The spectra of two-parameter quadratic operator pencils

Mathematical and Computer Modelling 54 (2011) 742–755

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Mathematical and Computer Modelling

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The spectra of two-parameter quadratic operator pencilsM. HasanovDogus University, Department of Mathematics, 34722, Kadikoy, Istanbul, Turkey

a r t i c l e i n f o

Article history:Received 6 April 2010Received in revised form 14 March 2011Accepted 15 March 2011

Keywords:EigenvaluesOperator pencilsVariational principlesWaveguidesCritical points

a b s t r a c t

This paper is devoted to a variational theory of the eigenvalue spectra of two-parameter,unbounded operator pencils, of the so-called waveguide type, which have importantphysical applications. It is known that the classical, nonlinear variational theory of thesespectra is only applicable to eigenvalues with a definite type (positive or negative),provided± type eigenvalues are separated. However, the classical theory has recently beenextended to mixed-type eigenvalues: first for linear matrix and operator pencils of theform L(λ) = λA − B, and later for self-adjoint operators in a Krein space (see Bindinget al. (2005) [17] and references therein). The real eigenvalues of the operator pencilsstudied in this paper can be split into five intervals according to type, the central rangecontainingmixed-type eigenvalues (see Fig. 3). Themain goal of this paper is to investigatethe variational principles for real eigenvalues in this zone. In addition, this paper studiesneutral (resonance) pairs and their existence criteria. We provide a complete analysisof the spectra of two-parameter, waveguide-type operator pencils. We use a variationalapproach for the spectrum of non-overdamped pencils, relying on Krein space techniquesand the Ljusternik-Schnirelman theory of critical points of nonlinear functionals. By usingthe Ljusternik-Schnirelman theory we investigate neutral eigenvalues in a mixed spectralzone.

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1. Introduction

In this paper we study the eigenvalue spectra of two-parameter operator pencils of the so-called waveguide type.Specifically, we investigate variational principles for their mixed-type, real eigenvalues. The practical interest in theseproblems arises from looking for special guided waves V of the form V (t, x) = uei(wt−kx), and for equations describingthe propagation of such waves in a cylindrical domain R × Ω .

For most waveguides encountered in practice, the dynamic model or evolution equation is a second-order differentialequation of the form (see [1–5])

RVtt − CVxx + iBVx + AV = 0, (1.1)

where V (x, t) : R × R → H and H is a Hilbert space. The operators R, A, B and C are symmetric, but not necessarilybounded; they might represent the material properties of the waveguide or directional derivatives perpendicular to thex-axis, for example (see Definition 2.1). Substituting V (t, x) = uei(wt−kx) into Eq. (1.1) results in the following two-parametereigenvalue problem:

L(k, w)u := (A + kB + k2C − w2R)u = 0. (1.2)

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M. Hasanov / Mathematical and Computer Modelling 54 (2011) 742–755 743

The higher-order linear waveguides (see Example 4.2) are governed by the dynamical model

Vtt + DVt +

n−s=1

(−1)sCs−1∂2sV∂x2s

+ in−1−s=0

(−1)sBs∂2s+1V∂x2s+1

+ AV = 0 (1.3)

with the associated two-parameter polynomial pencil

L(k, w) := A +

n−s=1

k2sCs−1 +

n−1−s=0

k2s+1Bs + iwD − w2I. (1.4)

The spectral structure and variational principles of the definite-type eigenvalues for polynomial waveguide pencils werestudied in our previous paper [6]. The most general two-parameter, analytic operator functions in the dissipative casehave been recently studied by Shkalikov in the paper [7], that also derives many interesting facts about dispersion curves(Rellich–Nagy type theorems) and perturbations to the discrete spectrum (see also [8]). However, the questions raised inthis paper are distinct from those studied in [6,7].

We are mainly interested in the quadratic case, but some facts can be easily extended to polynomial pencils. We willdiscuss an example of the higher-order waveguides at the end of this paper.

Now let us define the spectrum and some spectral sets for a general two-parameter operator pencil L(k, w). Let L :

C×C → L(H), whereC is the complex plane and L(H) is the space of all linear operators inH . The spectrum of L, denoted byσ(L), is defined as (k, w) ∈ σ(L) ⇔ 0 ∈ σ(L(k, w)). If the equation L(k, w)u = 0has anon-trivial solutionu ∈ H , then (k, w)is an eigenvalue corresponding to an eigenvector u. The set of all eigenvalues forms the point spectrum, denoted by σp(L).Analogously,we define the essential spectrumσess(L) by the relation (k, w) ∈ σess(L) ⇔ 0 ∈ σess(L(k, w)), and the resolventset ρ(L) by the relation (k, w) ∈ ρ(L) ⇔ 0 ∈ ρ(L(k, w)). Fixing the value of w, we define σw(L) = k|(k, w) ∈ σ(L);similarly, for a fixed kwe define σk(L) = w|(k, w) ∈ σ(L). Note that 0 ∈ σess(L(k, w)) means the operator L(k, w) is not aFredholm operator. Evidently, if L(λ) = A − λI then L is a linear pencil and σ(L) = σ(A).

The main subject of this paper is the following spectral sets: σwp (L) = k | (k, w) ∈ σp(L) and σR(L) = (k, w)|(k, w) ∈

σ(L), (k, w) ∈ R×R. We consider the space L(H) instead of the space of bounded operators B(H), because all the problemsconsidered in this paper come from dynamic models with unbounded operator coefficients of the form (1.1) or (1.3).

Now we will briefly discuss some known variational principles of the linear and nonlinear spectral theory. Denote theeigenvalues of a self-adjoint operator A below the minimum of its essential spectrum by

λ1 ≤ λ2 ≤ · · · ≤ λn ≤ · · ·

counted according to their multiplicities.Then the Poincaré–Ritz principle states that

λn = minL⊂D(A)dim L=n

maxu∈Lu=0

(Au, u)(u, u)

, n = 1, 2, . . . . (1.5)

Analogously, the Courant–Fischer–Weyl variational principle states that

λn = maxL⊂H

dim L=n−1

minu∈D(A)u⊥L

(Au, u)(u, u)

, n = 1, 2, . . . . (1.6)

If the number of eigenvalues is finite, λ1 ≤ λ2 ≤ · · · ≤ λn, then their values λ1, λ2, . . . , λn are characterized by theCourant–Fischer–Weyl and Poincaré–Ritz principles and the property

σi := minL⊂D(A)dim L=i

maxu∈Lu=0

(Au, u)(u, u)

= maxL⊂H

dim L=n−1

minu∈D(A)u⊥L

(Au, u)(u, u)

= min σess(A)

for i > n.Many authors have generalized these variational principles to λ-nonlinear eigenvalue problems of the form L(λ)u = 0.

The Rayleigh quotient p(u) =(Au,u)(u,u) of the linear problem Au = λu is replaced by the so-called Rayleigh functional p, which

is a homogeneous, nonlinear functional defined by the equation (L(p(u))u, u) = 0, for u = 0. In the framework of theclassical variational theory the following conditions have to be satisfied:(1) The equation (L(λ)u, u) = 0, u = 0 has a single root p(u) in the interval ∆, for all u ∈ H \ 0.(2) All functions (L(λ)u, u), u ∈ H \ 0 are decreasing or increasing in p(u).

