Differentiable Perturbation of Unbounded Operators

If A(t) is a C1,α-curve of unbounded self-adjoint operators with compact resolvents and common domain of definition, then the eigenvalues can be parameterized C1 in t. If A is C∞ then the eigenvalues can be parameterized twice differentiable. Theorem. Let t 7→ A(t) for t ∈ R be a curve of unbounded self-adjoint operators in a Hilbert space with common domain of definition and with compact resolvent. (A) If A(t) is real analytic in t ∈ R, then the eigenvalues and the eigenvectors of A(t) may be parameterized real analytically in t. (B) If A(t) is C∞ in t ∈ R and if no two unequal continuously parameterized eigenvalues meet of infinite order at any t ∈ R, then the eigenvalues and the eigenvectors can be parameterized smoothly in t, on the whole parameter domain. (C) If A(t) is C∞ in t ∈ R, then the eigenvalues of A(t) may be parameterized twice differentiable in t. (D) If A(t) is C for some α > 0 in t ∈ R then the eigenvalues of A(t) may be parameterized in a C way in t. Part (A) is due to Rellich [10] in 1940, see also [2] and [6], VII, 3.9. Part (B) has been proved in [1], 7.8, see also [8], 50.16, in 1997; there we gave also a different proof of (A). The purpose of this paper is to prove parts (C) and (D). Both results cannot be improved to obtain for the eigenvalues C with some β > 0 by the first example below. In our proof of (D) the assumption C cannot be weakened to C, see the second example. For finite dimensional Hilbert spaces part (D) has been proved under the assumption of C by Rellich [11], with a small inaccuracy in the auxiliary theorem on p. 48: Condition (4) has must be more restrictive, otherwise the induction argument on p. 50 is not valid, since the argument at the end of the proof on p. 52 relies an the fact that all values coincide at the point in question. A proof is also in [6], II, 6.8. We need a strengthened version of this result, thus our proof covers it also. We thank T. and M. Hoffmann-Ostenhof and T. Kappeler for their interest and hints. 2000 Mathematics Subject Classification. Primary 26C10..