qzke tertq v zadaçax pro mali koly-
vannq strun ta sterΩniv. Vstanovleno zakonomirnosti v roztaßuvanni vlasnyx znaçen\ takyx v’qzok.
Qkwo v’qzke tertq zoseredΩene v odnij toçci, to operator u linijnij za parametrom çastyni v’qzky [
odnovymirnym. Dlq c\oho vypadku znajdeno porqdok roztaßuvannq suto uqvnyx vlasnyx znaçen\.
1. Introduction. Pioneering results on direct and inverse problems of small transver-
sal vibrations of an inhomogeneous string with pointwise damping were obtained by
M. G. Krein and A. A. Nudelman [1, 2]. In these papers conditions were obtained nec-
essary and sufficient for a sequence of complex numbers to be the spectrum of a string
whose density belongs to the class of so-called S-strings. It should be mentioned that in
implicit form the necessary and sufficient conditions for a certain subclass were obtained
in [3]. Later vibrating systems with point-wise damping were considered in many pub-
lications [4 – 12]. One of the general approaches to abstract versions of such problems
is to use the theory of entire functions see [3, 13]. The spectra of strings are considered
there as the sets of zeros of function of Hermite – Biehler class (see [13, p. 307] for the
definition) or generalized Hermite – Biehler class. For compressed beam vibrations (see
[12]) one needs to use so-called shifted Hermite – Biehler functions (see [14]). Another
approach is to use the theory of quadratic operator pencils. Here an important step was
done in the famous paper by M. G. Krein and H. Langer [15]. This approach was used in
[4, 5, 8, 16] and in many other papers. In present paper some abstract results on quadratic
operator pencils are obtained and applied to boundary problems which have eigenvalues
in both upper and lower half-planes. In Section 2 we describe general results on loca-
tion of spectra of quadratic operator pencils of the form L(λ) = λ2M − iλK − A with
M ≥ 0, K ≥ 0, A = A∗ ≥ βI, −∞ < β < 0. In Section 3 we consider the case of
one-dimensional operator K. In Section 4 the results of Sections 2 and 3 are applied to
spectral problems which occur in physics.
2. Abstract results. Let us denote by B(H) the set of bounded operators acting on a
separable Hilbert space H. We deal here with the following quadratic operator pencil
L(λ) = λ2M − iλK −A,
where M ∈ B(H), K ∈ B(H) and A is a closed operator on H with the domain D(A)
dense in H. The domain of the pencil is chosen as usually: D(L(λ)) = D(M)∩D(K)∩
∩D(A) = D(A). Thus, it is independent of λ.
∗ This work is supported by grant UK2-2811-OD-06 of Ukrainian Ministry of Education and Science and
US Civil Research and Development Foundation (CRDF).
c© V. N. PIVOVARCHIK, 2007
702 ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 703
Definition 2.1. The pencil L(λ) is said to be monic if M = I, where I is the identity
operator.
In what follows we suppose the following conditions to be satisfied:
Conditions I. M ≥ 0 and K ≥ 0 are bounded operators, A = A∗ ≥ −βI (β is
a positive number); for some β1 > β there exists (A + β1I)−1 ∈ S∞, here by S∞ we
denote the set of compact operators on H; kerA ∩ kerK ∩ kerM = {0}.
Definition 2.2. The set of values λ ∈ C such that L−1(λ) exists in B(H) is said
to be the resolvent set ρ(L(λ)) of L(λ). The spectrum of the pencil L(λ) is denoted by
σ(L(λ)), i.e., σ(L(λ)) = C\ρ(L(λ)).
Definition 2.3. A number λ0 ∈ C is said to be an eigenvalue of the pencil L(λ) if
there exists a vector y0 ∈ D(A) (called an eigenvector of L(λ)) such that y0 = 0 and
L(λ0)y0 = 0. Vectors y1, . . . , ym−1 are called associated to y0 if
k∑
s=0
1
s!
dsL(λ)
dλs
∣∣∣∣
λ=λ0
yk−s = 0, k = 1,m− 1.
The number m is said to be the length of the chain composed of the eigen- and associated
vectors. The geometric multiplicity of an eigenvalue is defined to be the number of the
corresponding linearly independent eigenvectors. The algebraic multiplicity of an eigen-
value is defined to be the greatest value of the sum of the lengths of chains corresponding
to linearly independent eigenvectors. An eigenvalue is said to be isolated if it has some
deleted neighborhood contained in the resolvent set. An isolated eigenvalue λ0 of finite
algebraic multiplicity is said to be normal if the image Im L(λ0) is closed.
In the case of linear monic operator pencils (λI−A) with bounded operator A this def-
inition of a normal eigenvalue coincides with the corresponding definition in [17] (Chap-
ter I, Paragraph 2) for operators. Under Conditions I the theorem about analytic Fredholm
operator valued functions which can be found in [18] (Chapter XI, Corollary 8.4) implies
that the spectrum of L(λ) consists of normal eigenvalues only.
We denote by C+ (C−) the open upper (lower) half-plane.
The following lemma is a generalization of statement 2.40 in [15].
Lemma 2.1. 1. If A ≥ 0, then the spectrum of L(λ) (if not empty) is located in the
closed upper half-plane.
