CHAPTER 5 : NONLINEAR EIGENVALUE PROBLEMS

Transcription

1 CHAPTER 5 : NONLINEAR EIGENVALUE PROBLEMS Heinrich Voss voss@tu-harburg.de Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

2 Examples Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

3 Examples Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature On the one hand: Parameter dependent nonlinear (with respect to the state variable) operator equations T (λ, u) = 0 are discussed concerning positivity of solutions multiplicity of solution dependence of solutions on the parameter; bifurcation (change of ) stability of solutions TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

4 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

5 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

6 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

7 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

8 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

9 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates electronic behavior of semiconductor hetero-structures TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

10 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

11 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures accelerator design TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

12 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures accelerator design vibro-acoustics of piezoelectric/poroelastic structures TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

13 In this presentation Examples For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. Find λ D and x 0 such that T (λ)x = 0. (1) Then λ is called an eigenvalue of T ( ), and x a corresponding eigenvector. Nonlinear eigenproblems arise in dynamic/stability analysis of structures and in fluid mechanics regularization of total least squares problems vibration of sandwich plates electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures accelerator design vibro-acoustics of piezoelectric/poroelastic structures nonlinear integrated optics TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

14 Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are M q(t) + C q(t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

15 Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are M q(t) + C q(t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. Suppose that the system is exited by a time harmonic force f (t) = f 0 e iω 0t. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

16 Examples Example 1: Vibration of structures Equations of motion arising in dynamic analysis of structures (with a finite number of degrees of freedom) are M q(t) + C q(t) + Kq(t) = f (t) where q are the Lagrangean coordinates, M is the mass matrix, K the stiffness matrix, C the viscous damping matrix, and f an external force vector. Suppose that the system is exited by a time harmonic force f (t) = f 0 e iω 0t. If C = αk + βm (modal damping) and (x j, µ j ), j = 1,..., n denotes an orthonormal eigensystem of Kx = µmx (i.e. x T k Mx j = δ jk ) then the periodic response of the system is q(t) = e iω 0t n j=1 x T j f 0 µ j ω iω 0(αµ j + β) x j. Usually, one gets good approximations taking into account only a small number of eigenvalues in the vicinity of iω 0 (truncated mode superposition). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

17 Examples Example 2: Dynamic element method Discretizing a linear eigenproblem Lu(x) = λmu(x), x Ω, Bu(x) = 0, x Ω by finite elements which depend on the eigenparameter yields a nonlinear eigenproblem; cf. Przemieniecki (1968). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

18 Examples Example 2: Dynamic element method Discretizing a linear eigenproblem Lu(x) = λmu(x), x Ω, Bu(x) = 0, x Ω by finite elements which depend on the eigenparameter yields a nonlinear eigenproblem; cf. Przemieniecki (1968). If the ansatz functions φ j (x) + λψ j (x) depend linearly on the eigenparameter then one obtains a cubic eigenvalue problem (cf. V. 1987) where ( A 3 = Ω ( A 1 = Ω 3 λ j A j x = 0 j=0 ) ( ) ψ j Mψ k dx, A 2 = (ψ j Mφ k + φ j Mψ k ψ j Lψ k ) dx jk jk Ω ) (φ j Mφ k φ j Lψ k ψ j Lφ k ) dx jk (, A 0 = Ω ) φ j Lφ k dx jk TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

19 Examples Example 3: Conservative gyroscopic systems Simulation of acoustic behavior of rotating structures (rotating wheels, e.g.) results in M q + G q + Kq = 0 where M is positive definite, K is positive (semi-)definite, G skew-symmetric, i.e. G T = G. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

20 Examples Example 3: Conservative gyroscopic systems Simulation of acoustic behavior of rotating structures (rotating wheels, e.g.) results in M q + G q + Kq = 0 where M is positive definite, K is positive (semi-)definite, G skew-symmetric, i.e. G T = G. The corresponding eigenvalue problem has purely imaginary eigenvalues. ω 2 Mx + ωgx + Kx = 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

21 Examples Example 3: Conservative gyroscopic systems Simulation of acoustic behavior of rotating structures (rotating wheels, e.g.) results in M q + G q + Kq = 0 where M is positive definite, K is positive (semi-)definite, G skew-symmetric, i.e. G T = G. The corresponding eigenvalue problem has purely imaginary eigenvalues. ω 2 Mx + ωgx + Kx = 0 Wanted in simulation of rotating tires is a large number of (not necessarily extreme) eigenvalues, for instance for rolling wheels all eigenfrequencies between 500 Hz and 2000 Hz, an important interval of perception of the human ear (cf. Nackenhorst 2004). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

