Variational Principles for Nonlinear Eigenvalue Problems

Transcription

1 Variational Principles for Nonlinear Eigenvalue Problems Heinrich Voss Hamburg University of Technology Institute of Mathematics TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

2 Nonlinear eigenvalue problem The term nonlinear eigenvalue problem is not used in a unique way in the literature TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

3 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 u) operator equations are discussed concerning positivity of solutions multiplicity of solution T (λ, u) = 0 dependence of solutions on the parameter; bifurcation (change of ) stability of solutions TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

4 In this presentation For λ D C let T (λ) be a linear self-adjoint and bounded operator on a Hilbert space H. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

5 In this presentation 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 eigenelement. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

6 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

7 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

8 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

9 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

10 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates accelerator design TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

11 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates accelerator design vibro-acoustics of piezoelectric/poroelastic structures TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

12 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates accelerator design vibro-acoustics of piezoelectric/poroelastic structures nonlinear integrated optics TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

13 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates accelerator design vibro-acoustics of piezoelectric/poroelastic structures nonlinear integrated optics regularization of total least squares problems TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

14 In this presentation 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 eigenelement. Nonlinear eigenproblems of this type arise in dynamic/stability analysis of structures and in fluid mechanics electronic behavior of semiconductor hetero-structures vibration of fluid-solid structures vibration of sandwich plates accelerator design vibro-acoustics of piezoelectric/poroelastic structures nonlinear integrated optics regularization of total least squares problems stability of delay differential equations TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

15 Outline 1 Two examples 2 Variational characterization of eigenvalue problems Overdamped Problems Nonoverdamped problems 3 An unsymmetric linear eigenproblem 4 Concluding remarks TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

16 Outline Two examples 1 Two examples 2 Variational characterization of eigenvalue problems Overdamped Problems Nonoverdamped problems 3 An unsymmetric linear eigenproblem 4 Concluding remarks TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

17 Two examples Example 1: Electronic behavior 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

18 Two examples Example 1: Electronic behavior 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

19 Two examples Example 1: Electronic behavior 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

20 Problem Two examples 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

21 Problem ct. Two examples Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

22 Problem ct. Two examples Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinous across the heterojunction. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

23 Problem ct. Two examples Governing equation: Schrödinger equation ( 2 ) 2m(x, E) Φ + V (x)φ = EΦ, x Ω q Ω m where is the reduced Planck constant, m(x, E) is the electron effective mass, and V (x) is the confinement potential. m and V are discontinous across the heterojunction. Boundary and interface conditions Φ = 0 on outer boundary of matrix Ω m 1 Φ BenDaniel Duke condition m m n = 1 Φ Ωm m q n Ωq on interface TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

24 Variational form Two examples Find E R and Φ H 1 0 (Ω), Φ 0, Ω := Ω q Ω m, such that a(φ, Ψ; E) := Φ Ψ dx + m q (x, E) 2 Ω q Ω m 1 Φ Ψ dx m m (x, E) + V q (x)φψ dx + V m (x)φψ dx Ω q Ω m = E ΦΨ dx =: Eb(Φ, Ψ) for every Ψ H0 1 (Ω) Ω TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

25 Two examples Electron effective mass The dependence of m(x, E) on E 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 (E), x Ω m(x, E) = q m m (E), x Ω m ( 1 m j (E) = P2 j E + g j V j 1 E + 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

26 Two examples Electron effective mass The dependence of m(x, E) on E 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 (E), x Ω m(x, E) = q m m (E), x Ω m ( 1 m j (E) = P2 j E + g j V j 1 E + 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 E where 1/m(x, E) is monotonically decreasing with respect to E, and that s all we need. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

27 Properties Two examples a(,, E) bilinear, symmetric, bounded, H0 1 (Ω) elliptic for E 0 b(, ) bilinear, positive definite, bounded, completely continuous By the Lax Milgram lemma the variational eigenproblem is equivalent to T (E)Φ = 0 where T (E) : H0 1 (Ω) H0 1 (Ω), E 0, is a family of self-adjoint and bounded operators. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

