Eigensystem of an \(L ^{2}\)-perturbed harmonic oscillator is an unconditional basis
The authors investigate the spectrum and eigensystems of the perturbed operator \(L=L^0+B\) in \(L^2( \mathbb{R})\), where \(L^0=-\frac {d^2}{dx^2}+x^2\) is the harmonic oscillator, whereas \(B\) is a linear operator with certain properties, e.g., a multiplication operator \((Bf)(x)=b(x)f(x)\). Using that, the spectrum of \(L^0\) consists of the odd positive integers \(2k+1\), \(k\in \mathbb{Z}_+\), with the Hermite functions \(h_k\) as eigenfunctions. Define \(V=\{\phi \in L^2_{loc}(\mathbb R):(\|\phi h_k\|_2)_{k=0}^\infty \in c_0\}\), with \(\|\phi \|\) the (supremum) norm in \(c_0\) of \(\phi \in V\). It is shown that, for \(b\in V\), there is a positive integer \(n\) such that the spectrum of \(L\) is contained in \[ [-2n,2n]\times i[-Y,Y]\cup \bigcup_{k=n}^\infty D\left(2k+1,\frac{1}{16}\right), \] where \(Y=(8\|b\|+2\pi\|b\|^2)\), and each open disk \(D(2k+1,\frac{1}{16})\), \(k\geq n\), contains exactly one eigenvalue of \(L\). The main result is that the above localization of the spectrum leads to the spectral decomposition \[ f=S_*f+\sum _{k\geq n}P_kf,\;f\in L^2( \mathbb{R}), \] which converges unconditionally. It is shown that \(L_0^\infty \) as well as the weighted \(L^p\) spaces \(L(p,\alpha )\) with weight \((1+|x|^2)^{\alpha /2}\) and sufficiently large \(\alpha\) are subsets of \(V\). In a second case, the authors consider \(B\) to be a bounded operator with \(\|B\|<1\). Here, the spectrum is discrete with simple eigenvalues, and the system of corresponding eigenfunctions is an unconditional basis in \(L^2(\mathbb R)\). An example with \(\|B\|=1 \) is given where the eigenfunctions do not form a basis.
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