22 Examples Example 4: Controlled systems with delayed feedback The governing equation of a mechanical system with delayed feedback is (Elsgolts & Norkin 1974, Hale 1977) Mü + C u + Ku = G u u(t τ) + G u u(t τ). This delay normally originates from physical limitations like finite switching times in controllers or unavailability of the current state of the system. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

23 Examples Example 4: Controlled systems with delayed feedback The governing equation of a mechanical system with delayed feedback is (Elsgolts & Norkin 1974, Hale 1977) Mü + C u + Ku = G u u(t τ) + G u u(t τ). This delay normally originates from physical limitations like finite switching times in controllers or unavailability of the current state of the system. An ansatz u(t) = e λt x yields the eigenproblem λ 2 Mx + λcx + Kx e λτ G u x λe λτ G u x = 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

24 Examples Example 4: Controlled systems with delayed feedback The governing equation of a mechanical system with delayed feedback is (Elsgolts & Norkin 1974, Hale 1977) Mü + C u + Ku = G u u(t τ) + G u u(t τ). This delay normally originates from physical limitations like finite switching times in controllers or unavailability of the current state of the system. An ansatz u(t) = e λt x yields the eigenproblem λ 2 Mx + λcx + Kx e λτ G u x λe λτ G u x = 0 The system is stable if all eigenvalues have negative real part. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

25 Examples Example 5: Viscoelastic model of damping Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

26 Examples Example 5: Viscoelastic model of damping Using a viscoelastic constitutive relation to describe the material behavior in the equations of motion yields a rational eigenvalue problem in the case of free vibrations. Discretizing by finite elements yields (cf. Hager & Wiberg 2000) T (ω)x := ( ω 2 M + K K j=1 1 ) 1 + b j ω K j x = 0 where M is the consistent mass matrix, K is the stiffness matrix with the instantaneous elastic material parameters used in Hooke s law, and K j collects the contributions of damping from elements with relaxation parameter b j. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

27 Examples Example 6: Fluid solid vibrations can be modelled in terms of solid displacement and fluid pressure and one obtains the classical form of an eigenproblem div [σ(u)] + ω 2 ρ s u = 0 in Ω s, Ω s p + ω2 c 2 p = 0 in Ω f, σ(u) n pn = 0 on Γ I, Γ I Ω f Γ O p n + ω 2 ρ f u n = 0 on Γ I, u = 0 on Γ D, p n = 0 on Γ N, where u: solid displacement p: fluid pressure λ = ω 2 : eigenparameter σ(u): linearized stress tensor ρ s, ρ f : densities of solid and fluid TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

28 Examples Variational and operator form Find λ := ω 2 C and (u, p) HΓ 1 D (Ω s ) 3 H 1 (Ω f ) such that a s (v, u) + c(v, p) = λb s (v, u) and a f (q, p) = λ( c(u, q) + b f (q, p)). for every (v, q) HΓ 1 D (Ω s ) 3 H 1 (Ω f ). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

29 Examples Variational and operator form Find λ := ω 2 C and (u, p) H 1 Γ D (Ω s ) 3 H 1 (Ω f ) such that a s (v, u) + c(v, p) = λb s (v, u) and for every (v, q) H 1 Γ D (Ω s ) 3 H 1 (Ω f ). a f (q, p) = λ( c(u, q) + b f (q, p)). which (using the Lax-Milgram Lemma) can be transformed into a linear (but not self-adjoint) eigenvalue problem K s u + Cp = λm s u K f p = λ( C u + M f p) (2a) (2b) where K s : H 1 Γ D (Ω s ) 3 H 1 Γ D (Ω s ) 3 is self-adjoint, elliptic, bounded,... TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

30 Examples Rational form of fluid-solid eigenproblem Let 0 < σ 1 σ 2... denote the eigenvalues of the decoupled eigenproblem K s u = σm s u and denote by u 1, u 2,... corresponding orthonormal eigenfunctions. Then the spectral theorem yields (K s λm s ) 1 u = n=1 1 σ n λ u n, u u n. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

31 Examples Rational form of fluid-solid eigenproblem Let 0 < σ 1 σ 2... denote the eigenvalues of the decoupled eigenproblem K s u = σm s u and denote by u 1, u 2,... corresponding orthonormal eigenfunctions. Then the spectral theorem yields (K s λm s ) 1 u = n=1 1 σ n λ u n, u u n. If λ is not contained in the spectrum of the decoupled solid eigenproblem, then λ is an eigenvalue of the coupled fluid-solid problem if and only if it is an eigenvalue of the rational eigenvalue problem T (λ)p := K f p + λm f p + n=1 λ σ n λ C np, C n p := u n, Cp C u n. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

