Abstract
The problem of approximating the discrete spectra of families of self-adjoint operators that are merely strongly continuous is addressed. It is well-known that the spectrum need not vary continuously (as a set) under strong perturbations. However, it is shown that under an additional compactness assumption the spectrum does vary continuously, and a family of symmetric finite-dimensional approximations is constructed. An important feature of these approximations is that they are valid for the entire family uniformly. An application of this result to the study of plasma instabilities is illustrated.
Highlights
We present a method for obtaining finite-dimensional approximations of the discrete spectrum of families of self-adjoint operators
We are interested in operators that decompose into a system of two coupled Schrödinger operators with opposite signs (see (1.1) below)
One of the main driving forces behind the study of linear operators in the 20th century was the development of quantum mechanics
Summary
We present a method for obtaining finite-dimensional approximations of the discrete spectrum of families of self-adjoint operators. Acting in an appropriate subspace of L2(Rd ) ⊕ L2(Rd ), where {Kλ}λ∈[0,1] is a bounded, symmetric and strongly continuous family Is it possible to construct explicit finitedimensional self-adjoint approximations of Mλ whose spectrum in compact subsets of (−1, 1) converges to that of Mλ uniformly in λ?. (v) Compactification of the resolvent: There exist holomorphic forms {wλ±}λ∈D of type (a) and associated operators {W±λ }λ∈D of type (B) such that for λ ∈ [0, 1], W±λ are self-adjoint and non-negative.
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