28 Two examples Example 2: Free vibrations of fluid-solid structures 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, p + ω2 c 2 p = 0 in Ω f, σ(u) n pn = 0 on Γ I, p n + ω 2 ρ f u n = 0 on Γ I, u = 0 on Γ D, p n = 0 on Γ N, u: solid displacement p: fluid pressure λ = ω 2 : eigenparameter σ(u): linearized stress tensor ρ s, ρ f : densities of solid and fluid Γ I Ω f Ω s Interface conditions: equilibrium of accelerations and of force densities. Γ O TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

29 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

30 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

31 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

32 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

33 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

34 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

35 Two 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

36 Outline Variational characterization of eigenvalue problems 1 Two examples 2 Variational characterization of eigenvalue problems Overdamped Problems Nonoverdamped problems 3 An unsymmetric linear eigenproblem 4 Concluding remarks TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

37 Outline Variational characterization of eigenvalue problems Overdamped Problems 1 Two examples 2 Variational characterization of eigenvalue problems Overdamped Problems Nonoverdamped problems 3 An unsymmetric linear eigenproblem 4 Concluding remarks TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

38 Overdamped Problems Variational characterization for linear eigenproblems Let A : H H a bounded linear and self-adjoint operator in a Hilbert space H. Then those eigenvalues λ 1 λ 2... above the essential spectrum of A (if there are any) can be characterized by three fundamental variational principles, TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

39 Overdamped Problems Variational characterization for linear eigenproblems Let A : H H a bounded linear and self-adjoint operator in a Hilbert space H. Then those eigenvalues λ 1 λ 2... above the essential spectrum of A (if there are any) can be characterized by three fundamental variational principles,by Rayleigh s principle λ n = max{r(x) : x, x i = 0, i = 1,..., n 1}, R(x) := Ax, x x, x. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

40 Overdamped Problems Variational characterization for linear eigenproblems Let A : H H a bounded linear and self-adjoint operator in a Hilbert space H. Then those eigenvalues λ 1 λ 2... above the essential spectrum of A (if there are any) can be characterized by three fundamental variational principles,by Rayleigh s principle λ n = max{r(x) : x, x i = 0, i = 1,..., n 1}, R(x) := the maxmin characterization due to Poincaré λ n = max dim V =n min R(x). x V, x 0 Ax, x x, x. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

41 Overdamped Problems Variational characterization for linear eigenproblems Let A : H H a bounded linear and self-adjoint operator in a Hilbert space H. Then those eigenvalues λ 1 λ 2... above the essential spectrum of A (if there are any) can be characterized by three fundamental variational principles,by Rayleigh s principle λ n = max{r(x) : x, x i = 0, i = 1,..., n 1}, R(x) := the maxmin characterization due to Poincaré λ n = max dim V =n min R(x). x V, x 0 the minmax characterization due to Courant, Fischer, and Weyl. λ n = min dim V =n 1 max R(x). x V,x 0 Ax, x x, x. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

42 Variational characterizations are very powerful tools when studying self-adjoint linear operators on a Hilbert space H. Bounds for eigenvalues, comparison theorems, interlacing results and monotonicity of eigenvalues can be proved easily with these characterizations, to name just a few. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52 Variational characterization of eigenvalue problems Overdamped Problems Variational characterization for linear eigenproblems Let A : H H a bounded linear and self-adjoint operator in a Hilbert space H. Then those eigenvalues λ 1 λ 2... above the essential spectrum of A (if there are any) can be characterized by three fundamental variational principles,by Rayleigh s principle λ n = max{r(x) : x, x i = 0, i = 1,..., n 1}, R(x) := the maxmin characterization due to Poincaré λ n = max dim V =n min R(x). x V, x 0 the minmax characterization due to Courant, Fischer, and Weyl. λ n = min dim V =n 1 max R(x). x V,x 0 Ax, x x, x.

43 Rayleigh functional Overdamped Problems 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 (3) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

44 Rayleigh functional Overdamped Problems 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 (3) Then equation (3) 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

45 Rayleigh functional Overdamped Problems 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 (3) Then equation (3) 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. We further assume that for every x D, x 0 and λ J, λ p(x) it holds that f (λ, x)(λ p(x)) > 0 which generalizes the definiteness of the operator B for the generalized linear eigenproblem T (λ) := λb A. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

46 Overdamped problems 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

47 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 where M, C and K are symmetric and positive definite matrices. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

48 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 where M, C and K are symmetric and positive definite matrices. α = 0 all eigenvalues on imaginary axis TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