32 Examples Rational form of fluid-solid eigenproblem Let 0 < σ 1 σ 2... denote the eigenvalues of the decoupled eigenproblem K s u = σm s u and denote by u 1, u 2,... corresponding orthonormal eigenfunctions. Then the spectral theorem yields (K s λm s ) 1 u = n=1 1 σ n λ u n, u u n. If λ is not contained in the spectrum of the decoupled solid eigenproblem, then λ is an eigenvalue of the coupled fluid-solid problem if and only if it is an eigenvalue of the rational eigenvalue problem T (λ)p := K f p + λm f p + n=1 λ σ n λ C np, C n p := u n, Cp C u n. T (λ) : H 1 (Ω f ) H 1 (ω f ) is self-adjoint and bounded. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

33 Examples Quadratic form of fluid-solid eigenproblem Another self-adjoint form of the fluid-solid eigenproblem is obtained if the second equation in (2) is multiplied by ω and p is substituted by p =: ωw. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

34 Examples Quadratic form of fluid-solid eigenproblem Another self-adjoint form of the fluid-solid eigenproblem is obtained if the second equation in (2) is multiplied by ω and p is substituted by p =: ωw. Then problem (2) is equivalent to the quadratic eigenvalue problem (( ) ( ) ( )) ( ) Ks O O C + ω O C ω 2 Ms O u = 0. O O M f w K f TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

35 Examples Example 7: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

36 Examples Example 7: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

37 Examples Example 7: Electronic structure of quantum dots Semiconductor nanostructures have attracted tremendous interest in the past few years because of their special physical properties and their potential for applications in micro and optoelectronic devices. In such nanostructures, the free carriers are confined to a small region of space by potential barriers, and if the size of this region is less than the electron wavelength, the electronic states become quantized at discrete energy levels. The ultimate limit of low dimensional structures is the quantum dot, in which the carriers are confined in all three directions, thus reducing the degrees of freedom to zero. Therefore, a quantum dot can be thought of as an artificial atom. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

38 Examples Example 7: Electronic structure of quantum dots Electron Spectroscopy Group, Fritz-Haber-Institute, Berlin TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

39 Examples Electronic structure of quantum dots ct. Problem: Determine relevant energy states (i.e. eigenvalues) and corresponding wave functions (i.e. eigenfunctions) of a three-dimensional quantum dot embedded in a matrix. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

40 Examples Electronic structure of quantum dots ct. Governing equation: Schrödinger equation ( 2 ) 2m(x, λ) u + V (x)u = λu, x Ω q Ω m where is the reduced Planck constant, m(x, λ) is the electron effective mass, and V (x) is the confinement potential. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

41 Examples Electronic structure of quantum dots ct. Governing equation: Schrödinger equation ( 2 ) 2m(x, λ) u + V (x)u = λu, x Ω q Ω m where is the reduced Planck constant, m(x, λ) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinuous across the heterojunction. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

42 Examples Electronic structure of quantum dots ct. Governing equation: Schrödinger equation ( 2 ) 2m(x, λ) u + V (x)u = λu, x Ω q Ω m where is the reduced Planck constant, m(x, λ) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinuous across the heterojunction. Boundary and interface conditions u = 0 on outer boundary of matrix Ω m 1 u BenDaniel Duke condition m m n = 1 u Ωm m q n Ωq on interface TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

43 Variational form Examples Find λ R and u H 1 0 (Ω), u 0, Ω := Ω q Ω m, such that a(u, v; λ) := u v dx + m q (x, λ) 2 Ω q Ω m 1 u v dx m m (x, λ) + V q (x)uv dx + V m (x)uv dx Ω q Ω m = λ uv dx =: λb(u, v) for every v H0 1 (Ω) Ω TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

44 Examples Electron effective mass The dependence of m(x, λ) on λ can be derived from the eight-band k p analysis and effective mass theory. Projecting the 8 8 Hamiltonian onto the conduction band results in the single Hamiltonian eigenvalue problem with { mq (λ), x Ω m(x, λ) = q m m (λ), x Ω m ( 1 m j (λ) = P2 j λ + g j V j 1 λ + g j V j + δ j ), j {m, q} where m j is the electron effective mass, V j the confinement potential, P j the momentum, g j the main energy gap, and δ j the spin-orbit splitting in the jth region. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

