Schrödinger-Operators with Singular Interactions

ArticleinJournal of Mathematical Analysis and Applications 184(1):112–139 · May 1994with 136 Reads 
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    In this paper, the free motion of a particle on a manifold that consists of a one‐dimensional and a two‐dimensional part connected in one point is discussed. The class of admissible Hamiltonians is found using the theory of self‐adjoint extensions. Particular attention is paid to those Hamiltonians that allow the particle to pass through the point singularity; the reflection coefficient and other quantities characterizing scattering on the connection point are calculated. A possible application is also discussed.
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    The authors consider the motion of a charged quantum particle on a loop with two external leads which is placed into an electrostatic field. The loop Hamiltonian is chosen in the simplest possible way; in order to join it to the free Hamiltonians describing the leads, they employ a method based on self-adjoint extensions. Under a symmetry requirement, the resulting full Hamiltonian contains four free parameters; each junction is characterised by a pair of them. The system under consideration represents a model of metallic or semiconductor structure that can be fabricated by presently available technologies. Assuming the ballistic regime for electrons in such a structure, the authors calculate the resistance dependence on intensity of the external field. The results suggest the possibility of constructing quantum interference transistors whose size and switching voltage would be much smaller than in current microchips.
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    Diffusion and wave equations together with appropriate initial condition(s) are rewritten as integrodifferential equations with time derivatives replaced by convolution with tα−1/Γ(α), α=1,2, respectively. Fractional diffusion and wave equations are obtained by letting α vary in (0,1) and (1,2), respectively. The corresponding Green’s functions are obtained in closed form for arbitrary space dimensions in terms of Fox functions and their properties are exhibited. In particular, it is shown that the Green’s function of fractional diffusion is a probability density.
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    Two new analytically solvable models of relativistic point interactions in one dimension (being natural extensions of the nonrelativistic , resp. , interactions) are considered. Their spectral properties in the case of finitely many point interactions as well as in the periodic case are fully analyzed. Moreover, we explicitly determine the spectrum in the case of independent, identically distributed random coupling constants and derive the analog of the Saxon and Hutner conjecture concerning gaps in the energy spectrum of such systems.
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    We treat a free spinless quantum particle moving on a configuration manifold which consists of two identical parts connected in one point. Most attention is paid to the three-dimensional case when these parts are halfspaces with Neumann condition on the boundary; we also discuss briefly a more general boundary conditions. The class of admissible Hamiltonians is constructed by means of the theory of self-adjoint extensions. Among them, particularly important is a two-parameter family whose elements are invariant with respect to exchange of the halfspaces; we compute the transmission coefficient for each of these extensions. We discuss also the motion on two planes considered in our recent paper, obtaining another characterization of the admissible Hamiltonians. In conclusion, the two situations are compared as models for point-contact spectroscopical experiments in thin metal films.
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    We study the free motion of a particle on the manifold which consists of two planes connected at one point. The four-parameter family of admissible Hamiltonians is constructed by self-adjoint extensions of the free Hamiltonian with the singular point removed. The probability of penetration between the two parts of the configuration manifold is calculated. The results can be used as a model for quantum point-contact spectroscopy.
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    We study the Schrödinger Hamiltonian corresponding to a finite number of δ’-interactions supported by concentric spheres. In particular we discuss the self-adjointness of the Hamiltonian, derive the resolvent equation and study the spectral properties. Si studia l’hamiltoniana di Schrödinger che corrisponde a un numero fïnito di interazioni δ’ sostenute da sfere concentriche. In particolare si discute la capacità dell’Hamiltoniana di essere autoaggiunta, si deriva l’equazione risolutiva e si studiano le proprietà spettrali. Мы исследуем Гамильт ониан Шредингера, соответствующий конечному числу σ′-вз аимодействий, опираю щихся на концентрические сфе ры. В частности, мы обсужда ем самосопряженный Гамильтониан, выводи м уравнение резольвенты и исслед уем спектральные сво йства.
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    We sketch here two mathematical models intended to describe the point-contact spectroscopical experiments. A new item is added to the list of recently discovered applications of the self-adjoint extensions theory.
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    We consider the free motion of a quantum particle on the graph consisting of three half-lines whose ends are connected. It is shown that the time evolution can be described by a Hamiltonian and the class of all admissible Hamiltonians is constructed using the theory of self-adjoint extensions. Three subclasses are discussed in detail: (a) the one-parameter family of Hamiltonians whose domains contain functions continuous at the junction, (b) the wider four-parameter family with the wavefunctions continuous between two branches of the graph only, (c) the Hamiltonian invariant under permutations of the branches. For the class (c), generalization to the graphs consisting of n half-lines is given. The scattering problem of such a branching graph is also discussed.
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    Under certain conditions on the potential a one-dimensional Schrödinger operator has a unique bound state in the limit of weak coupling while under other conditions no bound state is present in this limit. This question is investigated for potentials obeying ∫ (1 + |χ|) | V(χ)|dχ<∞. An asymptotic formula for the bound state is proven.
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    The free Dirac operator defined on composite one‐dimensional structures consisting of finitely many half‐lines and intervals is investigated. The influence of the connection points between the constituents is modeled by transition conditions for the wave functions or equivalently by different self‐adjoint extensions of the Dirac operator. General relations between the parameters of the extensions and the eigenvaluesr e s p. the scattering coefficients are derived and then applied to the cases of a bundle of half‐lines, a point defect, a branching line, and an eye‐shaped structure.
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    Singular perturbations of — Δ in L²(R³) supported by points, regular curves and regular surfaces are considered. Using a renormalization technique the corresponding quadratic forms are constructed and a complete characterization of the domain and the action of the operators is given, together with explicit expressions for the resolvent. © 1990, Research Institute forMathematical Sciences. All rights reserved.