49 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

50 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

51 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

52 Overdamped problems 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. This notation is motivated by the finite dimensional quadratic eigenvalue problem T (λ)x = λ 2 Mx + λαcx + Kx = 0 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

53 Overdamped Problems Quadratic overdamped problems For quadratic overdamped systems the two solutions p ± (x) = 1 ( ) α Cx, x ± α 2 2 Cx, x 2 4 Mx, x Kx, x / Mx, x. of the quadratic equation T (λ)x, x = λ 2 Mx, x + λα Cx, x + Kx, x = 0 (4) are real, and they satisfy sup x 0 p (x) < inf x 0 p + (x). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

54 Overdamped Problems Quadratic overdamped problems For quadratic overdamped systems the two solutions p ± (x) = 1 ( ) α Cx, x ± α 2 2 Cx, x 2 4 Mx, x Kx, x / Mx, x. of the quadratic equation T (λ)x, x = λ 2 Mx, x + λα Cx, x + Kx, x = 0 (4) are real, and they satisfy sup x 0 p (x) < inf x 0 p + (x). Hence, equation (4) 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

55 Rayleigh s principle Overdamped Problems 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) (5) 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

56 Overdamped Problems Rayleigh s principle (Hadeler 1967,1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

57 Overdamped Problems Rayleigh s principle (Hadeler 1967,1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. Let T ( ) be continuously differentiable and suppose that T (λ)x, x λ > 0 for ever x 0. λ=p(x) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

58 Overdamped Problems Rayleigh s principle (Hadeler 1967,1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. Let T ( ) be continuously differentiable and suppose that T (λ)x, x λ > 0 for ever x 0. λ=p(x) 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..., where each eigenvalue is counted according to its multiplicity. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

59 Overdamped Problems Rayleigh s principle (Hadeler 1967,1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. Let T ( ) be continuously differentiable and suppose that T (λ)x, x λ > 0 for ever x 0. λ=p(x) 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..., where each eigenvalue is counted according to its multiplicity. The corresponding eigenvectors x 1, x 2,... can be chosen orthonormally with respect to the generalized scalar product (5), and the eigenvalues can be determined recurrently by λ n = min{p(x) : [x, x i ] = 0, i = 1,..., n 1, x 0}. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

60 Minmax principle 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

61 Minmax principle 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

62 Minmax principle 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

63 Overdamped Problems Minmax and Maxmin principle (Hadeler 1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

64 Overdamped Problems Minmax and Maxmin principle (Hadeler 1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. Let T ( ) be continuously differentiable and suppose that T (λ)x, x λ > 0 for ever x 0. λ=p(x) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

65 Overdamped Problems Minmax and Maxmin principle (Hadeler 1968) Let T (λ) : H H, λ J be a family of linear self-adjoint and bounded operators such that (1) is over-damped, and assume that for λ J there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact. Let T ( ) be continuously differentiable and suppose that T (λ)x, x λ > 0 for ever x 0. λ=p(x) Let the eigenvalues λ n of T (λ)x = 0 be numbered in non-decreasing order regarding their multiplicities. Then they can be characterized by the following two variational principles λ n = min dim V =n = max dim V =n 1 max p(x) x V, x 0 min p(x). x V, x 0 TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

66 Overdamped Problems Example 1: Quantum dot problem For the quantum dot problem the family of operators T (λ) satisfies the general conditions of the variational characterizations. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

67 Overdamped Problems Example 1: Quantum dot problem For the quantum dot problem the family of operators T (λ) satisfies the general conditions of the variational characterizations. For x 0 it holds that a(x, x, 0) > 0, b(x, x) > 0 and λ a(x, x, λ) is monotonically decreasing for λ 0. Hence, f (λ, x) = T (λ)x, x = λb(x, x) a(x, x, λ) = 0 has exactly one positive solution p(x), and f (λ, x) λ > 0. λ=p(x) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

68 Overdamped Problems Example 1: Quantum dot problem For the quantum dot problem the family of operators T (λ) satisfies the general conditions of the variational characterizations. For x 0 it holds that a(x, x, 0) > 0, b(x, x) > 0 and λ a(x, x, λ) is monotonically decreasing for λ 0. Hence, f (λ, x) = T (λ)x, x = λb(x, x) a(x, x, λ) = 0 has exactly one positive solution p(x), and f (λ, x) λ > 0. λ=p(x) Thus, the quantum dot problem has a countable number of non-negative eigenvalues which allow for all three variational characterizations. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