45 Examples Electron effective mass The dependence of m(x, λ) on λ can be derived from the eight-band k p analysis and effective mass theory. Projecting the 8 8 Hamiltonian onto the conduction band results in the single Hamiltonian eigenvalue problem with { mq (λ), x Ω m(x, λ) = q m m (λ), x Ω m ( 1 m j (λ) = P2 j λ + g j V j 1 λ + g j V j + δ j ), j {m, q} where m j is the electron effective mass, V j the confinement potential, P j the momentum, g j the main energy gap, and δ j the spin-orbit splitting in the jth region. Other types of effective mass (taking into account the effect of strain, e.g.) appear in the literature. They are all rational functions of λ where 1/m(x, λ) is monotonically decreasing with respect to λ, and that s all we need. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

46 Variational characterization Variational characterization of eigenvalues The variational characterization of eigenvalues is a powerful tool to analyze selfadjoint operators in Hilbert spaces. In this subsection we discuss generalizations to operators depending nonlinearly on the eigenparameter. Although we are mainly interested in numerical methods (and therefore on the finite dimensional case) we consider the infinite dimensional case. For linear selfadjoint eigenvalue problems the following characterizations of eigenvalues are well known (cf. Rektorys 1984): TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

47 Variational characterization Variational characterization of eigenvalues The variational characterization of eigenvalues is a powerful tool to analyze selfadjoint operators in Hilbert spaces. In this subsection we discuss generalizations to operators depending nonlinearly on the eigenparameter. Although we are mainly interested in numerical methods (and therefore on the finite dimensional case) we consider the infinite dimensional case. For linear selfadjoint eigenvalue problems the following characterizations of eigenvalues are well known (cf. Rektorys 1984): Theorem 1 Let A : H H be a selfadjoint and completely continuous operator on a real Hilbert space H with scalar product,, and denote by R A (x) := Ax, x / x, x the Rayleigh quotient of A at x 0. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

48 Variational characterization Variational characterization of eigenvalues The variational characterization of eigenvalues is a powerful tool to analyze selfadjoint operators in Hilbert spaces. In this subsection we discuss generalizations to operators depending nonlinearly on the eigenparameter. Although we are mainly interested in numerical methods (and therefore on the finite dimensional case) we consider the infinite dimensional case. For linear selfadjoint eigenvalue problems the following characterizations of eigenvalues are well known (cf. Rektorys 1984): Theorem 1 Let A : H H be a selfadjoint and completely continuous operator on a real Hilbert space H with scalar product,, and denote by R A (x) := Ax, x / x, x the Rayleigh quotient of A at x 0. Let λ 1 λ 2... be the positive eigenvalues of A ordered by magnitude and regarding their multiplicities, and let x 1, x 2,... be a system of orthogonal eigenelements where x j corresponds to λ j. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

49 Variational characterization Variational characterization of eigenvalues The variational characterization of eigenvalues is a powerful tool to analyze selfadjoint operators in Hilbert spaces. In this subsection we discuss generalizations to operators depending nonlinearly on the eigenparameter. Although we are mainly interested in numerical methods (and therefore on the finite dimensional case) we consider the infinite dimensional case. For linear selfadjoint eigenvalue problems the following characterizations of eigenvalues are well known (cf. Rektorys 1984): Theorem 1 Let A : H H be a selfadjoint and completely continuous operator on a real Hilbert space H with scalar product,, and denote by R A (x) := Ax, x / x, x the Rayleigh quotient of A at x 0. Let λ 1 λ 2... be the positive eigenvalues of A ordered by magnitude and regarding their multiplicities, and let x 1, x 2,... be a system of orthogonal eigenelements where x j corresponds to λ j. Assume that A has at least n positive eigenvalues. Then the following characterizations hold: TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

50 Variational characterization Variational characterization ct. (i) Rayleigh s principle λ n = max{r A (x) : x, x i = 0, i = 1,..., n 1}. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

51 Variational characterization Variational characterization ct. (i) Rayleigh s principle λ n = max{r A (x) : x, x i = 0, i = 1,..., n 1}. (ii) Maxmin characterization of Poincaré λ n = max dim V =n min R A(x). x V, x 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

52 Variational characterization Variational characterization ct. (i) Rayleigh s principle λ n = max{r A (x) : x, x i = 0, i = 1,..., n 1}. (ii) Maxmin characterization of Poincaré λ n = max dim V =n min R A(x). x V, x 0 (iii) Minmax characterization of Courant, Fischer, Weyl λ n = min dim V =n 1 max R A (x). x V,x 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