By condition (1), the functional p(u) is defined on the entire space H \ 0. This case is called the overdamped case. Foroverdamped spectral problems, the eigenvalues in ∆ below σess(L) will be described by the variational principles (1.5) and(1.6) if we replace (Au,u)

(u,u) by p(u) (see [9–11] and references therein). Overdamped spectral problems were the first stage inthe analysis of nonlinear variational theory.

The second stage in the variational theory of the spectrum of operator pencils is connected to non-overdamped problems(see [12,3,4,13,5,6,2,14–16]). In this case the functional p(u) is not defined on the entire space, only on a conic subset. Inaddition to conditions (1) and (2), the following condition is imposed:(3) κα := max dimE | 0 = u ∈ E, (L(α)u, u) < 0 < +∞, where α = inf∆.

744 M. Hasanov / Mathematical and Computer Modelling 54 (2011) 742–755

c+Im k

h (w) ≥ 0

c0 Re k

Fig. 1. The domain containing σwp (L) for real w.

Note that for overdamped problems κα = 0, so no index shift occurs in the variational principles. Non-overdampedvariational theory for eigenvalues in the infinite-dimensional case is rooted in the paper by Voss andWerner [14]. Moreover,themethod given byWerner [9] for overdamped pencils is also themostwidespread technique for solving non-overdampedspectral problems.

Consider a non-overdamped spectral problem for an operator pencil L in the interval ∆. Then for all eigenvaluesλ1, λ2, . . . , λn, . . . of L below σess(L) in ∆, we have

λn = minL⊂D(p)

dim L=n+κα

supu∈Lx=0

p(u), (1.7)

provided κα < +∞.The last stage in this field is connected to generalized spectral problems in a Hilbert space (problems of the form

Au = λBu) and spectral problems for self-adjoint operators in a Krein space ⟨K , [., .]⟩ (see [17]). For example, if the operatorB is bounded and invertible, setting [u, u] := (Bu, u) and C := B−1A will reduce the problem Au = λBu to an eigenvalueproblem for the operator C in the Krein spacewith the inner product [u, u] = (Bu, u) (formore about Krein spaces, see [18]).

Let us define the positive (or negative) cone as

C±= u ∈ H| ± (Bu, u) > 0

for problems of the form Au = λBu, and as

C±

K = u ∈ H| ± [u, u] > 0

for spectral problems in a Krein space ⟨K , [., .]⟩. We say that an eigenvalue λ of the problem Au = λBu is of the positive(negative) type if (Bu, u) > 0 ((Bu, u) < 0)for all non-zero eigenvectors corresponding to λ. Positive-type and negative-type eigenvalues are analogously defined in a Krein space. In addition to positive, neutral and negative-type eigenvalues,we can define the following sets, which contain mixed type eigenvalues:

B±= λ : ∃u, u = 0, Au = λBu and ± (Bu, u) > 0 (1.8)

and

B±

K = λ : ∃u, u = 0, Au = λu and ± [u, u] > 0. (1.9)

That is, all mixed-type eigenvalues for the generalized problem Au = λBu and spectral problem Au = λu in a Krein space Kbelong to the sets B± and B±

K respectively.It has been proved that under certain restrictions, some of the eigenvalues in B± are eliminated and the remaining

eigenvalues are described by the following variational principle (see [17]):

λ±

n = supL⊂H

dim L=n−1

infu∈C±

u⊥L

(Au, u)(Bu, u)

. (1.10)


ζ√

w

k

w = c0k

Fig. 2. The domain containing the real spectrum σR(L).

For a self-adjoint eigenvalue problem Au = λu in a Krein space ⟨K , [., .]⟩, with some restrictions on the operator A, someof the eigenvalues from B±

K are eliminated. The remaining eigenvalues are described by a similar variational principle(see [17]):

λ±

n = supL⊂H

dim L=n−1

infu∈C±

Ku⊥L

[Au, u][u, u]

. (1.11)

In this paper, we study the same questions for non-linear eigenvalue problems (1.2) with the Rayleigh functionals

p±(u, w) =−(Bu, u) ±

√d(u, w)

2(Cu, u),

where d(u, w) = (Bu, u)2 − 4((A − w2R)u, u)(Cu, u). In this case, mixed-type eigenvalues belong to the sets (+ or −)

B±

w (L) = k : ∃u, u = 0, L(k, w)u = 0 and ± (L′(p±(u, w))u, u) > 0. (1.12)

In addition to this Introduction (Section 1), this paper consists of three Sections. Section 2 gives some preliminary factsand general results concerning the spectra of the pencils studied in this paper. Domains containing the point spectrum σw

p (L)and the real spectrum σR(L), for a fixed real w, are given in Figs. 1 and 2 respectively (for more results on bounds of spectraof operator functions see also [19]). In Section 3 we analyze all real eigenvalues according to the distribution of Fig. 3. Fornon-overdamped pencils, we use variational techniques to determine the eigenvalues in positive and negative-type spectralzones.Wewill show that neutral-type eigenvalues at the ends of the numerical ranges can be evaluated using the functionalsp±(u, w) without resorting to a max–min principle. The Krein space technique is used to describe eigenvalues in B±

w (L) inthe mixed spectral zone, where the numerical ranges of two root functionals overlap. We use the Ljusternik–Schnirelmammethod to find criteria for the existence of a neutral eigenvector and the associated pair (k, w) of neutral wave number kand frequency w. The main question in this context could be stated as follows: At what values of w ∈ R does there existan eigenvalue k with a neutral eigenvector, and how can we find w and k? Finally, in Section 4 we consider two examples:one second-order waveguide, and one higher-order waveguide. We will also show that the variational principles obtainedin this paper can be used to prove the existence of classical, propagating and trapped waves.

2. The spectral structure of waveguide-type pencils: preliminary facts

In this sectionwe give some general facts about the spectra σw(L) and σk(L). First, wewill define two-parameter operatorpencils of the waveguide type (for the conditions on these pencils, see [1], p. 33). A definition of polynomial waveguide-typepencils is given in [6]. Note that the conditions below are satisfied for most regular waveguides.

Definition 2.1. A second-order, two-parameter operator pencil L(k, w) given by (1.2) is said to be an operator pencil of thewaveguide type (w.g.t.) iff

(I) A is a self-adjoint, non-negative operator such that (A + I)−1∈ S∞, where S∞ is the set of all compact operators.

(II) C and R are bounded and positive definite operators.


neutral eigenvalues

eigenvalues of negative type

middle spectrum

eigenvalues of positive type

neutral eigenvalues

k-' (w) k- (w) δ- (w) δ+ (w) k+ (w) k+' (w)

Wp- (w) Wp+ (w)

Fig. 3. The distribution of the real eigenvalues.

