Sharpening the norm bound in the subspace perturbation theory

Abstract

Let A be a self-adjoint operator on a Hilbert space H. Assume that σ is an isolated component of the spectrum of A, i.e. dist(σ,Σ)=d>0 where Σ=spec(A)\σ. Suppose that V is a bounded self-adjoint operator on H such that ||V||<d/2 and let L=A+V. Denote by P the spectral projection of A associated with the spectral set σ and let Q be the spectral projection of L corresponding to the closed ||V||-neighborhood of σ. We prove a bound of the form arcsin(||P-Q||)\leq M(||V||/d), M: [0,1/2)-->R^+, that is essentially stronger than the previously known estimates for ||P-Q||. In particular, the bound obtained ensures that ||P-Q||<1 and, thus, that the spectral subspaces Ran(P) and Ran(Q) are in the acute-angle case whenever ||V||<cd with c=0.454169... (the precise expression for c is also given). Our proof of the above results is based on using the triangle inequality for the maximal angle between subspaces and on employing the a priori generic \sin2θestimate for the variation of a spectral subspace. As an example, the boundedly perturbed quantum harmonic oscillator is discussed.

Some typos fixed; minor changes in the text; a new reference added

References in corpus (3)

• Bounds on variation of spectral subspaces under J-self-adjoint perturbations
• Bounds on the spectrum and reducing subspaces of a J-self-adjoint operator
• The a priori Tan Theta Theorem for spectral subspaces

Cited by in corpus (9)

• Notes on the sin2Θ theorem
• Quantum speed limits for time evolution of a system subspace
• Notes on the subspace perturbation problem for off-diagonal perturbations
• On an estimate in the subspace perturbation problem
• Optimal bounds on the speed of subspace evolution
• An alternative proof of the a priori $\tan \Theta$ Theorem
• On applying the subspace perturbation theory to few-body Hamiltonians
• Solvability of the operator Riccati equation in the Feshbach case
• On an integral formula for Fredholm determinants related to pairs of spectral projections