53 Variational characterization Variational characterization ct. Analogously the negative eigenvalues λ 1 λ 2... of A can be characterized by the three principles above if we replace min by max and vice versa. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58 (i) Rayleigh s principle λ n = max{r A (x) : x, x i = 0, i = 1,..., n 1}. (ii) Maxmin characterization of Poincaré λ n = max dim V =n min R A(x). x V, x 0 (iii) Minmax characterization of Courant, Fischer, Weyl λ n = min dim V =n 1 max R A (x). x V,x 0

54 Variational characterization Variational form Quite often the eigenvalue problem is given in variational form: TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

55 Variational characterization Variational form Quite often the eigenvalue problem is given in variational form: Let a : H H R and b : H H R be symmetric (i.e. a(u, v) = a(v, u), b(u, v) = b(v, u) for every u, v H) and bounded quadratic forms, i.e. there exist K a, K b > 0 such that a(u, v) K a u v and b(u, v) K b u v for ever u, v H. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

56 Variational characterization Variational form Quite often the eigenvalue problem is given in variational form: Let a : H H R and b : H H R be symmetric (i.e. a(u, v) = a(v, u), b(u, v) = b(v, u) for every u, v H) and bounded quadratic forms, i.e. there exist K a, K b > 0 such that a(u, v) K a u v and b(u, v) K b u v for ever u, v H. Let a be H-elliptic (i.e. a(u, u) α u 2 for some α > 0 for all u H), and let b be positive (i.e. b(u, u) > 0 for every u H, u 0) and completely continuous, i.e. u n u, v n v b(u n, v n ) b(u, v), where u n u denotes the weak convergence of u n to u. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

57 Variational characterization Variational form Quite often the eigenvalue problem is given in variational form: Let a : H H R and b : H H R be symmetric (i.e. a(u, v) = a(v, u), b(u, v) = b(v, u) for every u, v H) and bounded quadratic forms, i.e. there exist K a, K b > 0 such that a(u, v) K a u v and b(u, v) K b u v for ever u, v H. Let a be H-elliptic (i.e. a(u, u) α u 2 for some α > 0 for all u H), and let b be positive (i.e. b(u, u) > 0 for every u H, u 0) and completely continuous, i.e. u n u, v n v b(u n, v n ) b(u, v), where u n u denotes the weak convergence of u n to u. Find λ R and u H, u 0 such that a(u, v) = λb(u, v) for every v H. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

58 Variational characterization Variational form ct. By the Lax-Milgram theorem there exists a selfadjoint and (by Rellich s lemma) completely continuous operator A : H H such that b(u, v) = a(au, v) for every u, v H. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

59 Variational characterization Variational form ct. By the Lax-Milgram theorem there exists a selfadjoint and (by Rellich s lemma) completely continuous operator A : H H such that b(u, v) = a(au, v) for every u, v H. Hence, the variational eigenvalue problem is equivalent to a(u, v) = λa(au, v) v H u = λau, and λ is an eigenvalue of the variational eigenproblem if and only if λ 1 is an eigenvalue of A. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

60 Variational characterization Variational form ct. By the Lax-Milgram theorem there exists a selfadjoint and (by Rellich s lemma) completely continuous operator A : H H such that b(u, v) = a(au, v) for every u, v H. Hence, the variational eigenvalue problem is equivalent to a(u, v) = λa(au, v) v H u = λau, and λ is an eigenvalue of the variational eigenproblem if and only if λ 1 is an eigenvalue of A. Since A is positive definite (note that b(u, u) > 0 for u 0), the variational problems has positive eigenvalues and lim k λ k =, if dim H =. 0 < λ 1 λ 2 λ 3..., TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

61 Variational characterization Variational form ct. Since the Rayleigh quotient is given by R A (u) := a(au, u) a(u, u) it follows from Theorem 1 (cf. Weinberger 1974): = b(u, u) a(u, u) TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

62 Variational characterization Variational form ct. Since the Rayleigh quotient is given by R A (u) := a(au, u) a(u, u) it follows from Theorem 1 (cf. Weinberger 1974): Rayleigh s principle = b(u, u) a(u, u) a(u, u) λ j = min{ b(u, u) : u H \ {0}, b(u, uk ) = 0 for k = 1,..., j 1}, where u j, j N denotes the eigenelement corresponding to λ j, and the the eigenelements are chosen such that b(u j, u k ) = δ jk. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

63 Variational characterization Variational form ct. Since the Rayleigh quotient is given by R A (u) := a(au, u) a(u, u) it follows from Theorem 1 (cf. Weinberger 1974): Rayleigh s principle = b(u, u) a(u, u) a(u, u) λ j = min{ b(u, u) : u H \ {0}, b(u, uk ) = 0 for k = 1,..., j 1}, where u j, j N denotes the eigenelement corresponding to λ j, and the the eigenelements are chosen such that b(u j, u k ) = δ jk. Minmax characterization; Poincaré λ j = min dim V =j a(u, u) max u V,u 0 b(u, u). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

