Some quantum operators with discrete spectrum but classically continuous spectrum
This paper gives some examples for the breakdown of the following rule of thumb in quantum mechanics: If the volume (i.e. the Lebesgue measure) of the set \(\{(p,q)\in {\mathbb{R}}^{2\nu}| p^ 2+V(q)\leq E\}\) for some \(E<\infty\) is infinite then the classical intuition says that the spectrum of the Hamiltonian \(H=-\Delta +V(x)\) in \(L^ 2({\mathbb{R}}^{\nu})\) is not purely discrete. The author considers the two dimensional Hamiltonian \(H_ 1=-\partial^ 2/\partial x^ 2- \partial^ 2/\partial y^ 2+x^ 2y^ 2\) which is closely related to the Dirichlet operator \(H_ 2=-\Delta_ D^{\Omega}\) with zero boundary conditions on \(\Omega =\{(x,y)\in {\mathbb{R}}^ 2| | xy|\leq 1\}.\) He gives five proofs that \(H_ 2\) has discrete spectrum, where three of them work directly for \(H_ 1\). The different proofs have different virtues and use different reasoning. The arguments run from estimates for the zero point harmonic oscillator, Dirichlet-Neumann bracketing, path integral representations and a ''sliced bread'' inequality, to a theorem of Feffermann and Phong which estimates the number of eigenvalues by the number of cubes \(\Delta_ j^{\lambda}\) centered at \(\lambda^{- {1\over2}}j\) and of side \(\lambda^{-{1\over2}}\) with the property \(\max\{V(x)|\quad x\in\Delta_ j^{\lambda}\}\leq b\lambda\) (for suitable b).
scientific article; zbMATH DE number 3871879