Abstract
Non-standard Schrödinger operators have proved to be a powerful tool for modelling a wide variety of physical phenomena, cfr. [1], [2], [3], [4] and references given therein (a non-standard Schrödinger operator H is, by definition, a self-adjoint operator on L 2(ℝd, dx) (dx denotes the Lebesgue measure) with following properties:
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(i)
there exists a closed subset Γ of ℝd with Lebesgue measure zero such that C ∞0 (ℝd\Γ), the space of infinitely differentiable functions with compact support away from Γ, is in the domain D(H) of H,
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(ii)
H ≡ −∆ on C ∞0 (ℝd\Γ) where −∆ denotes the free Hamiltonian (cfr. [5], §IX.7) but
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(iii)
H ≠ −∆).
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Brasche, J.F., Karwowski, W. (1990). On Boundary Theory for Schrödinger Operators and Stochastic Processes. In: Exner, P., Neidhardt, H. (eds) Order,Disorder and Chaos in Quantum Systems. Operator Theory: Advances and Applications, vol 46. Birkhäuser Basel. https://doi.org/10.1007/978-3-0348-7306-2_21
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DOI: https://doi.org/10.1007/978-3-0348-7306-2_21
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