2. If A >> 0, i.e., A ≥ εI, ε > 0 and K > 0, then the spectrum of L(λ) (if not
empty) is located in the open upper half-plan.
3. If A >> 0 and λ2y − Ay = 0 for all real λ and all nonzero y ∈ kerK, then the
spectrum of L(λ) (if not empty) is located in the open upper half-plane.
Proof. Let y0 be an eigenvector corresponding to the eigenvalue λ0. Then
(L(λ0)y0, y0) = 0, y0 = 0,
and consequently,
((Reλ0)2 − (Imλ0)2)(My0, y0) + Imλ0(Ky0, y0) − (Ay,y0) = 0 (2.1)
and
Reλ0(2 Imλ0(My0, y0) − (Ky0, y0)) = 0. (2.2)
If Reλ0 = 0, then the inequality Imλ0 ≥ 0 follows from (2.2). If Reλ0 = 0, then (2.1)
implies Imλ0 ≥ 0. The first assertion is proved.
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704 V. N. PIVOVARCHIK
Now let A >> 0 and K > 0. Let us set Imλ0 = 0 into (2.1) and (2.2). Then
for Reλ0 = 0 (2.1) implies (Ay0, y0) = 0 what contradicts the inequality A >> 0. If
Reλ0 = 0, Imλ0 = 0, then (2.2) implies (Ky0, y0) = 0, a contradiction. The second
assertion is proved.
Let y0 ∈ kerK be an eigenvector corresponding to a real eigenvalue λ0. Then
L(λ0)y0 = λ2
0y0 −Ay0 = 0, a contradiction.
Now let A be not positive but still bounded below. Then the pencil L(λ) has eigen-
values in the open lower half-plane (see below).
Lemma 2.2. 1. The part of the spectrum of L(λ) located in the open lower half-
plane lies on the imaginary axis.
2. If K > 0, then the part of the spectrum of L(λ) located in the closed lower half-
plane lies on the imaginary axis.
Proof. Let y0 be an eigenvector corresponding to an eigenvalue λ0 with Imλ0 < 0.
Then for Reλ0 = 0 equation (2.2) implies
(My0, y0) = (Ky0, y0) = 0
and consequently, My0 = Ky0 = 0. Then Conditions I imply Ay0 = 0 and L(λ0)y0 =
= Ay0 = 0, a contradiction. The first assertion is proved.
If K > 0, then for Imλ0 ≤ 0 the equality Reλ0 = 0 follows from (2.2).
Lemma 2.2 and the following lemma were proved in [19] for the case of monic oper-
ator pencils, i.e., for M = I.
Lemma 2.3. 1. All the eigenvalues of L(λ) located in C−\{0} are semisimple, i.e.,
they do not possess associated vectors.
2. If K > 0, then all the eigenvalues of L(λ) located in the closed lower half-plane
are semisimple.
Proof. Let λ0 be an eigenvalue of L(λ) located in the open lower half-plane (on the
imaginary axis according to Lemma 2.2). Let us denote by y0 (one of) the corresponding
eigen- and by y1 the associated vector. By Definition 2.3
λ2
0My1 − iλ0Ky1 −Ay1 + 2λ0My0 − iKy0 = 0. (2.3)
Multiplying (2.3) by y0 we obtain
((λ2
0 − iλ0K −A)y1, y0) +
(
(2λ0M − iK)y0, y0
)
= 0. (2.4)
Taking into account that λ0 is pure imaginary we obtain from (2.4):(
y1, (λ2
0M − iλ0K −A)y0
)
+
(
(2λ0M − iK)y0, y0
)
= 0,
what means
i
(
(2 Imλ0M −K
)
y0, y0) = 0. (2.5)
Equality (2.5) is possible for Imλ0 < 0 only if (My0, y0) = (Ky0, y0) = 0, i.e., if
My0 = Ky0 = 0. In this case L(λ0)y0 = −Ay0 = 0 and, consequently, y0 ∈ kerM ∩
∩ kerK ∩ kerA. Then due to Conditions I we have y0 = 0, a contradiction.
Let now an eigenvalue λ0 ∈ R\{0}. Then (2.2) implies (Ky0, y0) = 0, and, conse-
quently, Ky0 = 0 and (λ2
0M −A)y0 = 0. Then (2.4) is equivalent to(
y1, (λ2
0M + iλ0K −A)y0
)
+ 2λ0(My0, y0) = 0,
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ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 705
or
(My0, y0) = 0.
That means My0 = 0. Hence, taking into account Ky0 = 0 we obtain Ay0 = 0, what
contradicts Conditions I. The first statement is proved.
Let K > 0 and let λ0 be a real eigenvalue of L(λ). Let y0 and y1 be the corresponding
eigen- and associated vectors. Then according to Lemma 2.2 λ0 = 0 or, what is the same,
y0 ∈ kerA and (2.3) can be written as follows
Ay1 + iKy0 = 0. (2.6)
Multiplying (2.6) by y0 we obtain
(Ay1, y0) + i(Ky0, y0) = (y1, Ay0) + i(Ky0, y0) = i(Ky0, y0) = 0, (2.7)
what contradicts K > 0.
Lemma 2.4. If M + K ≥ εI (ε > 0), dim kerA > 0 and dim(kerA ∩ kerK) =
= p ≥ 0, then the algebraic multiplicity of λ = 0 as an eigenvalue of L(λ) is equal to
p + dim kerA.