69 Overdamped Problems Example 2: Fluid-solid vibration For the rational eigenproblem governing free vibrations the family of operators T (λ) satisfies the general conditions of the variational characterizations in every interval J n := (σ n 1, σ n ). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

70 Overdamped Problems Example 2: Fluid-solid vibration For the rational eigenproblem governing free vibrations the family of operators T (λ) satisfies the general conditions of the variational characterizations in every interval J n := (σ n 1, σ n ). f (λ, p) := T (λ)p, p = K f p, p + λ M f p, p + n=1 λ σ n λ u n, Cp 2 is monotonically increasing, such that f (λ, p) = 0 has at most one solution in J n, but the Rayleigh functional is not defined on the entire space. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

71 Outline Variational characterization of eigenvalue problems Nonoverdamped problems 1 Two examples 2 Variational characterization of eigenvalue problems Overdamped Problems Nonoverdamped problems 3 An unsymmetric linear eigenproblem 4 Concluding remarks TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

72 Non-overdamped problems Nonoverdamped problems For nonoverdamped eigenproblems (i.e. D(p) H \ {0}) the natural ordering to call the smallest eigenvalue the first one, the second smallest the second one, etc., is not appropriate. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

73 Non-overdamped problems Nonoverdamped problems For nonoverdamped eigenproblems (i.e. D(p) H \ {0}) the natural ordering to call the smallest eigenvalue the first one, the second smallest 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 smallest eigenvalue of A. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

74 Non-overdamped problems Nonoverdamped problems For nonoverdamped eigenproblems (i.e. D(p) H \ {0}) the natural ordering to call the smallest eigenvalue the first one, the second smallest 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 smallest eigenvalue of A. Then all conditions are satisfied, p is the restriction of the Rayleigh quotient R A to D(p) := {x 0 : R A (x) J}, and inf x D(p) p(x) will not be an eigenvalue. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

75 Enumeration of eigenvalues Nonoverdamped 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 Variational Principles for Nonlinear EVPs Lausanne, August, / 52

76 Enumeration of eigenvalues Nonoverdamped 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). We assume that for every λ J it holds that the supremum of the essential spectrum of T (λ) is negative ( for instance: there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

77 Enumeration of eigenvalues Nonoverdamped 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). We assume that for every λ J it holds that the supremum of the essential spectrum of T (λ) is negative ( for instance: there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact). If λ J is an eigenvalue of T ( ) then there exists n N such that 0 = max dimv =n min x V, x 0 T (λ)x, x. x, x TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

78 Enumeration of eigenvalues Nonoverdamped 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). We assume that for every λ J it holds that the supremum of the essential spectrum of T (λ) is negative ( for instance: there exists ν(λ) > 0 such that T (λ) + ν(λ)i is compact). If λ J is an eigenvalue of T ( ) then there exists n N such that 0 = max dimv =n min x V, x 0 T (λ)x, x. x, x In this case we assign n to the eigenvalue λ of problem T (λ)x = 0 as its number and call λ an n-th eigenvalue of T ( ). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

79 Nonoverdamped problems Minmax characterization (V.&B.Werner 1982, V. 2010) Let T (λ), λ J be a family of linear self-adjoint and bounded operators on a Hilbert space H depending continuously on a parameter λ J where J is an open real (not necessarily bounded) interval. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

80 Nonoverdamped problems Minmax characterization (V.&B.Werner 1982, V. 2010) Let T (λ), λ J be a family of linear self-adjoint and bounded operators on a Hilbert space H depending continuously on a parameter λ J where J is an open real (not necessarily bounded) interval. Assume that for x 0 f (λ, x) := T (λ)x, x = 0 has at most one solution p(x) J, and let D be the domain of the Rayleigh functional p, (λ p(x))f (λ, x) > 0 for every x D and λ p(x), and that the supremum of the essential spectrum of T (λ) is negative for every λ J. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

