bilinear operators

The Heisenberg uncertainty principle is a consequence of the postulate that coordinate and momentum representations are related to each other by the Fourier transform. This postulate has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. We argue that the postulate is based neither on strong theoretical arguments nor on experimental data. A position operator proposed in our recent publication resolves inconsistencies of standard approach and sheds a new light on important problems of quantum theory. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.

The Heisenberg uncertainty principle is a consequence of the postulate that coordinate and momentum representations are related to each other by the Fourier transform. This postulate has been accepted from the beginning of quantum theory by analogy with classical electrodynamics. We argue that the postulate is based neither on strong theoretical arguments nor on experimental data. A position operator proposed in our recent publication resolves inconsistencies of standard approach and sheds a new light on important problems of quantum theory. We do not assume that the reader is an expert in the given field and the content of the paper can be understood by a wide audience of physicists.

The behavior of bilinear operators acting on interpolation of Banach spaces for the ρ method in relation to the compactness is analyzed. Similar results of Lions-Peetre, Hayakawa and Person's compactness theorems are obtained for the bilinear case and the ρ method.

Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with: (i) the hermitian (but not selfadjoint) Time operator in non-relativistic quantum mechanics and in quantum electrodynamics; with (ii) idem, the introduction of Time and Space operators; and with (iii) the problem of the four-position and fourmomentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles. Afterwards, other physical applications of non-selfadjoint (and even non-hermitian) operators are briefly discussed. We briefly mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a chronon. This paper is largely based on work developed, along the years, in collaboration with V.

Let Ω ∈ L ∞ (R n) × L 2 (S n−1) be a homogeneous function of degree zero. In this article, we obtain some boundedness of the parameterized Littlewood−Paley operators with variable kernels on weighted Herz-Morrey spaces with variable exponent. As a supplement, the boundedness of fractional integral operators with variable kernel is also obtained on these spaces.

We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.

We obtain boundedness for the bilinear spherical maximal function in a range of exponents that includes the Banach triangle and a range of L p with p < 1. We also obtain counterexamples that are asymptotically optimal with our positive results on certain indices as the dimension tends to infinity.

Commutators of a large class of bilinear operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be jointly compact. Under a similar commutation, fractional integral versions of the bilinear Hilbert transform yield separately compact operators.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

The dual purpose of this article is to establish bilinear Poincaré-type estimates associated to an approximation of the identity and to explore the connections between bilinear pseudo-differential operators and bilinear potential-type operators. The common underlying theme in both topics is their applications to Leibniz-type rules in Sobolev and Campanato-Morrey spaces under Sobolev scaling.

The behavior of bilinear operators acting on interpolation of Banach spaces for the ρ method in relation to the compactness is analyzed. Similar results of Lions-Peetre, Hayakawa and Person's compactness theorems are obtained for the bilinear case and the ρ method.

A notion of compactness in the bilinear setting is explored. Moreover, commutators of bilinear Calderón-Zygmund operators and multiplication by functions in a certain subspace of the space of functions of bounded mean oscillations are shown to be compact.

An analytical expression for the relativistic corrections to the energy spectra of particles completely confined in an one-dimensional limited length in real space is given, based upon the wave property of particles, the relativistic energy-momentum relation and two mathematical equations.

Aim of this paper is trying to show the possible significance, and usefulness, of various non-selfadjoint operators for suitable Observables in non-relativistic and relativistic quantum mechanics, and in quantum electrodynamics: More specifically, this work starts dealing with:  (i) the
hermitian (but not selfadjoint) TIME  operator in non-relativistic quantum
mechanics and in quantum electrodynamics; with  (ii) IDEM, the introduction of Time and Space operators; and with  (iii) the problem of the four-position and four-momentum operators, each one with its hermitian and anti-hermitian parts, for relativistic spin-zero particles.  Afterwards, other physical applications of non-selfadjoint (and even
non-hermitian) operators are briefly discussed.  We mention how non-hermitian operators can indeed be used in physics [as it was done, elsewhere, for describing Unstable States]; and some considerations are added on the cases of the nuclear optical potential, of quantum dissipation, and in particular of an approach to the measurement problem in QM in terms of a "chronon".  This paper is largely based on work developed, along the years, in collaboration with V.S.Olkhovsky, and, in smaller parts, with P.Smrz, with R.H.A.Farias, and with S.P.Maydanyuk.  [PACS numbers: 03.65.Ta; 03.65.-w; 03.65.Pm; 03.70.+k; 03.65.Xp; 03.65.Yz; 11.10.St; 11.10.-z; 11.90.+t; 02.00.00; 03.00.00; 24.10.Ht; 03.65.Yz; 21.60.-u; 11.10.Ef; 03.65.Fd; 02.40.Dr; 98.80.Jk.  (KEYWORDS: time operator, space-time operator, non-selfadjoint operators, non-hermitian operators, bilinear operators, time operator for discrete energy spectra, time-energy uncertainty relations,
quasi-hermitian Hamiltonians, Klein-Gordon equation, chronon, quantum
dissipation, decoherence, nuclear optical model, cosmology, projective relativity)