(III) The operator B is symmetric, and (A + I)−1/2B(A + I)−1/2∈ S∞. In particular, this condition means that D((A + I)1/2)

⊂ D(B).(IV) (the energetic stability principle) There exist real numbers ζ ≥ 0 and c0 > 0 such that for all k ∈ R and all

u ∈ D((A + I)1/2),

(Au, u) + k2(Cu, u) + k(Bu, u) ≥ (c02k2 + ζ )(Ru, u).

Under these conditions the pencil L(k, w) is well-defined on D(L) := D((A + I)1/2). Now we repeat a Theorem about thediscreteness of the spectra σw(L) and σk(L) [1] (for a similar theorem concerning polynomial pencils, see [6]). The main ideais based on a standard technique from the spectral theory of operator pencils (see [20,11]).

Theorem 2.1. (a) Let L(k, w) be a two-parameter, polynomial operator pencil of w.g.t. Then for all w ∈ C the spectrum σw(L)is discrete, i.e., σw(L) consists of isolated eigenvalues of finite multiplicity.

(b) Let L(k, w) satisfy conditions (I)–(III). Then for all k ∈ C, the spectrum σk(L) is discrete.

An especially important consequence of this theorem is the fact that σw(L) = σwp (L).

Now we will give a Theorem about the bounds of the spectral sets σw(L) and σR(L). In quadratic spectral problems,including the vibration of an elastic cylinder (see Examples 5.1, 5.2 and 5.3 in [1]), acoustics and electromagneticwaveguides,condition (IV) contains a term like c20k

2+ ζ as shown above. But in the bending vibration of an elastic plate (see [1,21]), we

have

A + kB + k2C ≥ (π2+ k)2R, for k ≥ 0.

To account for this case in addition to the aforementioned quadratic spectral problems, one canmodify the energetic stabilitycondition to

A + kB + k2C ≥ (c20 (k − c1)2 + ζ )R.

The following theorem follows from the conditions given in Definition 2.1, and its proof goes through exactly the samesteps as a related theorem from our previous paper [6](see also [1]). For this reason, here we state the theorem withoutproof.

Theorem 2.2. Let L(k, w) = A + kB + k2C − w2R be a two-parameter operator pencil of w.g.t., with the energetic stabilitycondition

(Au, u) + k(Bu, u) + k2(Cu, u) ≥ (c20k2+ ζ )(Ru, u), u ∈ D(L).

Then the following statements are true: (1) For every w ∈ R, the spectrum σw(L) lies in the hyperbola

c2+(Imk)2 − c20 (Rek)

2≥ ζ − w2. (2.1)

(2) The real spectrum σR(L) lies inside the hyperbola

w2− c20k

2≥ ζ . (2.2)

(3) (k, w) ∈ σ(L), and Imk = 0 implies Imw = 0.

The inequalities from Theorem 2.2 can localize the spectra σwp (L) (or the same σw(L)) and σR(L), which are plotted in

Figs. 1 and 2.

Corollary 2.1. (1) There are no real eigenvalues if |w| <√

ζ (see Fig. 2).(2) If w2

= ζ then the only real eigenvalue permitted is k = 0. Zero will be an eigenvalue only if ζ is an eigenvalue for theproblem Au = λRu.

(3) If w >√

ζ then there are a finite number of real eigenvalues in σw(L) (this follows from Theorems 2.1 and 2.2; see Fig. 2).


3. Variational principles for real eigenvalues

This section discusses only the real eigenvalues of a two-parameter operator pencil of w.g.t. We will concentrate onquadratic pencils, but many facts can easily be extended to polynomial pencils.

Let us fix w, and set Lw(k) := L(w, k) = A + kB + k2C − w2R. Define p±(u, w) =−(Bu,u)±

√d(u,w)

2(Cu,u) , where d(u, w) =

(Bu, u)2 − 4((A − w2)u, u)(Cu, u). The functionals p±(u, w) are the roots of the equation (Lw(k)u, u) = 0, u = 0. Clearly,if k ∈ R is an eigenvalue and u = 0 is its corresponding eigenvector, then either p+(u, w) = k or p−(u, w) = k. Moreover,at each eigenvector we have d(u, w) ≥ 0.

Now we define two conic subsets, which will play an important role in finding the real eigenvalues by variationalprinciples: G(w) = u ∈ D(L) | d(u, w) > 0 and G′(w) = u ∈ D(L) \ 0 | d(u, w) ≥ 0. The set G′(w) will alsobe the domain of the functionals p±(u, w). Let Wp±

(w) = p±(u, w)|u ∈ G(w) be the numerical range of p±(u, w) onG(w). Now define the bounds of the ranges of p±(x, w) on G(w) and G′(w) as

k−(w) = infu∈G(w)

Wp−(w), k′

−(w) = inf

u∈G′(w)p−(u, w),

k+(w) = supu∈G(w)

Wp+(w), k′

+(w) = sup

u∈G′(w)

p+(u, w),

δ−(w) = infu∈G(w)

Wp+(w), δ+(w) = sup

u∈G(w)

Wp−(w).

As we will show below (Theorem 3.2), the inequalities k′−(w) ≤ k−(w) ≤ δ−(w) ≤ δ+(w) ≤ k+(w) ≤ k′

+(w) hold for

all two-parameter, unbounded operator pencils of w.g.t. For the inequality δ−(w) ≤ δ+(w) of a one-parameter, boundedoperator pencil of w.g.t., see Abramov [4] (p. 1279). These inequalities will lead us to the distribution of real eigenvalues asin Fig. 3.

The goal is now to describe all real eigenvalues of Lw(k) for a fixed w, including the eigenvalues of mixed type. Based onthe distribution given in Fig. 3, we will study each spectral zone separately. Each zone will require a different method.1. Eigenvalues in ∆′(w) := [k′

−(w), k−(w)) ∪ (k+(w), k′

+(w)].

Define W ′p±

(w) = p±(u)|u ∈ G′(w). First we note that the numerical ranges W ′p±

(w), in general, are not connected(see Example 3.1, [6,4], p. 1285). Consequently, in general [k′

−(w), k−(w)) ⊂ W ′

p−(w) and (k+(w), k′

+(w)] ⊂ W ′

p+(w).

The following theorem gives a characterization of the eigenvalues in ∆′(w).

Theorem 3.1. The set W ′p±

(w)∩∆′(w) consists of a finite number of points, where each point is a neutral eigenvalue of L(k, w).Thus, the range of p±(u, w) in ∆′(w) and the set of eigenvalues of L(k, w) in ∆′(w) coincide.

Proof. We need to show that if k ∈ ∆′(w) and (L(k, w)u, u) = 0, u = 0, then k is a neutral eigenvalue and u is acorresponding eigenvector. By the definition of the numbers k±(w), we have L(k, w) ≥ 0 for all k ∈ ∆′(w). It is knownthat if a bounded, self-adjoint operator S is non-negative, then

‖Su‖2≤ ‖S‖(Su, u). (3.1)

In our case the operator pencil L(k, w) is non-negative for all k ∈ ∆′(w) and all real w. Since L(k, w) is unbounded, wetransform L(k, w) to the similar bounded operator pencil

L(k, w) = (A + I)−12 L(k, w)(A + I)−

12 .

