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Ultra-Weak Time Operators of Schrödinger Operators

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Abstract

In an abstract framework, a new concept on time operator, ultra-weak time operator, is introduced, which is a concept weaker than that of weak time operator. Theorems on the existence of an ultra-weak time operator are established. As an application of the theorems, it is shown that Schrödinger operators \({H_V}\) with potentials V obeying suitable conditions, including the Hamiltonian of the hydrogen atom, have ultra-weak time operators. Moreover, a class of Borel measurable functions \(f:{\mathbb {R}}\rightarrow {\mathbb {R}}\) such that \(f({H_V})\) has an ultra-weak time operator is found.

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Authors and Affiliations

  1. Department of Mathematics, Hokkaido University, Sapporo, 060-0810, Japan

    Asao Arai

  2. Faculty of Mathematics, Kyushu University, Fukuoka, 819-0395, Japan

    Fumio Hiroshima

Authors
  1. Asao Arai
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  2. Fumio Hiroshima
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Correspondence to Fumio Hiroshima.

Additional information

Communicated by David Perez-Garca.

This research was supported by KAKENHI 15K04888 from JSPS (A.A.), partially supported by CREST, JST, and Challenging Exploratory Research 15K13445 from JSPS (F.H).

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Arai, A., Hiroshima, F. Ultra-Weak Time Operators of Schrödinger Operators. Ann. Henri Poincaré 18, 2995–3033 (2017). https://doi.org/10.1007/s00023-017-0586-x

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