Global Perturbation of Nonlinear Eigenvalues

1 Introduction

The problem of the global perturbation of eigenvalues, whose origins go back to Rayleigh in the 19th century, has shown to have a wide range of applications in Mathematical Physics and Nonlinear Partial Differential Equations. Essentially, given a family T⁢(μ) of linear operators on a Banach space U on 𝕂∈{ℝ,ℂ} whose dependence on μ is holomorphic, the main classical problem consists in ascertaining whether, or not, the eigenvalues of T⁢(μ) also define holomorphic functions of μ. In other words, should it be true that the values λ⁢(μ) for which

also vary holomorphically with respect to the parameter μ? Through this paper, IU and GL⁢(U) stand for the identity map and the linear group of U, respectively.

In the finite-dimensional context, X=ℂN for some N∈ℕ and (1.1) can be expressed equivalently in the form

which is an algebraic equation of degree N in λ with holomorphic coefficients. Thus, by the theory of Algebraic Riemann Surfaces [17, 18], [11, Chapter 1, Section 8], the simple roots of (1.2) are branches of holomorphic functions with respect to μ. Therefore, for any algebraically simple eigenvalue λ0 of T⁢(μ0), there is an holomorphic function, λ⁢(μ), defined on a neighborhood of μ0, Ωμ0, such that λ⁢(μ0)=λ0 and λ⁢(μ) is an eigenvalue of T⁢(μ) for all μ∈Ωμ0. Adopting a global perspective, there are, at most, finitely many branching points on each compact subset of ℂ where the number of distinct eigenvalues of T⁢(μ) is less than N, i.e., points μ∈ℂ with multiple roots λ⁢(μ) of (1.2). Actually, if we denote by ℛ the set of these branching points, the set of eigenvalues λ⁢(μ) can be viewed as a Riemann surface of N-sheets with ramification points ℛ. Hence, as soon as μ0∉ℛ, there are N holomorphic local functions λi⁢(μ), i∈{1,…,N}, of eigenvalues of T⁢(μ) such that

These classical finite-dimensional results are covered, e.g., by [2, 5, 13, 16, 28]. The monograph of Kato [16] being paradigmatic from the point of view of the applications of perturbation theory in Mathematical Physics.

In the more general context when U is a complex Banach space and T⁢(μ) is a compact operator for μ in a neighborhood of μ0, a similar perturbation result holds at simple eigenvalues of T⁢(μ0). Indeed, for any simple eigenvalue λ0 of T⁢(μ0), there exists a holomorphic function, λ⁢(μ), in a neighborhood of μ0, Ωμ0, such that λ⁢(μ0)=λ0 and λ⁢(μ) is a simple eigenvalue of T⁢(μ) for all μ∈Ωμ0. Moreover, adopting a global perspective, any finite set of isolated eigenvalues of T⁢(μ) behaves much like in the finite-dimensional context (see, e.g., [15, 29, 31]). Most of these classical results were collected by T. Kato in his influential monograph [16].

Subsequently, we will denote by ℒc⁢(U) the set of compact perturbations of IU. Summarizing the previous discussion, classical perturbation theory deals with operator surfaces 𝔏:Ω¯×ℂ→ℒc⁢(U) of the special form

where Ω is an open neighborhood of λ0 in ℂ, and T⁢(μ) is assumed to be holomorphic in μ∈ℂ with 𝔏⁢(∂⁡Ω×ℂ)⊂GL⁢(U), i.e., T⁢(μ) cannot admit any eigenvalue on the boundary of Ω, ∂⁡Ω. Under these circumstances, the perturbation problem consists in establishing the existence of perturbed eigenvalues from any given λ0∈σ⁢(T⁢(μ0)). Most of the technical tools available to deal with this problem are algebraic and analytical, and rely upon the linear structure of the operators involved in the formulation of the perturbation problem as well as, very strongly, on the holomorphy with respect to μ. However, in many applications to Real World problems, one cannot enjoy the holomorphy with respect to μ and the parameters are real, instead of being complex. Although Kato [16] also dealt with the general case when T⁢(μ) depends continuously on the perturbation parameter μ, his analysis only covered the finite-dimensional setting (see [16, Chapter 2, Section 5]). More precisely, according to [16], for any given finite-dimensional operator surface 𝔏:Ω¯×[c,d]→ℒ⁢(ℂN) of the, very special, form

continuous in μ∈[c,d], with 𝔏⁢(∂⁡Ω×[c,d])⊂GL⁢(ℂN), and for every λ0∈σ⁢(T⁢(μ0)) with μ0∈(c,d), there exists a continuous function λ⁢(μ) of eigenvalues of T⁢(μ) defined on a neighborhood of μ0 and such that λ⁢(μ0)=λ0. Naturally, as T. Kato worked without any differentiability assumption on μ, his proof relies on topological techniques; in particular, on [16, Chapter 2, Theorem 5.2]. Our main goal in this paper is to deliver a generalized version of Katos’s result valid for a wide class of Fredholm surfaces 𝔏⁢(λ,μ) through the algebraic multiplicity of Esquinas and López-Gómez [10, 9, 19].

