5,446 research outputs found
A Pedestrian Introduction to Gamow Vectors
The Gamow vector description of resonances is compared with the S-matrix and
the Green function descriptions using the example of the square barrier
potential. By imposing different boundary conditions on the time independent
Schrodinger equation, we obtain either eigenvectors corresponding to real
eigenvalues and the physical spectrum or eigenvectors corresponding to complex
eigenvalues (Gamow vectors) and the resonance spectrum. We show that the poles
of the S matrix are the same as the poles of the Green function and are the
complex eigenvalues of the Schrodinger equation subject to a purely outgoing
boundary condition. The intrinsic time asymmetry of the purely outgoing
boundary condition is discussed. Finally, we show that the probability of
detecting the decay within a shell around the origin of the decaying state
follows an exponential law if the Gamow vector (resonance) contribution to this
probability is the only contribution that is taken into account.Comment: 25 RevTex pages, 3 figure
Endpoint Sobolev continuity of the fractional maximal function in higher dimensions
We establish continuity mapping properties of the non-centered fractional maximal operator in the endpoint input space for in the cases for which its boundedness is known. More precisely, we prove that for the map is continuous from to for if is radial and for for general . The results for extend to the centered counterpart . Moreover, if , we show that the conjectured boundedness of that map for implies its continuity
The role of the rigged Hilbert space in Quantum Mechanics
There is compelling evidence that, when continuous spectrum is present, the
natural mathematical setting for Quantum Mechanics is the rigged Hilbert space
rather than just the Hilbert space. In particular, Dirac's bra-ket formalism is
fully implemented by the rigged Hilbert space rather than just by the Hilbert
space. In this paper, we provide a pedestrian introduction to the role the
rigged Hilbert space plays in Quantum Mechanics, by way of a simple, exactly
solvable example. The procedure will be constructive and based on a recent
publication. We also provide a thorough discussion on the physical significance
of the rigged Hilbert space.Comment: 29 pages, 2 figures; a pedestrian introduction to the rigged Hilbert
spac
F-transforms for the definition of contextual fuzzy partitions
Fuzzy partitions can be defined in many different ways, but usually, it is done taking into account the whole universe. In this paper, we present a method to define fuzzy partitions according to those elements in the universe holding certain fuzzy attribute. Specifically, we show how to define those fuzzy partitions by means of F-transforms.Universidad de Málaga. Campus de Excelencia Internacional AndalucÃa Tech
This work has been partially supported by the Spanish Ministry of Science by the projects TIN15-70266-C2-P-1 and TIN2016-76653-
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