81 Nonoverdamped problems Minmax characterization (V.&B.Werner 1982, V. 2010) Let T (λ), λ J be a family of linear self-adjoint and bounded operators on a Hilbert space H depending continuously on a parameter λ J where J is an open real (not necessarily bounded) interval. Assume that for x 0 f (λ, x) := T (λ)x, x = 0 has at most one solution p(x) J, and let D be the domain of the Rayleigh functional p, (λ p(x))f (λ, x) > 0 for every x D and λ p(x), and that the supremum of the essential spectrum of T (λ) is negative for every λ J. Then the nonlinear eigenvalue problem T (λ)x = 0 has at most a countable set of eigenvalues in J, and it holds that: If λ n J is an n-th eigenvalue then λ n = min dim V =n, V D sup x D V p(x). (6) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

82 Nonoverdamped problems Minmax characterization (V.&B.Werner 1982, V. 2010) Let T (λ), λ J be a family of linear self-adjoint and bounded operators on a Hilbert space H depending continuously on a parameter λ J where J is an open real (not necessarily bounded) interval. Assume that for x 0 f (λ, x) := T (λ)x, x = 0 has at most one solution p(x) J, and let D be the domain of the Rayleigh functional p, (λ p(x))f (λ, x) > 0 for every x D and λ p(x), and that the supremum of the essential spectrum of T (λ) is negative for every λ J. Then the nonlinear eigenvalue problem T (λ)x = 0 has at most a countable set of eigenvalues in J, and it holds that: If λ n J is an n-th eigenvalue then If conversely λ n = λ n = min dim V =n, V D inf dim V =n, V D sup x D V sup x D V p(x). (6) p(x) J then λ n is an n-th eigenvalue of T (λ)x = 0 and (6) holds. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

83 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Step 1 (technical): Let λ J, and assume that V is a finite dimensional subspace of H such that V D. Then it holds that < < λ = sup p(x) min T (λ)x, x = > x V D(p) x V 0 (7) > TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

84 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Step 1 (technical): Let λ J, and assume that V is a finite dimensional subspace of H such that V D. Then it holds that < < λ = sup p(x) min T (λ)x, x = > x V D(p) x V 0 (7) > Step 2: If λ n is an n-th eigenvalue, then µ n (λ n ) = 0, and µ n (λ n ) = max dim V =n min T (λ n)x, x = x V, x =1 min T (λ n )x, x. x V, x =1 TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

85 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Step 1 (technical): Let λ J, and assume that V is a finite dimensional subspace of H such that V D. Then it holds that < < λ = sup p(x) min T (λ)x, x = > x V D(p) x V 0 (7) > Step 2: If λ n is an n-th eigenvalue, then µ n (λ n ) = 0, and µ n (λ n ) = max dim V =n min T (λ n)x, x = x V, x =1 min T (λ n )x, x. x V, x =1 Hence, min x V, x =1 T (λ n )x, x 0 for every V with dim V = n, and (7) implies sup x V D p(x) λ n = sup x V D p(x). Hence, λ n is a minmax value of p. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

86 Nonoverdamped problems Theorem (minmax for extreme eigenvalues) Assume that the conditions of the minmax characterization hold and that inf p(x) J. x D TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

87 Nonoverdamped problems Theorem (minmax for extreme eigenvalues) Assume that the conditions of the minmax characterization hold and that inf p(x) J. x D If λ n J for some n N then every V H j with V D(p) and λ j = sup x V D(p) p(x) is contained in D {0}, and the characterization (6) can be replaced by λ j = min dim V = j V D {0} max p(v), j = 1,..., n. (8) v V, x 0 TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

88 Nonoverdamped problems Maxmin characterization (V. 2003) Assume that the conditions of the minmax characterization are satisfied. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

89 Nonoverdamped problems Maxmin characterization (V. 2003) Assume that the conditions of the minmax characterization are satisfied. If there is an n-th eigenvalue λ n J of T (λ)x = 0, then λ n = max V H n 1 V D inf p(v), v V D and the maximum is attained by W := span{u 1,..., u n 1 } where u j denotes an eigenvector corresponding to the j-largest eigenvalue µ j (λ n ) of T (λ n ). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

90 Nonoverdamped problems Maxmin characterization (V. 2003) Assume that the conditions of the minmax characterization are satisfied. If there is an n-th eigenvalue λ n J of T (λ)x = 0, then λ n = max V H n 1 V D inf p(v), v V D and the maximum is attained by W := span{u 1,..., u n 1 } where u j denotes an eigenvector corresponding to the j-largest eigenvalue µ j (λ n ) of T (λ n ). Proof takes advantage of the following Lemma: Let λ J, and let V be a finite dimensional subspace of H such that V D. Then it holds that λ < = > inf x V D(p) p(x) max x V, x =1 T (λ)x, x < = > 0 TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