Proof. Let 0 = y0 ∈ kerA and let y1 be an associated to y0 vector. Then
dL(λ)
dλ
∣∣∣∣
λ=0
y0 + L(0)y1 = −iKy0 −Ay1 = 0. (2.8)
If y0 ∈ kerK then y1 can be chosen equal to 0. If y0 ∈ kerK, then (2.8) implies
−i(Ky0, y0) − (Ay1, y0) = −i(Ky0, y0) − (y1, Ay0) = −i(Ky0, y0) = 0.
Combining the last equality with the condition K ≥ 0 we obtain Ky0 = 0, a contradic-
tion. It remains to prove that the third vector of the chain does not exist. Suppose it does
exist and denote it by y2. Then
−Ay2 − iKy1 + My0 = 0.
Consequently,
0 = −(Ay2, y0) − i(Ky1, y0) + (My0, y0) = (My0, y0)
and (Ky0, y0) = (My0, y0) = 0 contradicts the conditions of Lemma 2.4.
Usually it is more convenient to deal with bounded operator pencils. Let us introduce
the following auxiliary pencil:
L̃(λ) = L(λ)(β1I + A)−1.
Since A ≥ −βI > −β1I, the pencil L1(λ) is bounded and the following lemma follows
from Lemma 20.1 in [20].
Lemma 2.5.
σ(L̃(λ)) = σ(L(λ)).
Let us introduce the following parameter-dependent operator pencil:
L(λ, η) = λ2M − iληK −A. (2.9)
It is clear that L(λ, 1) = L(λ). Lemma 2.5 enables us to use the results of [21] (see also
[22, 23]) established for bounded operator pencils. Adapted for our aims these results can
be given in the following form.
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706 V. N. PIVOVARCHIK
Theorem 2.1. Let the domain Ω contain the only eigenvalue λ0 of the pencil L(λ,
η0). Denote by m the algebraic multiplicity of λ0. Then there exist numbers ε > 0 and
m1 ∈ N, m1 ≤ m, such that the following assertions are true in the neighborhood
|η − η0| < ε :
1. L(λ, η) possesses exactly m1 different eigenvalues inside the domain Ω. Those
eigenvalues can be arranged in groups λij(η)
(
i = 1, l; j = 1, pi;
∑l
i=1
pi = m1
)
, in
such a way that the functions of the group, i.e., λi1, λi2, . . . , λipi
compose a complete set
of a pi-valued function. In this case those eigenvalues can be presented in the form of the
following series:
λij(η) = λ0 +
∞∑
k=1
aik
((
(η − η0)
1
pi
)
j
)k
, j = 1, 2, . . . , pi, (2.10)
where
(
(η−η0)
1
pi
)
j, j = 1, pi, is the complete set of branches of the function (η−η0)
1
pi ,
pi is the chain length of the eigenvector and associated vectors corresponding to the
eigenvalue λ0 and to the eigenvector yi0 of the pencil L(λ, η0).
2. The basis of the eigen-space corresponding to λij(η) can be presented in the fol-
lowing form:
y
(q)
ij (η) = y
(q)
i0 +
∞∑
k=1
y
(q)
ik (((η − η0)
1
pi )j)k, j = 1, 2, . . . , pi, q = 1, 2, . . . , αi,
(2.11)
where αi is the geometric multiplicity of λij(η), y
(q)
i0 belong to the eigen-subspace of
L(λ, η0) corresponding to the eigenvalue λ0.
It should be mentioned that this theorem is a generalization of the Weierstrass theorem
on function analytic in two variables [24, p. 476].
Definition 2.4. The total algebraic multiplicity of the spectrum of L(λ) lying in a
domain Ω is defined to be the number
∑n
i=1
mi, where mi, i = 1, p, are the algebraic
multiplicities of all the eigenvalues lying in Ω.
Theorem 2.2. 1. Let, in addition to Condition I, be M = I. Then the total alge-
braic multiplicity of the spectrum of L(λ) located in the open lower half-plane coincides
with the total algebraic multiplicity (here it is the same as total geometric multiplicity) of
the negative spectrum of the operator A.
2. If, in addition K > 0, then the total algebraic multiplicity of the spectrum of L(λ)
located in the closed lower half-plane coincides with the total algebraic multiplicity of
the nonpositive spectrum of A.
Proof. 1. We are going to prove that the total algebraic multiplicity of the spectrum
of the pencil L(λ) located in the open lower half-plane coincides with the total algebraic
multiplicity of the spectrum of the pencil λ2I − A located in the open lower half-plane,
or what is the same with the total algebraic multiplicity of the negative spectrum of the
operator A. We consider the pencil L(λ, η) as a perturbation of the pencil λ2I −A.