Evidently, σ(L) = σ(L). Since L(k, w) is bounded and nonnegative for k ∈ ∆′(w), we obtain

‖L(k, w)u‖2≤ ‖L(k, w)‖(L(k, w)u, u).

Thus, (L(k, w)u, u) = 0 implies L(k, w)u = 0. Consequently, k is an eigenvalue of L(k, w), and it is a neutral eigenvaluebecause L(k, w) ≥ 0.

2. Eigenvalues in [k−(w), δ−(w)) and (δ+, k+(w)].These parts of the spectrum were completely studied for two-parameter, unbounded operator pencils in our previous

paper [6], and for one-parameter, boundedwaveguide-type pencils in [4]. Some results for two-parameter, bounded pencilsare also given in [3] without proof. The results of Abrambov [4] for one parameter bounded operator pencils and our resultsin [6] for two parameter unbounded operator pencils allowus to state that the eigenvalues are distributed according to Fig. 2.Itmeans that eigenvalues in [k−(w), δ−(w)) are of the negative type, and eigenvalues in (δ+, k+(w)] are of the positive type.By Corollary 2.1, there are a finite number of eigenvalues in both intervals. Moreover,

k±(w) ∈ σwp (L), for every w ∈ R and w >

ζ . (3.2)

This fact demonstrates the existence of real eigenvalues for a fixed w.


Denote ∆−(w) := [k−(w), δ−(w)) and ∆+(w) = (δ+(w), k+(w)]. The spectral problem on ∆±(w) is not overdamped.As mentioned in the Introduction, in order to describe all eigenvalues in ∆±(w) by variational principles, three conditionsmust be satisfied. To interpret these conditions, define G−(w) = u|u ∈ G(w), p−(u, w) < δ−(w) and G+(w) = u|u ∈

G(w), p+(u, w) > δ+(w). The conditions on ∆−(w) are satisfied by the distribution of roots according to Fig. 3. Thus wehave

(1) The equation (Lw(k)u, u) = 0, u = 0 has a single root in ∆−(w) for all u ∈ G−(w),(2) All functions (Lw(k)u, u), u ∈ G−(w) are decreasing at p−(u, w),(3) inf∆−(w) = k−(w), Lw(k) is defined at k−(w), and

κ−(α) := max dimE|(L(α)x, x) < 0, x ∈ E \ 0 = 0, for α = k−(w).

One can check these conditions on the interval∆+(w) in the samemanner. Arranging the eigenvalues in∆−(w) in increasingorder gives

k1−(w) ≤ k2

−(w) ≤ · · · ≤ kn

−(w),

so by the conditions (1)–(3) above we have

ki−(w) = min

L⊂G−(w)

dim L=i

maxu∈L

p−(u, w), i = 1, 2, . . . , n. (3.3)

ki−(w) = max

L⊂Hdim L=i−1

infu∈G−(w)

u⊥L

p−(u, w) i = 1, 2, . . . , n. (3.4)

3. The middle spectrum: σwp (L) ∩ [δ−(w), δ+(w)].

As mentioned in the Introduction, this part of the spectrum is themain interest of this paper. We start with the followingtheorem.

Theorem 3.2. (a) Every mixed-type eigenvalue in [δ−(w), δ+(w)] has a neutral eigenvector.(b) There is a number w∗

∈ R such that G(w) = ∅ and G(w) = H \ 0, for all w > w∗.(c) δ−(w) ≤ δ+(w), for w ≥ w∗.

Proof. (a) Let k ∈ [δ−(w), δ+(w)] be an eigenvalue of mixed type. Assume that k has a positive-type eigenvector u+ anda negative-type eigenvector u−. Then the vectors ut = tu+

+ (1 − t)u−, t ∈ [0, 1] are eigenvectors corresponding to thesame eigenvalue k. We also know that d(u+, w) > 0 and d(u−, w) < 0. Consequently, by continuity of d(ut , w), there existsa number t∗ ∈ (0, 1) such that d(ut∗ , w) = 0, which means that the vector ut∗ is a neutral eigenvector corresponding to theeigenvalue k.

(b) By conditions (I) and (II) of Definition 2.1, the eigenvalues ν2n of A − λR are non-negative and ν2

n → ∞. Thus,there are always two eigenvalues ν2

m and ν2k such that ν2

m < w2 < ν2k . Let um and uk be the eigenvectors of A − λR

corresponding to ν2m and ν2

k respectively. Here we can assume that all operators are bounded. Otherwise, we use thetransformation L(k, w) = (A + I)−

12 L(k, w)(A + I)−

12 . Now, d(um, w) = (Bum, um)2 − ((A − w2R)um, um)(Cum, um) =

(Bum, um)2 − [(Aum, um) − ν2m(Rum, um) + (ν2

m − w2)(Rum, um)](Cum, um) = (Bum, um)2 + (w2− ν2

m)(Rum, um)(Cum, um)

and d(uk, w) = (Buk, uk)2− (ν2

k −w2)(Ruk, uk)(Cuk, uk). Using these and the fact that B (or more precisely, the transformedoperator B) is bounded, we validate assertion (b).

(c) Letw > w∗. We have just proved thatG(w) = H \0. This implies thatG′(w)\G(w) = 0. Choose a point u1 ∈ G(w)and 0 = u2 ∈ G′(w) \ G(w). Define ut = tu1 + (1 − t)u2, t ∈ [0, 1]. Now d(u2, w) = 0 and d(u1, w) > 0. Since d(ut , w) isa continuous function on [0, 1], there exists the biggest number t∗ ∈ [0, 1] such that d(ut∗ , w) = 0. Then d(ut , w) > 0 forall t ∈ (t∗, 1]. This fact yields the sequence

δ−(w) = infG(w)

p+(u, w) ≤ limt→t∗+0

p+(ut , w) = p+(u∗

t , w) = p−(u∗

t , w)

= limt→t∗+0

p−(ut , w) ≤ supG(w)

p−(u, w) = δ+(w).

An example of a matrix pencil of w.g.t. with δ−(w) < δ+(w) was given in [6].All eigenvalues from themiddle spectrum [δ−(w), δ+(w)], except for neutral-type eigenvalues, clearly belong to the sets

B+w (L) or B−

w (L) defined as:

B±

w (L) = k : ∃u, u = 0, L(k, w)u = 0 and ± (L′(p±(u, w))u, u) > 0.

For example, B+w (L) (B−

w (L)) contains eigenvalues which have at least one positive (negative) eigen-pair. Evidently, the setB+

w (L) (B−w (L)) contains all positive (negative) type eigenvalues too. Now we will use the Krein space approach described

in [17] to investigate eigenvalues in B±w (L). Eigenvalues with at least one neutral eigenvector will be studied separately,

using the Ljusternik–Schnirelman theory for critical points.The first step is to construct a ‘‘bridge’’ between the spectrum of self-adjoint operators in the Krein space and two-

parameter operator pencils of w.g.t. Then we will establish a connection between the root functionals of the pencils studied


in this paper and the numerical ranges of their linearizations. This connection will allow us to obtain analogous results inthe mixed spectral zone [δ−(w), δ+(w)]. The most suitable linearization for this context is that given by Krein and Langerin their well-known paper [22].