64 Variational characterization Variational form ct. Since the Rayleigh quotient is given by R A (u) := a(au, u) a(u, u) it follows from Theorem 1 (cf. Weinberger 1974): Rayleigh s principle = b(u, u) a(u, u) a(u, u) λ j = min{ b(u, u) : u H \ {0}, b(u, uk ) = 0 for k = 1,..., j 1}, where u j, j N denotes the eigenelement corresponding to λ j, and the the eigenelements are chosen such that b(u j, u k ) = δ jk. Minmax characterization; Poincaré λ j = min dim V =j Maxmin characterization, Courant, Fischer λ j = max dim V =j 1 a(u, u) max u V,u 0 b(u, u). a(u, u) min u V,u 0 b(u, u). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

65 Variational characterization Nonlinear eigenvalue problem Consider a family of bounded and selfadjoint operators T (λ) : H H, λ J, where J R is an open interval which may be unbounded. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

66 Variational characterization Nonlinear eigenvalue problem Consider a family of bounded and selfadjoint operators T (λ) : H H, λ J, where J R is an open interval which may be unbounded. Let f : J H R, (λ, x) T (λ)x, x be continuous, and assume that for every fixed x H, x 0, the real equation has at most one solution in J. f (λ, x) = 0 (2) Then equation (2) implicitly defines a functional p on some subset D of H \ {0} which we call the Rayleigh functional, and which is exactly the Rayleigh quotient in case of a linear eigenproblem T (λ) = λi A. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

67 Variational characterization Nonlinear eigenvalue problem We assume further that (λ p(x))f (λ; x) > 0 for every x D(p) and λ p(x) which generalizes the definiteness of the operator B for the generalized linear eigenproblem T (λ) := λb A, TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

68 Variational characterization Nonlinear eigenvalue problem We assume further that (λ p(x))f (λ; x) > 0 for every x D(p) and λ p(x) which generalizes the definiteness of the operator B for the generalized linear eigenproblem T (λ) := λb A, and which is satisfied if T is differentiable and λ f (λ, x) λ=p(x) > 0 for every x D (3) TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

69 Variational characterization Nonlinear eigenvalue problem We assume further that (λ p(x))f (λ; x) > 0 for every x D(p) and λ p(x) which generalizes the definiteness of the operator B for the generalized linear eigenproblem T (λ) := λb A, and which is satisfied if T is differentiable and λ f (λ, x) λ=p(x) > 0 for every x D (3) The implicit function theorem gives that D is an open set and p is continuously differentiable on D. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

70 Variational characterization Nonlinear eigenvalue problem We assume further that (λ p(x))f (λ; x) > 0 for every x D(p) and λ p(x) which generalizes the definiteness of the operator B for the generalized linear eigenproblem T (λ) := λb A, and which is satisfied if T is differentiable and λ f (λ, x) λ=p(x) > 0 for every x D (3) The implicit function theorem gives that D is an open set and p is continuously differentiable on D. Differentiating the governing equation f (p(x), x) = T (p(x)x, x 0 yields that the eigenelements of problem T ( ) are the stationary vectors of the Rayleigh functional p. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

71 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

72 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

73 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. α = 0 all eigenvalues on imaginary axis TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

74 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. α = 0 increase α all eigenvalues on imaginary axis eigenvalues go into left half plane as conjugate complex pairs TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

75 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. α = 0 increase α increase α all eigenvalues on imaginary axis eigenvalues go into left half plane as conjugate complex pairs complex pairs reach real axis, run in opposite directions TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

76 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. α = 0 increase α increase α increase α all eigenvalues on imaginary axis eigenvalues go into left half plane as conjugate complex pairs complex pairs reach real axis, run in opposite directions all eigenvalues on the negative real axis TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

77 Variational characterization Overdamped problems If the Rayleigh functional p is defined on the entire space H \ {0} then the eigenproblem T (λ)x = 0 is called overdamped. The notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 (4) where M, C and K are symmetric and positive definite matrices. α = 0 increase α increase α increase α increase α all eigenvalues on imaginary axis eigenvalues go into left half plane as conjugate complex pairs complex pairs reach real axis, run in opposite directions all eigenvalues on the negative real axis all eigenvalues going to the left are smaller than all eigenvalues going to the right system is overdamped TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

