We construct a new type of quantum walks on simplicial complexes as a natural extension of the well-known Szegedy walk on graphs. One can numerically observe that our proposing quantum walks possess linear spreading and localization as in the case of the Grover walk on lattices. Moreover, our numerical simulation suggests that localization of our quantum walks reflect not only topological but also geometric structures. On the other hand, our proposing quantum walk contains an intrinsic problem concerning exhibition of nontrivial behavior, which is not seen in typical quantum walks such as Grover walks on graphs.
We discuss the description of eigenspace of a quantum walk model U with an associating linear operator T in abstract settings of quantum walk including the Szegedy walk on graphs. In particular, we provide the spectral mapping theorem of U without the spectral decomposition of T. Arguments in this direction reveal the eigenspaces of U characterized by the generalized kernels of linear operators given by T.
We show that discrete-time quantum walks on the line, Z, behave as "the quantum tunneling." In particular, quantum walkers can tunnel through a double-well with the transmission probability 1 under a mild condition. This is a property of quantum walks which cannot be seen on classical random walks, and is different from both linear spreadings and localizations.
The ordinary (holomorphic) N = 2 Wess-Zumino model in supersymmetric quantum mechanics is extended to the case where the superpotential V(z) is a meromorphic function on CU{ CO}. The extended model is analyzed in a mathematically rigorous way. Self-adjoint extensions and the essential self-adjointness of the supercharges are discussed. The supersymmetric Hamiltonian defined by one of the self-adjoint extensions of the supercharges has no fermionic zero-energy states ("vanishing theorem"). It is proven that if V(z) has only one pole at z = 0 in @, then the supercharges are essentially self-adjoint on Cc (IX*\ {O};c") . The special case where V(z) = ;lz-P (#+I&Q=\{O]) is analyzed in detail to prove the following facts: (i) the number of the bosonic zero-energy ground state(s) is equal top -1; (ii) the supercharges are not Fredholm.
It is investigated that the structure of the kernel of the Dirac–Weyl operator D of a charged particle in the magnetic-field B=B0+B1, given by the sum of a strongly singular magnetic field B0(⋅)=Σνγνδ(⋅−aν) with some singular points aν and a magnetic-field B1 with a bounded support. Here the magnetic field B1 may have some singular points with the order of the singularity less than 2. At a glance, it seems that, following “Aharonov–Casher Theorem” [Phys. Rev. A 19, 2461 (1979)], the dimension of the kernel of D, dim ker D, is a function of one variable of the total magnetic flux (=Σνγν+∫R2B1dxdy) of B. However, since the influence of the strongly singular points works, dim ker D indeed is a function of several variables of the total magnetic flux and each of γν’s.
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