We start with Lw(k) = A+ kB+ k2C −w2R. First show that Lw(k) is a translated invertible. According to the definition ofoperator pencils of w.g.t., we can reduce an unbounded pencil of w.g.t. Lw(k) = A+ kB+ k2C − w2R to a bounded operatorpencil of the form Tw(λ) = λ2M + 2λS + I , where T , S ∈ S∞, T > 0 and S = S∗. Indeed, we have

Lw(k) = (A + I)−12 Lw(k)(A + I)−

12 = k2C + kB + I − (A + I)−1

− w2R,

where C = (A + I)−12 C(A + I)−

12 , B = (A + I)−

12 B(A + I)−

12 , and R = (A + I)−

12 R(A + I)−

12 . It follows from the properties

of A, B, C , and R given in Definition 2.1 that B, C, R ∈ S∞ and that C > 0, R > 0, B = B∗. The energetic stability conditionyields

(A(k)u, u) ≥ (c20k2+ ζ )(Ru, u), u ∈ H and ζ ≥ 0,

where A(k) = I − (A + I)−1+ kB + k2C . Then

(Lw(k)u, u) = ((A(k) − w2)Ru, u) ≥ (c20k2+ ζ − w2)(Ru, u).

For a fixed w, let the number k∗ satisfy the inequality c20 (k∗)2 − w2 > 0, i.e., |k∗

| > |wc0

|. Then we obtain (Lw(k∗)u, u) ≥

δ(u, u), δ > 0, u ∈ H . This implies that Lw(k∗) is positive definite, and consequently invertible. Substituting k = λ + k∗,one can write

Lw(λ + k∗) = λ2C + 2λk∗C +

12B

+ Lw(k∗).

Finally, setting

Tw(λ) := L−

12

w (k∗)Lw(λ + k∗)L−

12

w (k∗) (3.5)

we obtain

Tw(λ) = λ2M + 2λN + I,

where M = L−

12

w (k∗)C L−

12

w (k∗) and N = L−

12

w (k∗)k∗C +

12 BL−

12

w (k∗). In what follows, we fix w and write T (λ) for Tw(λ).We therefore have

T (λ) = λ2M + 2λN + I,

where M, N ∈ S∞, M > 0 and N = N∗.Notice that many properties of this pencil were studied in the paper of Krein and Langer [22]. A simple connection

between the eigenvalues and associated vectors (for the definition see [11]) of T (λ) and Lw(k) follows immediately from

(3.5). Namely, (λ0, u0) is an eigenpair for T (λ) if and only if (λ0 + k∗, L−

12

w (k∗)u0) is an eigenpair for Lw(k).In the following discussion, our starting point is the pencil

T (λ) = λ2M + 2λN + I

and its linearization: L(λ) = I − λT in the space H2= H ⊕ H , where

T =

0 M

12

−M12 −2N

.

A proof of the following proposition is based on the definitions of eigenvectors and associated vectors by using the abovegiven linearization.

Proposition 3.1. σ(T ) = σ(L). Moreover, if the vectors u0, u1, . . . .un form a chain of eigenvectors and associated vectors (e.a.v.)

corresponding to the eigenvalue λ0 of the pencil T (λ), then the vectors[

λ0M12 u0

u0

],

[λ0M

12 (λ0u1 + u0)

u1

], . . . ,

[M

12 (λ0un + un−1)

un

]form a chain of e.a.v. corresponding to the same eigenvalue λ0 of the linear pencil L(λ) = I − λT. Conversely, if

u10u20

,u11u21

,u1nu2n

form a chain of e.a.v. corresponding to the eigenvalue λ0 of L(λ) = I − λT, then the vectors u2

0, u21, . . . , u

2n form a chain of e.a.v.

corresponding to same eigenvalue λ0 of the pencil T (λ).


Evidently, the eigenvalues of the linear pencil L(λ) = I − λT and the characteristic values of the operator T are thesame. The operator T is not self-adjoint in the Hilbert space H ⊕ H . But if we define J =

I 00 −I

, then (JT)∗ = JT. This

relationship means that the operator T is a self-adjoint operator in the Krein space K := H ⊕ H with the inner product[u, v]J := (J u, v). According to Proposition 3.1, u0 is an eigenvector corresponding to the eigenvalue λ0 of the pencil T (λ)

if and only if the vector u0 =

[λ0M

12 u0

u0

]is an eigenvector corresponding to the characteristic value λ0 of the operator T.

Moreover, these eigenvectors are of the same type. Indeed, [u0, u0]J = (J u0, u0) =

λ0M

12 u0, λ0M

12 u0

− (u0, u0) =

λ20(Mu0, u0) − (u0, u0) = λ2

0(Mu0, u0) + λ20(Mu0, u0) + 2λ0(Nu0, u0) = λ0(T ′(λ0)u0, u0).

Thus,

[u0, u0]J = λ0(T ′(λ0)u0, u0) (3.6)

establishes a connection between the types of eigenvectors corresponding to the same eigenvalue. It follows from thisformula that (u0, λ0) is an eigenpair of T of positive, negative or neutral type if and only if (u0, λ0) is an eigenpair of T (λ) ofthe same type. Before continuing, please take into account the following remark.

Remark. We are dealing with the translated pencil T (λ), so the spectrum of this pencil is a translation of the spectrum ofLw(k) by −k∗. By correctly choosing the number k∗, we can define the operator pencil T (λ) so that σ(T ) is always positive.Therefore, λ0 in (3.6) is positive.

In what follows, we write [u, v] for [u, v]J in the Krein space K . Variational principles for real eigenvalues in a Kreinspace K are given using the forms [Tu,u]

[u,u] . To find the characteristic values, we use the form [u,u][Tu,u] . Note that [u,u]

[Tu,u] is the rootfunctional of L(λ) = I − λT in the space K ; i.e., [L(λ)u, u] = [u, u] − λ[Tu, u] = 0 and t(u) := λ =

[u,u][Tu,u] .

To establish variational principles for the real eigenvalues of T (λ) = λ2M + 2λN + I by using the traditional Rayleighfunctionals

p±(u) =−(Nu, u) ±

(Nu, u)2 − (Mu, u)

(Mu, u),

we give a connection between t(u) =[u,u][Tu,u] and p±(u).

Proposition 3.2. Let (T (p±(u))u, u) = 0 and u± =

p±(u)M

12 u

u

. Then t(u±) = p±(u), or equivalently [u±, u±] − p±(u)

[T u±, u±] = 0.

Proof. [u±, u±] = (J u±, u±)H2 = (p±(u)M12 u, p±(u)M

12 u) − (u, u) = |p±(u)|2(Mu, u) − (u, u). On the other hand,

[T u±, u±] = (JT u±, u±)H2 and

JT u± =

0 M

12

M12 2N

[p±(u)M

12 u

u

]=

[M

12 u

p±(u)Mu + 2Nu

].