To appreciate the strength, versatility and generality of the main theorem of this paper, the reader should note that classical spectral theory focuses attention on the eigenvalues of linear operator curves of the form T-λ⁢IU, where the dependence on the spectral parameter λ also is holomorphic, for as it is linear. This is a serious shortage of the existing theory from the point of view of its applications, because many engineering problems involve an intricate nonlinear dependence with respect to the spectral parameter λ and, as a byproduct, classical spectral theory cannot be applied. Actually, T. Kato in [16, Chapter 7, Section 6] tried to generalize the classical theory to cover the simplest prototypes of operator surfaces of the form

However, since the spectrum of these operators differs from the classical spectrum, new problems arose. Following Kato’s words (see [16, pp. 416–417]):

“In applications one often encounters problems of a more general form

(1.4) T ⁢ u = λ ⁢ A ⁢ u

where T and A are operators in U or, more generally, operators from U to another Banach space V. There are several ways of dealing with this general type of eigenvalue problem, etc. But it is not clear what is meant by an isolated eigenvalue of (1.4) or the algebraic multiplicity of such an eigenvalue.”

These open problems were solved some time later by Esquinas and López-Gómez [10, 9, 19], enhancing the birth of nonlinear spectral theory, which will play a central role in this paper. In Section 2 we will collect all the necessary results to read comfortably this paper. In particular, we will shortly remember how the classical spectrum is replaced by the most general generalized spectrum and the classical eigenvalues by the generalized eigenvalues. Problem (1.4) has been widely studied in recent years trying to understand the behavior of such eigenvalues in more general situations where the operator family A⁢(μ) needs not to be invertible (see, e.g., [2, 3, 6, 7, 8, 27]).

As already commented above, this paper sharpens, very substantially, [16, Theorem 5.2 of Chapter 2] to cover the most general case when 𝔏:Ω¯×[c,d]→Φ0⁢(U,V) is an operator surface depending continuously on the perturbation parameter μ and analytically on the spectral parameter λ, where Φ0⁢(U,V) stands for the set of Fredholm operators of index zero between U and V. Naturally, this setting covers the case of operators surfaces in Φ0⁢(U,V) of the form

where Tn∈𝒞⁢([c,d],ℒ⁢(U,V)) for each integer n≥0. The huge gap between the framework of this paper and the classical setting becomes apparent by simply comparing the operators involved in the formulations of (1.3) and (1.5). In particular:

1. Instead of compact perturbations of the identity map, we are dealing with Fredholm operators in Φ0⁢(U,V), which are the more natural generalizations of the finite-dimensional ones, for as they respect the dimension formula. In particular, compact perturbations of identity operators are Fredholm of index zero by Fredholm’s Theorem, i.e., ℒc⁢(U,V)⊂Φ0⁢(U,V) if U⊂V.

2. Instead of imposing the analyticity with respect to μ of 𝔏⁢(λ,μ), we only require continuity, which was a real challenge in perturbation theory and it seems an important advance.

3. We allow the dependence on the spectral parameter λ to be of general analytic type, rather than linear, T-λ⁢IU, as in classical perturbation theory, which provides us with a huge flexibility to apply our perturbation theorem in applied sciences and engineering, where the dependence on λ can be nonlinear.

In particular, this type of Fredholm operator surfaces covers nonlinear elliptic eigenvalue problems as those described next. Let Ω be a bounded domain of ℝN of class 𝒞2 whose boundary consists of two disjoint open and closed subsets, Γ0 and Γ1, and consider the next family of elliptic operators

where the dependence of

is analytic in λ∈[a,b] and continuous in μ∈[c,d]. For every p∈[1,+∞], W2,p⁢(Ω) stands for the Sobolev space of the functions u∈Lp⁢(Ω) such that Dα⁢u∈Lp⁢(Ω), in a distributional sense, for every α∈ℕN with |α|≤2. Let 𝔅:𝒞⁢(Γ0)⊗𝒞⁢(Γ1)→𝒞⁢(∂⁡Ω) be the boundary operator defined by

where ν=A⁢𝐧 is the conormal vector field, i.e., 𝐧 is the normal outward vector field of Ω, and β∈𝒞⁢(Γ1) satisfies either β⪈0 if Γ0=∅, or β≥0 if Γ0≠∅. Then, setting

we can consider, as soon as p>N, the operator surface 𝔏⁢(λ,μ):W𝔅2,p⁢(Ω)→Lp⁢(Ω) defined by

where c⁢(λ,μ,⋅)∈L∞⁢(Ω) for every (λ,μ)∈[a,b]×[c,d] and it is analytic in λ and continuous in μ. Since (1.6) satisfies the general requirements of the abstract perturbation results of this paper, we can apply them in this rather general context where the dependence on the spectral parameter λ is nonlinear. No previous perturbation result seems to be available under these general assumptions.

