×

zbMATH — the first resource for mathematics

Some quantum operators with discrete spectrum but classically continuous spectrum. (English) Zbl 0547.35039
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).
Reviewer: H.Cycon

MSC:
35J10 Schrödinger operator, Schrödinger equation
35P15 Estimates of eigenvalues in context of PDEs
47B25 Linear symmetric and selfadjoint operators (unbounded)
PDF BibTeX Cite
Full Text: DOI
References:
[1] Fefferman, C.; Phong, D., Commun. pure appl. math., 34, 285-331, (1981)
[2] {\scC. Fefferman and D. Phong}, unpublished.
[3] {\scJ. Goldstone and R. Jackiw}, private communication.
[4] Reed, M.; Simon, B., ()
[5] Simon, B., Adv. math., 30, 268-281, (1978)
[6] Simon, B., ()
[7] {\scB. Simon}, J. Fune. Anal., to appear.
[8] Temple, G., (), 276-293
[9] Yajima, K., J. math. soc. Japan, 29, 729-743, (1977)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.