78 Variational characterization Quadratic overdamped problems For quadratic overdamped systems the two solutions p ± (x) = 1 2 Mx, x of the quadratic equation ( α Cx, x ± α 2 Cx, x 2 4 Mx, x Kx, x T (λ)x, x = λ 2 Mx, x + λα Cx, x + Kx, x = 0 (5) are real, and they satisfy sup x 0 p (x) < inf x 0 p + (x). ). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

79 Variational characterization Quadratic overdamped problems For quadratic overdamped systems the two solutions p ± (x) = 1 2 Mx, x of the quadratic equation ( α Cx, x ± α 2 Cx, x 2 4 Mx, x Kx, x T (λ)x, x = λ 2 Mx, x + λα Cx, x + Kx, x = 0 (5) are real, and they satisfy sup x 0 p (x) < inf x 0 p + (x). ). Hence, equation (5) defines two Rayleigh functionals p and p + corresponding to the intervals J := (, inf p +(x)) and J + := (sup p (x), ). x 0 x 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

80 Variational characterization Rayleigh s principle For general (not necessarily quadratic) overdamped problems Hadeler (1967 for the finite dimensional case, and 1968 for dim H = ) generalized Rayleigh s principle proving that the eigenvectors are orthogonal with respect to the generalized scalar product T (p(x)) T (p(y)) x, y, if p(x) p(y) [x, y] := p(x) p(y) (6) T (p(x))x, y, if p(x) = p(y) which is symmetric, definite and homogeneous, but in general is not bilinear. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

81 Variational characterization Rayleigh s principle Theorem 2 (Hadeler) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

82 Variational characterization Rayleigh s principle Theorem 2 (Hadeler) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. Then problem T (λ)x = 0 has at most a countable set of eigenvalues in J which we assume to be ordered by magnitude λ 1 λ 2... regarding their multiplicities. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

83 Variational characterization Rayleigh s principle Theorem 2 (Hadeler) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. Then problem T (λ)x = 0 has at most a countable set of eigenvalues in J which we assume to be ordered by magnitude λ 1 λ 2... regarding their multiplicities. The corresponding eigenvectors x 1, x 2,... can be chosen orthonormally with respect to the generalized scalar product (6),and the eigenvalues can be determined recursively by λ n = max{p(x) : [x, x i ] = 0, i = 1,..., n 1, x 0}. (7) TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

84 Variational characterization minmax and maxmin for overdamped problems Poincaré s maxmin characterization was first generalized by Duffin (1955) to overdamped quadratic eigenproblems of finite dimension, and for more general overdamped problems of finite dimension it was proved by Rogers (1964). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

85 Variational characterization minmax and maxmin for overdamped problems Poincaré s maxmin characterization was first generalized by Duffin (1955) to overdamped quadratic eigenproblems of finite dimension, and for more general overdamped problems of finite dimension it was proved by Rogers (1964). Infinite dimensional eigenvalue problems were studied by Turner (1967), Langer (1968), and Weinberger (1969) who proved generalizations of both, the maxmin characterization of Poincaré and of the minmax characterization of Courant, Fischer and Weyl for quadratic (and by Turner (1968) for polynomial) overdamped problems. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

86 Variational characterization minmax and maxmin for overdamped problems Poincaré s maxmin characterization was first generalized by Duffin (1955) to overdamped quadratic eigenproblems of finite dimension, and for more general overdamped problems of finite dimension it was proved by Rogers (1964). Infinite dimensional eigenvalue problems were studied by Turner (1967), Langer (1968), and Weinberger (1969) who proved generalizations of both, the maxmin characterization of Poincaré and of the minmax characterization of Courant, Fischer and Weyl for quadratic (and by Turner (1968) for polynomial) overdamped problems. The corresponding generalizations for general overdamped problems of infinite dimension were derived by Hadeler (1968). Similar results (weakening the compactness or smoothness requirements) are contained in Rogers (1968), Werner (1971), Abramov (1973), Hadeler (1975), Markus (1985), Maksudov & Gasanov (1992), and Hasanov (2002). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

87 Variational characterization minmax and maxmin for overdamped problems Theorem 3 (Hadeler 1968) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

88 Variational characterization minmax and maxmin for overdamped problems Theorem 3 (Hadeler 1968) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. Let the eigenvalues λ n of T (λ)x = 0 be numbered in nonincreasing order according to multiplicities. Then they can be characterized by the following two variational principles λ n = max dim V =n = min dim V =n 1 min p(x) x V, x 0 max p(x). x V, x 0 TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

