We address a conceptual issue of reconciling the traditional canonical quantization framework of quantum theory with the spatially restricted quantum dynamics and the related spectral problems for confined and global observables of the quantum system.
MotivationModern technologies enable one to enslave individual quantum particles in various traps for long time intervals. Manipulations with nanostructures involve a fine tuned control of quantum wells shape and depth that has a decisive influence on blocking or enabling various transport (in fact, tunelling) phenomena. In all those cases a fairly pragmatic usage of the traditional quantum mechanical formalism shows an acceptable explanatory/predictive power.A proliferation of papers on various aspects of the infinite potential well, [1] -[7], and on sophisticated "exercises in exact quantization" on half-line, [8] motivates our renewed interest in reconciling the canonical quantization principles with the sole analysis of well posed spectral problems for the Hamilton operator, with Dirichlet or Neumann boundary data. The latter, purely spectral (spectroscopy oriented) attitude *
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