Thus, [T u±, u±] = (M12 u, p±M

12 u) + (p±(u)Mu, u) + 2(Nu, u) and

[u±, u±] − p±(u)[T u±, u±] = |p±(u)|2(Mu, u) − (u, u) − |p±(u)|2(Mu, u) − p2±(u)(Mu, u) − 2p±(u)(Nu, u) = 0.

Finally, the spectral problem of the operator pencil Lw(k) in [δ−(w), δ+(w)] is reduced to the spectral problem of theself-adjoint operator T in a Krein space. In this case one can use the results given in the paper of Binding et al. [17] (andreferences therein). These results are briefly summarized below.

If C±= u ∈ K | ± [u, u] > 0 and

σ±(T) := supL

codimL=k−1

infL∩C±

[Tu, u][u, u]

, (3.7)

then some eigenvalues of the operator T in [δ−(w), δ+(w)] are described by σ±(T). They also describe an algorithm toeliminate pairs of positive and negative eigenvalues which cannot be found using classical variational principles (3.7).

Note that the results given in Propositions 3.1 and 3.2 regarding the connection between the spectra of T and T (λ)and their root functionals allow us to state that the same part of the spectrum (see [17]) of the operator pencil T (λ) in[δ−(w), δ+(w)] is described by

σ±(T ) := supL

codimL=k−1

infL∩G(w)

p±(u),


where

p±(u) =−(Nu, u) ±

(Nu, u)2 − (Mu, u)

(Mu, u)

and G = u|(Nu, u)2 − (Mu, u) > 0.(b) Ljusternik–Schnirelman theory for critical pointsRecall that our main question in this context can be stated as follows: For what values of w ∈ R does there exist an

eigenvalue k with a neutral eigenvector, and how can we find w and k? This problem can be restated as the requirementthat w and k satisfy the following equations:

k2Cu + kBu + Au = w2Ru

(L′

w(k)u, u) = (Bu, u) + 2k(Cu, u) = 0,(3.8)

where the first equation is the two parameter spectral problem (1.2) and the second defines a neutral eigenvector.For the sake of convenience we are studying this problems in a real Hilbert space. Actually, almost all dynamic models of

waveguides come from real-valued partial differential equations, which can be reformulated in a suitable complex Hilbertspace (see Example 4.1).

By the method given in the previous section, we can transform the pencil Lw(k) into a pencil with bounded coefficients.There is a relationship between the critical points of the Rayleigh functionals p±(w, u) of Lw(k) and the eigenvalues of Lw(k).

We fix w and write p(u, w) for p±(u, w). Then

p2(Cu, u) + p(Bu, u) + (Au, u) = w2(Ru, u). (3.9)

Let δp(u, w, h) =ddt p(u + th, w)|t=0 be the first variation of the functional p(u, w). The gradient is defined as a bounded

operator ∇p(u, w) satisfying

δp(u, w, h) = (∇p(u, w), h).

For example, a simple computation gives∇(Au, u) = 2Au for a bounded operator A. Now, using the gradient rules we obtainfrom (3.9) that

2p∇p(Cu, u) + 2p2Cu + ∇p(Bu, u) + 2pBu + 2Au = 2w2Ru

and ∇p(2p(Cu, u) + (Bu, u)) = −2Lw(p(u, w))u. Consequently,

∇p(u, w) =−2Lw(p(u, w))u(L′

w(p(u, w))u, u). (3.10)

The following proposition follows from (3.10).

Proposition 3.3. ∇p(u, w) = 0 if and only if p(u, w) is an eigenvalue and (p(u, w), u) is corresponding positive-type ornegative-type eigenpair for problem (1.2).

This relation does not hold for a neutral eigenpair because of (L′w(p(u, w))u, u) = 0. On the other hand, from the second

equation in (3.8) we find k = −(Bu,u)2(Cu,u) . Now setting this value of k into the first equation in (3.8) reduces the sytem of Eqs.

(3.8) to a nonlinear spectral problem of the form

(Bu, u)2

4(Cu, u)2Cu −

(Bu, u)2(Cu, u)

Bu + Au = w2Ru. (3.11)

Denoting Φ(u) :=(Bu,u)2

4(Cu,u)2Cu −

(Bu,u)2(Cu,u)Bu + Au, we arrive at the nonlinear generalized eigenvalue problem

Φ(u) = w2Ru. (3.12)

Now, (w2, u) is an eigenpair of the nonlinear spectral problem (3.12) if and only if for k = −(Bu,u)2(Cu,u) the pair (k, u) is a

neutral eigenpair of problem (3.8) for a fixed w. Clearly, ∇G(u) = Φ(u), where G(u) =12 [(Au, u) −

(Bu,u)2

4(Cu,u) ]. We also have

∇F(u) = Ru, where F(u) =(Ru,u)

2 . Thus, problem (3.12) is equivalent to the problem of finding the critical points of thefunctional F(u) restricted to the condition G(u) = 1, on the level set N1 = x|G(u) = 1. More precisely, we obtain theeigenvalue problem

∇F(u) = λ∇G(u)

on the level set N1, where λ =1

w2 .


To find λ and u, we use the Ljusternik–Schnirelman theory for critical points (see [23], Chapter 44). This problem hasbeen studied in the framework of Ljusternik–Schnirelman theory only for special functionals. For this reason we first checkthat the functionals F(u) and G(u) satisfy the conditions set out in the Ljusternik–Schnirelman theory. These conditions arestated as H1–H4 below (see [23], p. 325, and also Theorem 44A on p.326):

H1. Let X be a reflexive Banach space. F and G are even functionals such that F , G ∈ C1(X, R) and F(0) = G(0) = 0.H2. F ′

: X → X∗ is strongly continuous (i.e. un u implies F ′(un) → F ′(u)) and ⟨F ′(u), u⟩ = 0, u ∈ coSG implies F(u) = 0,where coSG denotes the closure of the convex hull of the set SG.

Moreover, for a functional v ∈ X∗ and u ∈ X the symbol ⟨v, u⟩ denotes the value of v at uH3. G′

: X → X∗ is continuous, bounded and satisfies the following condition:

un u, G′(un) v, ⟨G′(un), un⟩ → ⟨v, u⟩

implies un → u as n → ∞, where un u denotes the weak convergence in X .H4. The level set SG is bounded and u = 0 implies

⟨G′(u), u⟩ > 0, limt→+∞

G(tu) = +∞, infu∈SG

⟨G(u), u⟩ > 0.

To check the conditions H1–H4 in the present case we need some definitions and facts. Let X := D((A + I)12 ). By

Definition 2.1 A + I is a positive definite operator and consequently X ia a complete Hilbert space with respect to the scalarproduct

(u, v)X :=(A + I)

12 u, (A + I)

12 v.

Define also ‖u‖2X :=

(A + I)

12 u, (A + I)

12 u.

