In this paper we prove and apply a spectral expansion theorem for Schwartz linear operators showing a Schwartz linearly independent eigenfamily - that is an ordered system of eigenvectors, indexed by some Euclidean space, which is Schwartz regular. This type of spectral expansion is a perfect analogous of the spectral expansion of self-adjoint operators on separable Hilbert spaces, when the eigenbasis is indexed by some real Euclidean space instead of the set of positive integers. Moreover, our theorem appears formally identical to the spectral expansion of a non-defective operator in the nite dimensional case, where continuous superpositions (feasibly and naturally dened) take the place of the nite sums. The Schwartz expansion we present is a possible rigorous and simply manageable mathematical model for the spectral expansions, in the continuous case, used frequently in Quantum Mechanics, since it appears in a form extremely similar to its current elementary formulations in Physics. The key point consists into bringing again at the center of the stage the eigenbases, which are essentially lost in non-separable innite dimensional topological vector spaces. In fact, when the topological vector spaces are not separable, very hardly we can nd explicit substitutes of the Hilbert bases of separable Hilbert spaces. On the contrary, as it is well known, the fundamental recipe of Quantum Mechanics, adopted to nd the probabilities in the measuring processes, requires to nd a basis of the Physical state space formed by eigenvectors of the operators representing the observables to be measured. For instance, in dimension 1, the position operator shows a continuos eigenbasis formed by all the Dirac delta's on the real line, this continuous Dirac basis, on one hand, belongs completely to our framework and, on the other hand, is far to represent a classical basis in any Hilbert or Banach or topological vector space.

S-diagonalizable operators and their spectrum

CARFI', David
2011

Abstract

In this paper we prove and apply a spectral expansion theorem for Schwartz linear operators showing a Schwartz linearly independent eigenfamily - that is an ordered system of eigenvectors, indexed by some Euclidean space, which is Schwartz regular. This type of spectral expansion is a perfect analogous of the spectral expansion of self-adjoint operators on separable Hilbert spaces, when the eigenbasis is indexed by some real Euclidean space instead of the set of positive integers. Moreover, our theorem appears formally identical to the spectral expansion of a non-defective operator in the nite dimensional case, where continuous superpositions (feasibly and naturally dened) take the place of the nite sums. The Schwartz expansion we present is a possible rigorous and simply manageable mathematical model for the spectral expansions, in the continuous case, used frequently in Quantum Mechanics, since it appears in a form extremely similar to its current elementary formulations in Physics. The key point consists into bringing again at the center of the stage the eigenbases, which are essentially lost in non-separable innite dimensional topological vector spaces. In fact, when the topological vector spaces are not separable, very hardly we can nd explicit substitutes of the Hilbert bases of separable Hilbert spaces. On the contrary, as it is well known, the fundamental recipe of Quantum Mechanics, adopted to nd the probabilities in the measuring processes, requires to nd a basis of the Physical state space formed by eigenvectors of the operators representing the observables to be measured. For instance, in dimension 1, the position operator shows a continuos eigenbasis formed by all the Dirac delta's on the real line, this continuous Dirac basis, on one hand, belongs completely to our framework and, on the other hand, is far to represent a classical basis in any Hilbert or Banach or topological vector space.
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11570/1912849
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