The main theorems of this paper, Theorems 3.1 and 4.1, can be summarized into the next one, where χ and Σ stand for the generalized algebraic multiplicity and the generalized spectrum of Esquinas and López-Gómez [10, 9, 19], respectively.

Any of the two assumptions in (1.7) are imperative for the validity of the theorem. Indeed, if the first requirement of (1.7) fails, then the (generalized) eigenvalues of 𝔏⁢(λ,μ) might be lost through the lateral boundary of the rectangle [a,b]×[c,d], through {a,b}×[c,d]. And Section 3 ends with a simple example establishing the necessity of the oddity of χ⁢[𝔏⁢(⋅,c),[a,b]]. When, in addition, χ⁢[𝔏⁢(⋅,μ),λ] is odd for all (λ,μ)∈𝒞, then the pathological structures shown in Figure 2 cannot occur. Thus, the curve γ defined by (1.8) is continuous.

The distribution of this paper is the following one. Section 2 collects the main ingredients of the theory of [10, 9, 19], which are necessary to prove Theorem 1.1. In Section 3, we deliver the proof of the first part of Theorem 1.1. In Section 4, we prove the second part of Theorem 1.1. In Section 5 we adapt Theorem 1.1 to a complex setting. Finally, Section 6 delivers an interesting application of Theorem 1.1 to the intricate bi-parametric weighted eigenvalue problem

Our condition to guarantee that γ⁢(μ) is a curve in the setting of Theorem 1.1 seems far from optimal, but it should not be forgotten that we are within the classical topological dichotomy of connected and path connected spaces where sometimes things are more involved that they look. Theorem 1.1 is providing us with the simplest sufficient condition to guarantee the existence of a path joining two extremal points in a component 𝒞.

2 Nonlinear Spectral Theory

In this section we give a brief self-contained introduction to nonlinear spectral theory. Classical spectral theory deals with special curves 𝔏∈𝒞⁢(Ω,ℒc⁢(U,V)) of the form

where U and V are two 𝕂-Banach spaces such that U⊂V, 𝕂∈{ℝ,ℂ}, Ω is an open domain of 𝕂, ℒc⁢(U,V) stands for the space of linear continuous operators that are compact perturbations of the identity map, IU, and K is a given compact operator. Adopting a geometrical perspective, classical spectral theory studies the intersections of the straight line 𝔏⁢(λ) with the space of singular operators

where GL⁢(U,V) denotes the space of invertible operators. In this context, λ0∈Ω is said to belong to the spectrum of the straight line 𝔏⁢(λ)=λ⁢IU-K if 𝔏⁢(λ0)∈𝒮⁢(U,V), i.e., if λ0∈σ⁢(K), the spectrum of K. Note that, in particular, these linear paths lie in the set of Fredholm operators of index zero, denoted in this paper by Φ0⁢(U,V). More generally, nonlinear spectral theory deals with general continuous paths in Φ0⁢(U,V), 𝔏∈𝒞⁢(Ω,Φ0⁢(U,V)), generalizing the classical theory not only because it deals with arbitrary continuous curves, not necessarily straight lines, but also because these paths can lie in Φ0⁢(U,V) remaining outside ℒc⁢(U,V)⊊Φ0⁢(U,V). In this general context, given a Fredholm path, 𝔏∈𝒞⁢(Ω,Φ0⁢(U,V)), the generalized spectrum of 𝔏, Σ⁢(𝔏), is defined through

and each λ∈Σ⁢(𝔏) is called a generalized eigenvalue. In the remaining, when we refer to an eigenvalue, we will refer to a generalized eigenvalue. As in the classical case, Σ⁢(𝔏) also consists of the intersection points of the curve 𝔏⁢(λ) with the singular manifold 𝒮⁢(U,V)∩Φ0⁢(U,V). Naturally, for every compact operator K,

A pivotal concept in nonlinear spectral theory is the concept of generalized algebraic multiplicity, which assigns to every pair (𝔏,λ0) with 𝔏∈𝒞∞⁢(Ω,Φ0⁢(U,V)) and λ0∈Σ⁢(𝔏), an integer χ⁢[𝔏,λ0]∈[1,+∞] such that, whenever dim⁡U=dim⁡V<∞,