Here, X is the energetic space for problem (1.2) and X = W 1,2(Ω) or X = W 1,20 (Ω) for almost all waveguides from the

real world, whereW 1,2(Ω) is the Sobolev space andW 1,20 (Ω) is the closure of the test functions in the norm ofW 1,2(Ω).

Now we check these conditions in the present case.H1. Evidently F ,G : X → R are even functionals and F(0) = 0. Set G(0) = 0. Now, show that F ,G ∈ C(X, R). Let un → u

in X . Then un → u and F(un) =(Run,un)

2 →12 (Ru, u) = F(u). Moreover, ⟨F ′(un), v⟩ = (Run, v) → (Ru, v) = ⟨F ′(u), v⟩.

Thus F ∈ C ′(X, R). A straightforward computation gives G ∈ C ′(X, R). We need only to prove continuity at u = 0. Let usprove that un → 0 in X implies G(un) → 0. We have (Aun, un) =

(A + I)

12 un, (A + I)

12 un

− (un, un). By the definition

un → 0 in X ⇔(A + I)

12 un, (A + I)

12 un

→ 0. On the other hand the imbedding X → H is continuous. Hence, un → 0in X implies un → 0 in H . Thus (un, un) → 0 and consequently (Aun, un) → 0. By condition (III) of Definition 2.1 one canwrite |(Bu, u)| ≤ ‖u‖X .‖u‖. Then using c2

+(u, u) ≤ (Cu, u) we obtain

(Bu, u)2

4(Cu, u)≤

‖u‖2X .‖u‖

2

4c2−‖u‖2=

‖u‖2X

4c2−.

Consequently, (Bun,un)2

4(Cun,un)≤

‖un‖2X4c2

−

→ 0 provided that un → 0 in X . This, together with (Aun, un) → 0, as un → 0 in X implies

G(un) → 0.H2. Let us show that F ′ is strongly continuous, i.e. un → u in X implies ⟨F ′(un), v⟩ → ⟨F ′(u), v⟩ for all v ∈ X . Indeed,

⟨F ′(un), v⟩ = (Run, v) → (Ru, v) = ⟨F ′(u), v⟩ as n → ∞. We have 12 ⟨F

′(u), v⟩ = F(u). From this relation we obtain thesecond statement of H2.

H3. We have to show that G′: X → X∗ is continuous, bounded and satisfies the following condition:

un u, G′(un) v, ⟨G′(un), un⟩ → ⟨v, u⟩

implies un → u as n → ∞, where un u denotes the weak convergence in X . For sake of simplicity we check only thecase when u = 0 i.e.

un 0, G′(un) v, ⟨G′(un), un⟩ → ⟨v, u⟩

implies un → 0 in X . From the given conditions we get

⟨G′(un), un⟩ → 0. (3.13)

The derivative G′(u) is a functional, defined by ⟨G′(u), v⟩ =(Bu,u)2

4(Cu,u)2(Cu, v) −

(Bu,u)2(Cu,u) (Bu, v) + (Au, v). Comparing this with

(L(k)u, u), one may writeL

(Bu, u)

−2(Cu, u)

u, u

= ⟨G′(u), u⟩. (3.14)


We have (L(k)u, u) ≥ c‖u‖2X , for all u ∈ X and k ∈ R. This inequality together with (3.14) yields

⟨G′(u), u⟩ ≥ c‖u‖2X for all u ∈ X . (3.15)

Hence, ⟨G′(un), un⟩ ≥ c‖un‖2X . Taking the limit as n → ∞ and using (3.13) we arrive at ‖un‖X → 0, i.e. un → 0 in X as

n → ∞.H4. We have to check that: The level set SG is bounded and u = 0 implies

⟨G′(u), u⟩ > 0, limt→+∞

G(tu) = +∞, infu∈SG

⟨G(u), u⟩ > 0.

First, show that the level set is bounded. By inequality (3.15) we have ⟨G′(u), u⟩ ≥ c‖u‖2X for all u ∈ X . On the other

hand 12 ⟨G

′(u), u⟩ = G(u). Thus G(u) ≥c2‖u‖

2X , u ∈ X . Consequently, ‖un‖X → ∞ implies G(un) → ∞. It means

that the level set SG = u|G(u) = 1 is bounded. Let us show that ⟨G′(u), u⟩ > 0. Indeed, ⟨G′(u), u⟩ = 2G(u) =

(Au, u) −(Bu,u)2

4(Cu,u) =4(Au,u)(Cu,u)−(Bu,u)2

4(Cu,u) =−d(u)4(Cu,u) , where d(u) = (Bu, u)2 − 4(Au, u)(Cu, u). It follows from condition (4)

of Definition 2.1 that d(u) < 0. It means ⟨G′(u), u⟩ > 0. The fact that limt→+∞ G(tu) = +∞ follows from G(tu) = t2G(u).Finally, infu∈SG⟨G(u), u⟩ ≥ infu∈SG 2G(u) = 2.

These conditions show that we are indeed faced with a Ljusternik–Schnirelman critical point problem. Let C+(F) =

u|F(u) > 0, and by K+m denote the class of all compact, symmetric subsets K of N1 such that genK ≥ m and K ⊂ C+(F).

Here genK is defined as the smallest natural number n ≥ 1 for which there exists an odd and continuous functionf : K → Rn

\ 0. Letcm = sup

L⊂K+m

infu∈L

F(u).

Now we come to a theorem about the existence of critical points of the functional F(u) restricted to the conditionG(u) = 1 on the level set N1 = x|G(u) = 1. This result is analogous to Theorem 44A from [23, p. 326].

Theorem 3.3. If cm > 0, (a) then the problem

∇F(u) = λ∇G(u), u ∈ N1

possesses an eigenvector um with eigenvalue λm = 0 and F(um) = cm. Moreover, λm = cm, i.e.,

λm = supL⊂K+

m

infu∈L

F(u).

(b) if supm|cm > 0 = +∞, then cm → 0.Proof. Actually, according to Theorem 44A from [23, p. 326], by checking conditions H1–H4 we have already proved thistheorem.

As mentioned above, a critical point u of ∇F(u) = λ∇G(u), u ∈ N1, with λ > 0, is a neutral eigenvector of the problemAu + kBu + k2Cu = w2Ru, where λ =

1w2 and k = −

(Bu,u)2(Cu,u) . By using this relation and rewriting G(u) in the form

G(u) = −12d(u,w)−w2(Ru,u)(Cu,u)

4(Cu,u) we obtain the following corollary of Theorem 3.3, which gives not also existence criteria,but also a formula for finding k and w, with a neutral eigenvector.

Corollary 3.1. If cm > 0 then the problem Au+ kBu+ k2Cu = w2Ru has a neutral eigenvector um corresponding to the neutraleigenpair (km, wm) such that(a) F(um) = cm,(b) 1

w2m

= supL⊂K+minfu∈L F(u),

(c) km = −2(Bum,um)

(Cum,um).

Finally, we give two examples from second and higher order waveguides.