Let η0 ∈ [0, 1] and let λ0 (Reλ0 = 0, Imλ0 < 0) be an eigenvalue of L(λ, η0). Then
due to Lemma 2.3 this eigenvalue is semisimple. Then formulae (2.10), (2.11) for the
eigenvalues of L(λ, η) in the lower half-plane can be simplified:
λi(η) = λ0 +
∞∑
k=1
aik(η − η0)k, (2.12)
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ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 707
y
(q)
i (η) = y
(q)
i0 +
∞∑
k=1
y
(q)
ik (η − η0)k, q = 1, αi. (2.13)
Differentiating the equality
L(λi(η), η)y
(q)
i (η) = 0
with respect to η and multiplying the resulting equation by y
(q)
i we obtain for η = η0:
ai1 =
iλ0
(
Ky
(q)
i0 , y
(q)
i0
)
2λ0
∥∥y(q)
i0
∥∥2 − iη0
(
Ky
(q)
i0 , y
(q)
i0
) =
i Imλ0
(
Ky
(q)
i0 , y
(q)
i0
)
2 Imλ0
∥∥y(q)
i0
∥∥2 − η0
(
Ky
(q)
i0 , y
(q)
i0
) . (2.14)
It is clear that Re ai1 = 0 and Im ai1 > 0. It means that the eigenvalues of L(λ, η) located
in the open lower half-plane (on the imaginary axis) move upwards along the imaginary
axis when η grows from 0 to 1. To show that eigenvalues do not came from −i∞ let
us find γ > 0 such that (−i∞,−iγ) ∈ ρ(L(λ, η)) for η ∈ [0, 1]. Let −iτ, τ > 0,
be an eigenvalue of L(λ, η0), where η0 ∈ [0, 1] and let y be one of the corresponding
eigenvectors. Then
τ2‖y‖2 + η0τ(Ky, y) + (Ay, y) = 0 (2.15)
and consequently τ ≤
√
|β|. Consequently, the eigenvalues does not come from −i∞. It
means that
N(0) ≥ N(1), (2.16)
where by N(η) we denote the total algebraic multiplicity of the spectrum of L(λ, η)
located in the open lower half-plane. Lemma 2.4 shows that the algebraic multiplicity of
λ = 0 as of an eigenvalue of L(λ, η) does not depend on η for η ∈ (0, 1]. That means
that moving upwards as η changes from 0 to 1 the eigenvalues of L(λ, η) on the negative
imaginary half-axis do not cross the origin.
2. If K > 0 then according to Lemma 2.2 the only real eigenvalue can be at λ = 0.
According to Lemma 2.3 this eigenvalue is semisimple. Therefore, the multiplicity of
λ = 0 as an eigenvalue of L(λ) is the same as its multiplicity as an eigenvalue of A.
Combining this result with statement 1 of Theorem 2.2 we obtain statement 2.
Statement 1 of this theorem remains true for operators A admitting essential spectrum
on [0,∞) under more restrictive conditions as it was shown in [25, 19]. There exist
many related theorems [26 – 29]. This theorem remains true in the case of not symmetric
operator K but such that Re K >> 0 (under some additional restrictions) see [30, 31].
Theorem 2.3. Let us assume that, in addition to Condition I, M + K ≥ εI, ε > 0.
Then Statement 1 of Theorem 2.2 is true.
Proof. Due to Statement 1 of Theorem 2.2 the assertion of Theorem 2.3 is true for
the operator pencil
L̃(λ, 1) = λ2I − iλK −A.
Let us prove that it is true for each η ∈ [0, 1] for the pencil
L̃(λ, η) = λ2((1 − η)I + ηM) − iλK −A.
If iτ, τ < 0, is an eigenvalue of L̃(λ, η) and y is the corresponding eigenvector, then
ISSN 1027-3190. Ukr. mat. Ωurn., 2007, t. 59, # 5
708 V. N. PIVOVARCHIK
τ =
(Ky, y) −
√
(Ky, y)2 − 4((1 − η)I + ηM)y, y)(Ay, y)
2((1 − η)I + ηM)y, y)
=
=
−2(Ay, y)
(Ky, y) +
√
(Ky, y)2 − 4(((1 − η)I + ηM)y, y)(Ay, y)
.
Due to inequalities K ≥ 0, M ≥ 0, M + K ≥ ε > 0 and A ≥ βI, β > −∞, we obtain
that τ(η) ≥ −∞ for each η ∈ [0, 1]. It means that eigenvalues do not come from and do
not leave at −i∞. On the other, hand they do not cross the origin what can be proved in
the same way as in proof of Theorem 2.2.
In what follows we consider the spectrum of L(λ) in the upper half-plane.
Lemma 2.6. 1. Let M ≥ εI, ε > 0. Then if the eigenvalue λk is not pure imaginary
or if it is not semisimple then Imλk ∈ [0,m1], where
m1 =
1
2
sup
0 �=y∈D(A)
(Ky, y)
(My, y)
.
2. Let M >> 0 and K >> 0 on D(A). Then if the eigenvalue λk is not pure
imaginary or if it is not semisimple then Imλk ∈ [m2,m1], where
m2 =
1
2
inf
0 �=y∈D(A)
(Ky, y)
(My, y)
.
Proof. Let λk be a not pure imaginary eigenvalue of L(λ). Then due to Lemma 2.2
it lies in the closed upper half-plane. Assertion 1 of Lemma 2.6 follows from (2.2). If λk
is a pure imaginary eigenvalue in the closed upper half-plane having a chain of length 2
then Assertion 1 of Lemma 2.6 follows from (2.5). The proof of Assertion 2 is analogous.
We can consider θ = η−1 as the spectral parameter instead of λ when it is convenient.