89 Variational characterization minmax and maxmin for overdamped problems Theorem 3 (Hadeler 1968) Let T (λ) : H H, λ J be a family of selfadjoint and bounded operators. Assume that the problem T (λ)x = 0 is overdamped and that for every λ J there exists ν(λ) > 0 such that T (λ) ν(λ)i is completely continuous. Let the eigenvalues λ n of T (λ)x = 0 be numbered in nonincreasing order according to multiplicities. Then they can be characterized by the following two variational principles λ n = max dim V =n = min dim V =n 1 min p(x) x V, x 0 max p(x). x V, x 0 It is obvious that these principles can be transformed to minmax and maxmin principles for variational forms of nonlinear eigenvalue problems. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

90 Variational characterization Variational form of QD problem Multiplying the Schrödinger equation of the quantum dot problem by v H0 1 (Ω) one obtains the weak form TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

91 Variational characterization Variational form of QD problem Multiplying the Schrödinger equation of the quantum dot problem by v H0 1 (Ω) one obtains the weak form Find λ R and u H 1 0 (Ω), u 0, Ω := Ω q Ω m, such that a(u, v; λ) := u v dx + m q (x, λ) 2 Ω q Ω m 1 u v dx m m (x, λ) + V q (x)uv dx + V m (x)uv dx Ω q Ω m = λ uv dx =: λb(u, v) for every v H0 1 (Ω) Ω TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

92 Properties Variational characterization a(,, λ) bilinear, symmetric, bounded, H0 1 (Ω) elliptic for λ 0 b(, ) bilinear, positive definite, bounded, completely continuous By the Lax Milgram lemma the variational eigenproblem is equivalent to T (λ)u = 0 where T (λ) : H0 1 (Ω) H0 1 (Ω), λ 0, is a family of bounded operators such that... TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

93 Properties ct. Variational characterization For it holds f (λ; u) := T (λ)u, u = λb(u, u) a(u, u; λ) f (0; u) < 0 < lim f (λ; u) = for every u 0 λ λf (λ; u) > 0 for every u 0 and λ 0 For fixed λ 0 the eigenvalues of the linear eigenproblem T (λ)u = µu satisfy a maxmin characterization. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

94 Properties ct. Variational characterization For it holds f (λ; u) := T (λ)u, u = λb(u, u) a(u, u; λ) f (0; u) < 0 < lim f (λ; u) = for every u 0 λ λf (λ; u) > 0 for every u 0 and λ 0 For fixed λ 0 the eigenvalues of the linear eigenproblem T (λ)u = µu satisfy a maxmin characterization. Hence, the nonlinear Schrödinger equation has an infinite number of eigenvalues which can be characterized as minmax values of the Rayleigh functional. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

95 Variational characterization Nonoverdamped systems Nonoverdamped eigenvalue problems For nonoverdamped eigenproblems the natural ordering to call the largest eigenvalue the first one, the second largest the second one, etc., is not appropriate. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

96 Variational characterization Nonoverdamped systems Nonoverdamped eigenvalue problems For nonoverdamped eigenproblems the natural ordering to call the largest eigenvalue the first one, the second largest the second one, etc., is not appropriate. This is obvious if we make a linear eigenvalue T (λ)x := (λi A)x = 0 nonlinear by restricting it to an interval J which does not contain the largest eigenvalue of A. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

97 Variational characterization Nonoverdamped systems Nonoverdamped eigenvalue problems For nonoverdamped eigenproblems the natural ordering to call the largest eigenvalue the first one, the second largest the second one, etc., is not appropriate. This is obvious if we make a linear eigenvalue T (λ)x := (λi A)x = 0 nonlinear by restricting it to an interval J which does not contain the largest eigenvalue of A. Then all conditions are satisfied, p is the restriction of the Rayleigh quotient R A to D := {x 0 : R A (x) J}, and sup x D p(x) will not be an eigenvalue. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

98 Variational characterization Enumeration of eigenvalues Nonoverdamped eigenvalue problems λ J is an eigenvalue of T ( ) if and only if µ = 0 is an eigenvalue of the linear problem T (λ)y = µy. The key idea is to orientate the number of λ on the location on the eigenvalue µ = 0 in the spectrum of the linear operator T (λ). TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58

99 Variational characterization Enumeration of eigenvalues Nonoverdamped eigenvalue problems λ J is an eigenvalue of T ( ) if and only if µ = 0 is an eigenvalue of the linear problem T (λ)y = µy. The key idea is to orientate the number of λ on the location on the eigenvalue µ = 0 in the spectrum of the linear operator T (λ). To this end we assume that for every λ J there exists ν(λ) > 0 such that the linear operator S := T (λ) ν(λ)i is completely continuous. TUHH Heinrich Voss Nonlinear eigenvalue problems Eigenvalue problems / 58