4. Applications to second-order and higher-order waveguides

Example 4.1. The hyperbolic equation in a cylinder.The problem is formulated as follows:

ρ∂2v

∂t2−

3−i,j=1

∂

∂xi

ai,j(x2, x3)

∂v

∂xj

= 0, (4.1)

v := v(t, x1, x2, x3), t ∈ R, x ∈ R, (x2, x3) ∈ Ω, ρ > 0

v(t, x1, x2, x3)|S = 0, t ∈ R, S = R × ∂Ω.

We impose the hyperbolic condition, which means that the matrix (ai,j)ni,j=1 is positive definite.


Let x := x1, and write Eq. (4.1) in the form

ρ∂2v

∂t2− a11

∂2v

∂x2−

3−

j=2

a1j∂2v

∂x∂xj+

3−j=2

∂

∂xj

aj1

∂v

∂x

−

3−i,j=2

∂

∂xi

aij

∂v

∂xj

= 0.

Defining u := u(x2, x3), Ru = ρu, Cu = a11u, B0u =∑3

j=2

a1j ∂u

∂xj+

∂∂xj

aj1u

, and Au = −

∑3i,j=2

∂∂xi

aij ∂u

∂xj

, we can write

Eq. (4.1) in the form

RVtt − CVxx − B0Vx + AV = 0.

The space of classical solutions is C2(Ω). To find a weak or generalized solution, we have to choose the space of real-valued,measurable functions LR

2 (Ω) as a basis and define the operators A, B0, C and R by their bilinear forms in the energetic spaceW 1

2 (Ω). Here W 12 (Ω) denotes the closure of the space of test functions C∞

0 (Ω) in the Sobolev space W 12 (Ω).

(Au, v) =

∫Ω

3−i,j=2

aij∂u∂xj

∂v

∂xidx2dx3,

(B0u, v) =

∫Ω

3−j=2

a1j

∂u∂xj

+∂

∂xj

aj1u

v,

(Cu, v) =

∫Ω

a11uv dx2dx3,

(Ru, v) = ρ

∫Ω

uv dx2dx3.

Notice that operators A and C are positive definite by the hyperbolic condition. We thus have A = A∗, C = C∗ and R = R∗,but (B0u, v) = −(u, B0v) and (B0u, u) = 0. For this reason, we transform this problem to the space of complex-valued,measurable functions L2(Ω) = f |

Ω

|f |2 dx < ∞. Then one can write the dynamic equation (2.1) in the form

RVtt − CVxx + i(iB0)Vx + AV = 0.

By denoting iBo := B, we recover Eq. (2.1)

RVtt − CVxx + iBVx + AV = 0,

where the operator B is a symmetric operator in L2(Ω). To look for a solution in the form V (x, t) = uei(wt−kx), we now havea spectral problem for the two-parameter operator pencil of w.g.t. L(k, w) = A + kB + k2C − w2R. It is easy to check that(see [1]) the operators A, B, C , and R all satisfy the conditions for operator pencils of w.g.t.

The existence of propagating waves, i.e., solutions of the form V (x, t) = u(x2, x3)ei(wt−kx), (k, w) ∈ R × R and theexistence of trapped waves follow from (3.2) and Theorem 3.3. The existence of classical waves (i.e., real-valued solutions)follows from the fact that the coefficients of the equation are real-valued functions. Also, the fact that V (x, t) is a solutionimplies that V (x, t) is also a solution. The next example considers a higher-order operator pencil of w.g.t.

Example 4.2. Bending vibration of thin elastic strip (see [21,1]).Let D =

(x, y)|x ∈ R, 0 < y < 1

, v := v(t, x, y),

∂2v

∂t2+ ρ11v = 0,

v|y=0,1 =∂v

∂y

y=0,1

= 0.

Let us write the dynamic equation.

∂2v

∂t2+ ρ11v =

∂2v

∂t2+ ρ

∂4v

∂x4+ 2

∂4v

∂x2∂y2+

∂4v

∂y4

= 0.

Define V (t, x) : R × R → L2(0, 1). Then according to Eq. (1.3), we can write this equation in the following form:

Vtt + C0d2Vdx2

+ C1d4Vdx4

= 0,

where C0 = 2ρu′′(y), C1u = ρu, and Au = u(4)(y). Supposing that ρ = 1, the solution is defined as a generalized solutionin the space W 2

2 (0, 1) ⊂ L2(0, 1). Setting V (x, t) = uei(wt−kx) in the equation, we get

Au − k2C0u + k4C1u = w2u,


where the operators A, C0 and C1 are defined by their bilinear forms: (Au, u) = 10 u′′v′′dy, (C0u, u) = −2

10 u′v′dy, and

(C1u, u) = 10 uvdy. In this problem the associated two-parameter operator pencil is L(k, w) = A − k2C0 + k4C1 − w2I ,

which is reduced to the quadratic pencil L(k, w) = A + λB + λ2C − w2I by substituting k2 = λ, B = −C0 and C = C1.

References

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Mathematics 33 (1986) 1025–1065.[3] Yu. Abramov, Two parameter operator pencils of waveguide type, Russian Academy of Sciences. Doklady Mathematics 286 (4) (1986) 777–781.[4] Yu. Abramov, Pencils of waveguide type and related extremal problems, Journal of Soviet Mathematics 64 (6) (1993) 1278–1289.[5] M. Hasanov, On the spectrum of a weak class of operator pencils of waveguide type, Mathematische Nachrichten 279 (2006) 843–853. 71, (2002) pp.

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Mathematical Society, Providence, RI, 1988.[12] E.M. Barston, A minimax principle for nonoverdamped systems, International Journal of Engineering Science 12 (1974) 413–421.[13] D. Eschwé, M. Langer, Variational principles for eigenvalues of self-adjoint operator functions, Integral Equations Operator Theory 49 (3) (2004)

287–321.[14] H. Voss, B. Werner, A minimax principle for nonlinear eigenvalue problems with applications to nonoverdamped systems, Mathematical Methods in

the Applied Sciences 4 (1982) 415–424.[15] H. Voss, A rational spectral problem in fluid–solid vibration, Electronic Transactions on Numerical Analysis 16 (2003) 94–106.[16] H. Voss, A maxmin principle for nonlinear eigenvalue problems with application to a rational spectral problem in fluid–solid vibration, Applications

of Mathematics 48 (6) (2003) 607–622.[17] P. Binding, R. Hryniv, B. Najman, A variational principle in Krein space. II, Integral Equations Operator Theory 51 (4) (2005) 477–500.[18] J. Bognár, Indefinite Inner Product Spaces, Springer-Verlag, New York, 1974.[19] M.I. Gil, Operator Functions and Localization of Spectra, in: Lecture Notes in Mathematics, vol. 1830, Springer-Verlag, Berlin, 2003, xiv+256 pp.[20] I. Gohberg, M. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space, Amer. Math. Soc., Providence, RI, 1969.[21] R. Porter, Trapped waves in thin elastic plates, Wave Motion 45 (2007) 3–15.[22] M.G. Krein, H. Langer, On some mathematical principles in the linear theory of damped oscillations of continua. I, II, Integral Equations Operator

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The spectra of two-parameter quadratic operator pencils