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By Lefschetz S.

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Low Dimensional Topology

During this quantity, that's devoted to H. Seifert, are papers in keeping with talks given on the Isle of Thorns convention on low dimensional topology held in 1982.

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The energy eigenvalues, for the discrete or the continuum spectra, are precisely the indexes labeling the family elements, the wavefunctions or kets. Thus, bound states belong to separable Hilbert spaces while scattering states require non-separable Hilbert spaces. There are nevertheless new problems in this continuum-label case: the summations ∞ i=1 used in the expansions become integrals. 3. It is possible to show that En is the cartesian topological product of E1 taken n times, and so that En+m = En × Em .

This is quite inequivalent to starting with a non-compact space (recall that going from Fourier series to Fourier integrals requires some extra “smoothing” assumptions). Or, alternatively, by choosing periodic boundary conditions we somehow manage to make the boundary to vanish. We shall come to this later. More recently, it has become fashionable to “compactify” non-compact spaces. For example: field theory supposes that all information is contained in the fields, which represent the degrees of freedom.

4). 2 Mathematics deals basically with sets and functions. The sequences previously introduced as countable subsets {pn } of a topological space S are better defined as functions p : N → S , n → pn , from the set of natural numbers N into S. Only to quote a famous fundamental case, two sets have the same power if there exists some bijective function between them. In this sense, set theory uses functions in “counting”. 18 that the power set P (S) of a set S is the set of all its subsets. For S finite with n points, P (S) will have 2n (> n) elements and for this reason P (S) is sometimes indicated by 2S .