Denjoy-carleman Differentiable Perturbation of Polynomials and Unbounded Operators

Let t 7→ A(t) for t ∈ T be a C -mapping with values unbounded operators with compact resolvents and common domain of definition which are self-adjoint or normal. Here C stands for C (real analytic), a quasianalytic or non-quasianalytic Denjoy-Carleman class, C∞, or a Hölder continuity class C. The parameter domain T is either R or R or an infinite dimensional convenient vector space. We prove and review results on C -dependence on t of the eigenvalues and eigenvectors of A(t). Theorem. Let t 7→ A(t) for t ∈ T be a parameterized family of unbounded operators in a Hilbert space H with common domain of definition and with compact resolvent. If t ∈ T = R and all A(t) are self-adjoint then the following holds: (A) If A(t) is real analytic in t ∈ R, then the eigenvalues and the eigenvectors of A(t) can be parameterized real analytically in t. (B) If A(t) is quasianalytic of class C in t ∈ R, then the eigenvalues and the eigenvectors of A(t) can be parameterized C in t. (C) If A(t) is non-quasianalytic of class C in t ∈ R and if no two different continuously parameterized eigenvalues (e.g., ordered by size) meet of infinite order at any t ∈ R, then the eigenvalues and the eigenvectors of A(t) can be parameterized C in t. (D) If A(t) is C in t ∈ R and if no two different continuously parameterized eigenvalues meet of infinite order at any t ∈ R, then the eigenvalues and the eigenvectors of A(t) can be parameterized C in t. (E) If A(t) is C in t ∈ R, then the eigenvalues of A(t) can be parameterized twice differentiably in t. (F) If A(t) is C in t ∈ R for some α > 0, then the eigenvalues of A(t) can be parameterized C in t. If t ∈ T = R and all A(t) are normal then the following holds: (G) If A(t) is real analytic in t ∈ R, then for each t0 ∈ R and for each eigenvalue z0 of A(t0) there exists N ∈ N>0 such that the eigenvalues near z0 of A(t0 ± s ) and their eigenvectors can be parameterized real analytically in s near s = 0. (H) If A(t) is quasianalytic of class C in t ∈ R, then for each t0 ∈ R and for each eigenvalue z0 of A(t0) there exists N ∈ N>0 such that the eigenvalues near z0 of A(t0 ± s ) and their eigenvectors can be parameterized C in s near s = 0. (I) If A(t) is non-quasianalytic of class C in t ∈ R, then for each t0 ∈ R and for each eigenvalue z0 of A(t0) at which no two of the different continuously Date: April 24, 2012. 2000 Mathematics Subject Classification. 26C10, 26E10, 47A55.