Lemma 2.7. Let all eigenvalues of L(λ, θ−1) be of geometric multiplicity 1 for all
θ and let K ≥ νI, ν > 0. Let θ0 ∈ R\{0} be an eigenvalue of the operator-function
Q(λ, θ) = I − θ
(
iλ−1K− 1
2AK− 1
2 − iλK− 1
2MK− 1
2
)
for λ0 ∈ R\{0}. Then this eigenvalue is holomorphic as a function of λ in some real
neighborhood λ ∈ (λ0 − ε, λ0 + ε), ε > 0:
θ(λ) = θ0 +
∞∑
k=p
bk(λ− λ0)k, (2.17)
where p ∈ N, bp ∈ R\{0}.
Proof. The spectrum of Q(λ0, θ) in the domain C\{0} consists of normal eigen-
values only what follows from the above mentioned theorem about analytic operator-
function [18] (Chapter XI, Corollary 8.4). The geometric multiplicity of each eigenvalue
of Q(λ0, θ) is equal to 1 because it coincides with that of the corresponding eigenvalue of
L(λ0, θ
−1) = −iθ−1λ0K
1
2Q(λ0, θ)K
1
2 . Now it is possible to apply the Rellich – Nagy
theorem [32] to finish the proof.
Lemma 2.8. Let the conditions of Lemma 2.7 be satisfied. Let λ0 = iτ0, τ0 > 0,
and η0 > 0, then in some neighborhood of (λ0, η0), i.e., in
{
(λ, η) : |λ − λ0| < ε,
|η − η0| < δ, ε > 0, δ > 0
}
all the eigenvalues are given by the following formula:
λj(η) = λ0 +
∞∑
k=1
βk
(
(η − η0)
1
r
j
)k
, j = 1, 2, . . . , r, (2.18)
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ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 709
where β1 = 0 is a real or pure imaginary, (η−η0)
1
r
j , j = 1, 2, . . . , r, means the complete
set of branches of the root.
Proof. We obtain this result immediately after inverting (2.17).
Theorem 2.4. Let, in addition to Condition I, be M = I, then for every (pure
imaginary) eigenvalue λ−k of L(λ) from the closed lower half-plane there exists a pure
imaginary eigenvalue (denote it λk) such that
Im(λk + λ−k) ≥ 0. (2.19)
Proof. The eigenvalues of L(λ, η) are piecewise analytic functions of η. They may
loose analyticity only when they coincide. This follows from the results above. The
eigenvalues located on (−i∞, 0) are analytic functions of η > 0 (see (2.12)) and move
upwards along the imaginary axis when η increases. We identify λj(η) as the eigenvalue
satisfying the conditions λ−j (0) = −λj(0), where Im λ−j(0) < 0 and Re λ−j(0) = 0.
For sufficiently small η > 0 we have Im λj(η) > 0, Re λj(η) = 0 due to the symmetry
of the problem, Lemma 2.3 and the fact that all the normal eigenvalues of the pencil are
semisimple when η = 0. It is easy to derive (see (2.14)) the following formula for the
derivative:
λ′
j(η) =
iλj(η) (Kyj(η), yj(η))
2λj (η) ‖yj (η) ‖2 − iη (Kyj (η) , yj (η))
. (2.20)
This formula implies Im λ′
j (η) ≥ 0 and Re λ′
j(η) = 0 for η ≥ 0 small enough. Hence,
our theorem is true for η ≥ 0 small enough. While η > 0 increases, λ′
j (η) can change
its sign only when the denominator in the right-hand side of (2.20) vanishes, i.e., when
eigenvalues coalesce. If such a coalescence takes place on the interval (0, i∞) , then
the eigenvalues involved behave according to formula (2.18). Such a coalescence on the
interval (0, i∞) is of one of the following three types. The first one has r odd in (2.18).
In this case we identify the eigenvalue moving upwards along the imaginary axis after
the coalescence as the one which moved upwards along the imaginary axis before the
coalescence. By a coalescence of the second type we mean one which has r even and β1
purely imaginary (β1 = 0) in (2.18). After such a coalescence two new purely imaginary
eigenvalues appear which are moving in opposite directions along the imaginary axis,
and such a coalescence cannot violate Theorem 2.2. The third type of coalescence has
even r and real β1 = 0. Let λj(η) take part in such a coalescence at η = η0 ∈ (0, 1].
Then a coalescence of the second type indeed occurred at some η ∈ (0, η0) in some point
λ× ∈ (0, λj(η0)) on the imaginary axis. In this case the eigenvalue that has arisen after
this coalescence and is moving upwards is identified as λj(η).
To finish the proof we will show that for all η ∈ [0, 1] the pure imaginary eigenvalues
lie on some interval [−iγ, iγ] (γ ∈ (0,∞)). Let iτ (τ ∈ R) be an eigenvalue of L(λ, η)
and y be the corresponding eigenvector. Then
τ2‖y‖2 − τη(Ky, y) + (Ay, y) = 0
and, consequently,
|τ | ≤ 1
2
(
η‖K‖ +
√
η2‖K‖2 + 4β
)
. (2.21)
Remark 2.1. In [33] it was proved that under more restrictive condition K >> 0
the inequality Im(λk + λ−k) > 0 is valid.
