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List of Works
Current investigation
1) Scattering Theory:
For these years, I have been investigating scattering theory for the pair (H0,H) of Hamiltonians H0=-/2 and H=H0+VS+VL with perturbation by a long-range potential VL in addition to a short-range potential VS. In my recent researches I have investigated the scattering theory for the Hamiltonian H=H0+V, where the unperturbed Hamiltonian H0 is given by the fractional power H0=-1(-)/2 (ȁ1) of negative Laplacian - and the perturbation V is a sum of a short-range potential VS and a long-range potential VL, and have given in 2010 a proof of the existence and asymptotic completeness of the wave operators for the pair (H0,H) of the Hamiltonians H0 and H by Lax-Phillips-Enss method. The Hamiltonian of this form includes the relativistic Hamiltonian (-)1/2 with vanishing mass.
In a subsequent paper I have further given a new proof of the asymptotic completeness including the above-mentioned case of fractional power of negative Laplacian, which is simpler than the existing proofs by Lax-Phillips-Enss method. Namely Ruelle's theorem implies that the evolution exp(-itH) (t>0 or t<0) by the Hamiltonian H=H0+V with the perturbation V=VS+VL lets the scattering particle move toward the strongly outgoing or incoming region of the phase space where the quantum mechanical momentum and configuration are almost parallel or anti-parallel. The preceding proof of the asymptotic completeness by Lax-Phillips-Enss method shows the asymptotic completeness by utilizing the propagation estimates on those regions for the evolution exp(-itH0) by the unperturbed Hamiltonian H0. Ruelle's theorem however implies not only that the quantum mechanical momentum and configuration are almost parallel or anti-parallel in those regions but also that the quantum mechanical configuration is asymptotically equal to the product of quantum mechanical velocity and time. It is possible to show in a sort of tautological way with utilizing this consequence of Ruelle's theorem a very weak estimate corresponding to the Cauchy criterion of convergence. This estimate makes it possible to prove the convergence of wave operators and the inverse wave operators, whence gives a new proof of the asymptotic completeness. The asymptotic completeness is therefore a direct consequence of Ruelle's theorem, and it has been shown that it is no overstatement to say that the mathematical scattering theory is already completed by Ruelle's theorem.
2) Mathematical Quantum Mechanics:
As an extended research arisen from the investigation of the scattering theory, I have been also investigating a mathematical formulation of Quantum mechanics. Quantum mechanics is formulated on the basis of two quantities which satisfy the non-commutation relation called canonical commutation relation. Those two quantities are identified as configuration operator and momentum operator. In usual formulation of quantum mechanics, it is assumed that the third quantity called time coordinate exists in addition to these two quantities. However, a careful examination of the formulation of quantum mechanics shows that the two quantities, configuration operator and momentum operator, are sufficient in formulating the quantum mechanics, which proves that the time coordinate is a redundant quantity. The formulation of quantum mechanics based on this fact introduces the concept of time as a quantity defined in terms of configuration and momentum operators, and we see that the usual uncertainty occurs as an uncertainty of time. This formulation of quantum mechanics gives a rigorous notion of time which has been considered as an a priori given quantity in physics in somewhat ambiguous manner since the age of Isaac Newton. Consequences of these considerations are that (1) quantum mechanics must be considered as a theory describing the internal motion of a local system consisting of a finite number of particles, and (2) the concept of time is a ‘local’ notion proper to each local system having two quantities satisfying the canonical commutation relation. That every local system has its own local time means that each local system is an unceasingly changing system with the inside components always moving. This local internal motion is the origin of the local time of each local system.
The rigorous formulation of these things is given through the investigation of the asymptotic behavior of the solutions of Schrödinger equations, namely through developing the above-mentioned scattering theory for the Schrödinger equations with using the concepts of pseudodifferential and Fourier integral operators. The origin of the local time, namely the existence of the local motion inside each local system is explained by reducing it to the result in metamathematics of the existence of undecidable propositions.
(Initially created on August 15, 1996)
Copyright © 1996-2011 Hitoshi Kitada, All Rights Reserved
Reproduction in whole or in part without permission is prohibited
Communications in Mathematical Analysis
Journal of Abstract Differential Equations and Applications
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