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)
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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 Fefferman and D. Phong}, unpublished.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)
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