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710 V. N. PIVOVARCHIK
3. Pencils with one-dimensional linear part. Let us consider the quadratic operator
pencil L(λ, η) with operators M, K, A acting in the Hilbert space H ⊕ C and satisfying
Condition I, moreover, let the following condition be valid:
Condition II.
K =
(
0 0
0 1
)
.
Problems related to such pencils occur in the theory of vibrations of strings and elastic
beams with pointwise damping [1 – 6, 11, 12, 26, 34 – 36]. The so-called Regge problem
(see [36 – 41] and others) can be also reduced to the spectral problems for such pencils.
We consider the order in location of pure imaginary eigenvalues of L(λ). The order
was found for different cases in [12, 35, 42 – 45]. In [14] it was shown that the spectra
of the problems in all these papers are the sets of zeros of the so-called shifted Hermite –
Biehler functions or, as in case of [12], of shifted generalized Hermite – Biehler functions.
These functions possess the mentioned order in location of pure imaginary zeros. But in
some cases it is not so easy to prove that the spectrum of the problem coincides with the
set of zeros of a shifted Hermite – Biehler function. Another approach to describe the
spectra of such problems is to use the methods of operator theory.
Lemma 3.1. 1. Let Conditions I and II be valid. Let λ = −iτ and λ = iτ with
τ ∈ R\{0} be eigenvalues of the operator pencil L(λ, η0), where η0 ∈ (0, 1]. Then
λ = −iτ and λ = iτ are eigenvalues of L(λ, η) for each η ∈ [0, 1].
2. Let λ ∈ R\{0} be an eigenvalue of the operator pencil L(λ, η0), where η0 ∈
∈ (0, 1]. Then λ and −λ are eigenvalues of L(λ, η) for each η ∈ [0, 1].
Proof. 1. Suppose λ = −iτ and λ = iτ with τ ∈ R\{0} are eigenvalues of
the operator pencil L(λ, η0). Then denote by Y1 =
(
y11
y12
)
, the eigenvector of L(λ, η0)
corresponding to λ = iτ and by Y2 =
(
y21
y22
)
the eigenvector corresponding to λ = −iτ.
Then
(−τ2M + τη0K −A)Y1 = 0,
(−τ2M − τη0K −A)Y2 = 0
and consequently
−τ2(Y2,MY1) + τη0(Y2,KY1) − (Y2, AY1) = 0,
−τ2(MY2, Y1) − τη0(KY2, Y1) − (AY2, Y1) = 0.
Taking into account the symmetry of the operators we obtain by subtracting:
(KY2, Y1) = y12y22 = 0.
If y12 = 0, then Y1 is the eigenvector corresponding to the both eigenvalues iτ and −iτ.
These eigenvalues are independent of η ≥ 0 and they are located symmetrically with
respect to the real axis.
2. Let λ = 0 be a real eigenvalue and Y be the corresponding eigenvector, then
(see the proof of Assertion 1 of Lemma 2.3) KY = 0 and, consequently, −λ is also an
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ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 711
eigenvalue. If Y =
(
y1
y2
)
, then KY = 0 means y2 = 0 and
L(λ, η)Y = λ2MY − iληKY −AY = λ2MY −AY = 0.
This means λ and −λ are eigenvalues for all η > 0.
Definition 3.1. Let Conditions I and II be valid.
1. An eigenvalue λ = 0 of L(λ, η0), η0 ∈ (0, 1], is said to be of type I if λ2 is real
and −λ is also an eigenvalue of L(λ, η0).
2. Let λ = 0 be an eigenvalue of A of geometric multiplicity n ≥ 1, then either
dim kerA ∩ dim kerK = n or dim kerA ∩ dim kerK = n − 1. In the first case we say
that the pencil L(λ, η) has 2n eigenvalues of type I and no eigenvalues of type II at λ = 0
for each η ∈ (0, 1]. In the second case we say that that there are 2m − 2 eigenvalues of
type I and one eigenvalue of type II at λ = 0 for each η ∈ (0, 1].
3. All the other eigenvalues of L(λ, η0), η0 ∈ (0, 1], are said to be of type II (we
consider an eigenvalue of multiplicity p as p coinciding eigenvalues which can be of
different types).
Corollary 3.1. Let, in addition to Conditions I and II, be M +K ≥ εI, ε > 0, then
Lemma 3.1 shows that there can exist a sequence (finite or infinite) of eigenvalues of type I
and these eigenvalues are independent of η. This sequence consists of pure imaginary
eigenvalues (of finite number counting with multiplicities) and of real eigenvalues (of
finite with account of multiplicities or infinite number). This sequence is symmetric with
respect to the real and imaginary axes. Eigenvalues of type I possess no associated vectors
(except of possible eigenvalue at λ = 0).
Proof. Let ζ0 be an eigenvalue of the linear operator pencil ζM − A of multiplicity
s ≥ 1, then the basis of the corresponding subspace can be chosen in such a way that either
1) all s linearly independent eigenvectors are of the form Yj = (y1, 0)T , j = 1, . . . , s, or
2) s − 1 eigenvectors are of the form Yj = (y1, 0)T and one eigenvector is of the form
Ys = (0, y2)T with y2 = 0. Then it is clear, that ±
√
ζ0 are eigenvalues of type I of the
pencil L(λ, η) for each η ∈ [0, 1] and the corresponding eigenvectors are Yj (j = 1, . . . , s
in case 1) and j = 1, . . . , s−1 in case 2). The condition M+K ≥ εI guarantee finiteness
of the set of pure imaginary eigenvalues of type I.