The algebraic multiplicity of a generalized eigenvalue χ⁢[𝔏,λ0] can be seen as the intersection index of the curve 𝔏 with the singular surface 𝒮⁢(U,V) at λ0 as the authors have recently shown in [23]. Although a number of constructions of χ⁢[𝔏,λ0] had been done by Göhberg and Sigal [12], Magnus [24], Ize [14], Esquinas and López-Gómez [10], Esquinas [9] and Rabier [26], it was not until 2001 that López-Gómez [19, Chapters 4 and 5] characterized whether all these generalized algebraic multiplicities are finite through the pivotal concept of algebraic eigenvalue. For any 𝔏⁢(λ)∈𝒞⁢(Ω,Φ0⁢(U,V)), a given λ0∈Ω∩Σ⁢(𝔏) is said to be a κ-algebraic eigenvalue if there exist ε>0 and C>0 such that 𝔏⁢(λ) is an isomorphism whenever 0<|λ-λ0|<ε, and

with κ the minimal integer for which (2.1) holds. Throughout this paper, the set of κ-algebraic eigenvalues of 𝔏⁢(λ) is denoted by Algκ⁡(𝔏), and the set of algebraic eigenvalues of 𝔏 is defined by

It turns out that the multiplicities of [24, 10, 9] are well defined if, and only if, the path 𝔏⁢(λ) is of class 𝒞r for some r≥1 and λ0∈Algκ⁡(𝔏) for some 1≤κ≤r. Moreover, by [19, Theorems 4.4.1 and 4.4.4], when 𝔏 is analytic and 𝔏⁢(Ω) contains some invertible operator, then Σ⁢(𝔏) is discrete, and every λ0∈Σ⁢(𝔏) is an algebraic eigenvalue of 𝔏. Thus, through this paper, for any given domain Ωλ0 containing λ0, it is convenient to denote by 𝒜λ0⁢(Ωλ0,Φ0⁢(U,V)) the set of curves 𝔏∈𝒞r⁢(Ωλ0,Φ0⁢(U,V)) such that λ0∈Algκ⁡(𝔏)∩Ωλ0 with 1≤κ≤r for some r∈ℕ. According to [19, Chapter 4], the generalized algebraic multiplicity χ⁢[𝔏,λ0] is well defined if and only if 𝔏∈𝒜λ0⁢(Ωλ0,Φ0⁢(U,V)).

Short time later, in 2004, the theory of algebraic multiplicities for 𝒞∞-Fredholm paths was axiomatized by Mora-Corral [25] by establishing that, modulus a normalization condition, given an open connected neighborhood of λ0 in 𝕂, denoted by Ωλ0, the algebraic multiplicity χ is the unique map

satisfying the product formula

for all 𝔏,𝔐∈𝒞∞⁢(Ωλ0,Φ0⁢(U)), regardless whether, or not, λ0 is singular. These findings were collected by López-Gómez and Mora-Corral in [21], where, in addition, the theory of Göhberg and Sigal [12] was substantially generalized to a non-holomorphic framework, and the existence of the local Smith form was established in the class 𝒜λ0⁢(Ωλ0,Φ0⁢(U,V)) through the lengths of the Jordan chains of 𝔏⁢(λ).

A key stone in the development of the results of this paper is the behavior of the algebraic multiplicity under appropriate operator perturbations. Precisely, an homotopy H:Ω¯×[0,1]→Φ0⁢(U,V) is said to be a 𝒞-homotopy if H⁢(∂⁡Ω×[0,1])⊂GL⁢(U,V). According to [22, Theorem 4.8], if 𝔏:[a,b]×[0,1]→Φ0⁢(U,V) is a 𝒞-homotopy with analytic μ-sections, then sign⁡χ⁢[𝔏⁢(⋅,μ),[a,b]] is constant for all μ∈[0,1], where

and

More generally, for every integer n≥0, we will denote

According to [21, Theorem 8.4.3], the next complex counterpart of this result can be easily obtained. If Ω is a subdomain of ℂ and 𝔏:Ω¯×[0,1]→Φ0⁢(U,V) is a 𝒞-homotopy with holomorphic μ-sections, then χ⁢[𝔏⁢(⋅,μ),Ω] is constant for all μ∈[0,1], where, rather naturally, χ⁢[𝔏,Ω] stands for

Adopting this point of view, given an operator surface 𝔏:Ω¯×[c,d]→Φ0⁢(U,V) continuous in μ∈[a,b] and analytic in λ∈Ω, the problem that we are addressing in this paper can be viewed as the problem of analyzing the global topological properties of the intersection of 𝔏⁢(Ω¯×[c,d]) with the singular manifold 𝒮⁢(U,V).

3 Perturbation of Eigenvalues

In this section, we give an optimal condition in terms of χ to guarantee the existence of a spectral continuum connecting the spectra of two given operators. Throughout the rest of this paper, the curve 𝔏μ will stand for

regardless the value of μ. The main result of this section reads as follows.