91 Nonoverdamped problems Rayleigh s principle for nonoverdamped problems Under the conditions of the minmax characterization assume that J contains n 1 eigenvalues λ 1 λ n (where λ i is an ith eigenvalue) with corresponding [, ] orthogonal eigenvectors x 1,..., x n. If there exists x D with [x i, x] = 0 for i = 1,..., n then J contains an (n + 1)th eigenvalue, and λ n+1 = inf{p(x) : [x j, x] = 0, i = 1,..., n}. (9) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

92 Nonoverdamped problems Rayleigh s principle for nonoverdamped problems Under the conditions of the minmax characterization assume that J contains n 1 eigenvalues λ 1 λ n (where λ i is an ith eigenvalue) with corresponding [, ] orthogonal eigenvectors x 1,..., x n. If there exists x D with [x i, x] = 0 for i = 1,..., n then J contains an (n + 1)th eigenvalue, and λ n+1 = inf{p(x) : [x j, x] = 0, i = 1,..., n}. (9) The generalized scalar product (5) has to be modified according to T (p(x)) T (p(y)) x, y, if p(x) p(y) [x, y] := p(x) p(y) x, y, if p(x) = p(y) if T is not differentiable at p(x) = p(y). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

93 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Let {y k } be a minimizing sequence of (9) such that y k = 1, [x j, y k ] = 0, j = 1,..., n, p(y k ) λ n+1. TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

94 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Let {y k } be a minimizing sequence of (9) such that y k = 1, [x j, y k ] = 0, j = 1,..., n, p(y k ) λ n+1. There exist unique n ỹ k = y k + c kj x j such that T (λ n+1 )x j, ỹ k = 0, j = 1,..., n, and it holds that y k ỹ k 0 and p(ỹ k ) λ n+1. j=1 TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

95 Sketch of proof Variational characterization of eigenvalue problems Nonoverdamped problems Let {y k } be a minimizing sequence of (9) such that y k = 1, [x j, y k ] = 0, j = 1,..., n, p(y k ) λ n+1. There exist unique n ỹ k = y k + c kj x j such that T (λ n+1 )x j, ỹ k = 0, j = 1,..., n, and it holds that y k ỹ k 0 and p(ỹ k ) λ n+1. j=1 For V k := span{x 1,..., x n, ỹ k } it can be shown that lim sup{p(z) : z V k} = λ n+1, k and it follows from the minmax characterization that λ n+1 is an eigenvalue of T ( ). TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

96 Further literature Nonoverdamped problems Minmax characterizations for non-overdamped nonlinear eigenvalue problems were proved (independently from our work) by Barston (1974) for some extreme eigenvalues of finite dimensional quadratic eigenproblems TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

97 Further literature Nonoverdamped problems Minmax characterizations for non-overdamped nonlinear eigenvalue problems were proved (independently from our work) by Barston (1974) for some extreme eigenvalues of finite dimensional quadratic eigenproblems and for the infinite dimensional case by Griniv & Meln ik (1996): T (λ) = A(λ) I, A(λ) compact TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

98 Further literature Nonoverdamped problems Minmax characterizations for non-overdamped nonlinear eigenvalue problems were proved (independently from our work) by Barston (1974) for some extreme eigenvalues of finite dimensional quadratic eigenproblems and for the infinite dimensional case by Griniv & Meln ik (1996): T (λ) = A(λ) I, A(λ) compact Binding, Eschwé & H. Langer (2000) TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52

99 Further literature Nonoverdamped problems Minmax characterizations for non-overdamped nonlinear eigenvalue problems were proved (independently from our work) by Barston (1974) for some extreme eigenvalues of finite dimensional quadratic eigenproblems and for the infinite dimensional case by Griniv & Meln ik (1996): T (λ) = A(λ) I, A(λ) compact Binding, Eschwé & H. Langer (2000) Eschwé & M. Langer (2004) unbounded operators TUHH Heinrich Voss Variational Principles for Nonlinear EVPs Lausanne, August, / 52