Let us describe the location of the rest of eigenvalues which we relate to the subse-
quence of type II. It is clear that their geometric multiplicity is 1.
Theorem 3.1. Let, in addition to Conditions I and II, M + K ≥ εI, ε > 0, then
the eigenvalues of type II of the operator pencil L(λ, η), η ∈ (0, 1], possess the following
properties:
1. All but κ2 terms of the sequence lie in the open upper half-plane.
2. All terms in the closed lower half-plane are purely imaginary and occur only once.
If κ2 ≥ 1, we denote them as λ−j = −i|λ−j |, j = 1, . . . , κ2. We assume that |λ−j | <
< |λ−(j+1)|, j = 1, . . . , κ2 − 1.
3. If κ2 ≥ 1, the numbers i|λ−j |, j = 1, . . . , κ2 (with the exception of λ−1 if it
equals zero), are not terms of the sequence.
4. If κ2 ≥ 2, then the number of terms in the intervals (i|λ−j |, i|λ−(j+1)|), j =
= 1, . . . , κ2 − 1, is odd.
5. If |λ−1| > 0, then the interval (0, i|λ−1|) contains no terms at all or an even
number of terms.
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712 V. N. PIVOVARCHIK
Proof. To prove this theorem it is enough to use the arguments in the proof of Theo-
rem 2.3 keeping in mind that by Definition 3.1 and Lemma 3.1 if λ = −iτ, τ > 0, is an
eigenvalue of type II, then λ = iτ is not.
Theorem 3.2. Let, in addition to Conditions I and II, be M >> 0. Then:
1) if κ2 ≥ 1, then the interval (i|λ−κ2 |, i∞) contains an odd number of terms;
2) if κ2 = 0, then the sequence has an odd number of positive imaginary terms.
Proof. It remains to show that for all η ∈ [0, 1] the pure imaginary eigenvalues lie on
some interval [−iγ, iγ], γ < ∞. But it has been proved already (see (2.21)).
4. Examples. 1. Regge problem [36 – 41].
This problem occurs in the scattering theory when the potential is supposed to have
finite support
−y′′ + q(x)y = λ2y, (4.1)
y(0) = 0, (4.2)
y′(a) + iλy(a) = 0. (4.3)
Here λ is the spectral parameter and the potential q is real-valued and belongs to L2(0, a).
Let us introduce the operators A,K and M acting in the Hilbert space H = L2(0, a)∪
∪ C according to the formulae
A
(
v(x)
c
)
=
(−v′′ (x) + q (x) v (x)
v′(a)
)
, (4.4)
D (A) =
{(
v (x)
c
)
: v (x) ∈ W 2
2 (0, a) , v (0) = 0, c = v(a)
}
, (4.5)
M =
(
I 0
0 0
)
, K =
(
0 0
0 1
)
.
It is easy to show that A is selfadjoint and bounded below and that there exists −β1 <
< −β ≤ ‖y‖−2(Ay, y) such that (A + β1I)−1 is a compact operator.
Let us consider the nonmonic quadratic operator pencil of the form
L (λ) = λ2M − iλK −A.
We identify the spectrum of problem (4.1) – (4.3) with the spectrum of the pencil
L (λ) . It is clear that P ≥ 0, and K ≥ 0. The spectrum of the pencil consists of normal
eigenvalues (see Section 2).
Let us prove that all of these eigenvalues are of type II. Suppose a real λ = 0 is an
egenvalue (being real nonzero it must be of type I). Then −λ is also an eigenvalue and
according to the proof of Lemma 3.1 c = v(a) = 0 in (4.4). Therefore, the second
component of the equation L(λ0)Y = 0 gives v′(a) = 0 what contradicts v(a) = 0.
In the same way, one can prove that there are no symmetrically located pure imaginary
eigenvalues and that the possible eigenvalue at the origin is simple.
Thus, the conditions of Theorem 3.1 are satisfied and all the eigenvalues are of type II
and therefore statements 1 – 5 of Theorem 3.1 are valid.
2. The problem of small vibrations of a damped smooth inhomogeneous string in a
particular case of point mass at the right end can be reduced to the following problem:
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ON SPECTRA OF A CERTAIN CLASS OF QUADRATIC OPERATOR PENCILS . . . 713
y′′(λ, x) + (λ2 − i λp− q(x))y(λ, x) = 0, (4.6)
y(λ, 0) = 0, (4.7)
y′(λ, a) + (−mλ2 + i α λ + β) y(λ, a) = 0, (4.8)
where p > 0, m > 0, α > 0 are constants, q(x) ∈ L2(0, a) is a real-valued function. It
follows from the physical meaning of this problem that all the eigenvalues of this problem
lie in the open upper half-plane [35].
Transformation of the spectral parameter z = λ− i
p
2
leads to the following problem:
y′′(z, x) +
(
z2 +
p2
4
− q(x)
)
y(z, x) = 0, (4.9)
y(z, 0) = 0, (4.10)
y′(z, a) + (−mz2 + i(α−mp)z + β1)y(z, a) = 0, (4.11)
where β1 = β +
p2m
4
− αp
2
.
The spectrum of problem (4.9) – (4.11) coincides with the spectrum of the following
operator pencil:
L(z) = z2M1 − izK1 −A1 (4.12)
acting in L2(0, a) ⊕ C, where
D(L) = D(A1) =
{(
y(x)
y(a)
)
: y(x) ∈ W 2
2 (0, a), y(0) = 0
}
,
A1
(
y(x)
y(a)
)
=
−y′′ + q(x)y − p2
2
y
y′(a) + β1y(a)
(4.13)
and
M1 =
(
I 0
0 m2
)
, K1 =
(
0 0
0 (α− pm)I
)
. (4.14)
Let α > mp. As in previous problem we conclude that the pencil L possesses only
normal eigenvalues of type II. It allows to apply again Theorem 3.1 to the pencil L and
after inverse transformation λ = z + i
p
2
to obtain the result obtained in [14] by another
method:
1) all (if any) eigenvalues in the closed half-plane Imλ ≤ p
2
are pure imaginary and
simple
(
we denote them {λ−j}, j = 1, 2, . . . , κ, in the following order:
∣∣∣∣p2 − λ−j
∣∣∣∣ <
<
∣∣∣∣p2 − λ−j−1
∣∣∣∣
)
;
2) all the points i(p − |λ−j |), j = 1, 2, . . . , κ, do not belong to the spectrum of
(4.6) – (4.8) except of i(p− |λ−1|) if λ−1 = i
p
2
;
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714 V. N. PIVOVARCHIK
3) each interval
(
i(p − |λ−j |
)
, i
(
p − |λ−j−1|
)
, j = 1, 2, . . . , κ − 1, contains odd
number (with account of multiplicities) of the eigenvalues;
4) if λ−1 = i
p
2
, then the interval
(
i
p
2
, i(p− |λ−1|)
)
contains even number (with
account of multiplicities) of the eigenvalues.
The case of α < mp can be considered in the same way.
3. Forth order problem. In [12] the following spectral problem was considered:
y(4) − (g(x)y′)′ = λ2y, (4.15)
y(0) = y′′(0) = 0, (4.16)
y(a) = 0, (4.17)
y′′(a) + iαλy′(a) = 0 (4.18)
describing small transversal vibrations of an elastic beam. Here g(x) is a continuously
differentiable real function describing the distributed stretching or compressing force.
The left end of the beam is hinge connected and the right end is hinge connected with
damping.
We associate with problem (4.15) – (4.18) the following operator pencil:
L2(λ) = λ2M2 − iλK2 −A2,
where
D(L2) = D(A2) =
{(
y(x)
y′(a)
)
: y(x) ∈ W 4
2 (0, a), y(0) = y′′(0) = y(a) = 0
}
,
A2
(
y(x)
y′(a)
)
=
(
y(4) − (g(x)y′)′
y′′(a)
)
and
M2 =
(
I 0
0 0
)
, K2 =
(
0 0
0 α
)
.
It is clear that geometric multiplicity of an eigenvalue of this pencil can be 1 or 2 because
of conditions y(0) = y′′(0) = 0. Therefore, L2(λ) can have eigenvalues of the both
types I and II. Thus, we have deduced one of the results of [12] (see Theorem 3.1 there).
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716 V. N. PIVOVARCHIK
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Received 15.11.2006
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| institution | Ukrains’kyi Matematychnyi Zhurnal |
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| resource_txt_mv | umjimathkievua/74/25ef99f6d50569448597bc4645749a74.pdf |
| spelling | umjimathkievua-article-33392020-03-18T19:51:39Z On spectra of a certain class of quadratic operator pencils with one-dimensional linear part Про спектри певного класу квадратичних операторних в'язок з одновимірною лінійною частиною Pivovarchik, V. N. Пивоварчик, В. Н. We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found. Розглянуто певний клас квадратичних операторних в'язок, що виникають у багатьох задачах фізики. Лінійна за спектральним параметром частина в'язки описує в'язке тертя в задачах про малі коливання струн та стержнів. Встановлено закономірності в розташуванні власних значень таких в'язок. Якщо в'язке тертя зосереджене в одній точці, то оператор у лінійній за параметром частині в'язки є одновимірним. Для цього випадку знайдено порядок розташування суто уявних власних значень. Institute of Mathematics, NAS of Ukraine 2007-05-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3339 Ukrains’kyi Matematychnyi Zhurnal; Vol. 59 No. 5 (2007); 702–716 Український математичний журнал; Том 59 № 5 (2007); 702–716 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/3339/3422 https://umj.imath.kiev.ua/index.php/umj/article/view/3339/3423 Copyright (c) 2007 Pivovarchik V. N. |
| spellingShingle | Pivovarchik, V. N. Пивоварчик, В. Н. On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title | On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title_alt | Про спектри певного класу квадратичних операторних в'язок з одновимірною лінійною частиною |
| title_full | On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title_fullStr | On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title_full_unstemmed | On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title_short | On spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| title_sort | on spectra of a certain class of quadratic operator pencils with one-dimensional linear part |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/3339 |
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