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Incompatible Coulomb hamiltonian extensions.

ABSTRACT: We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb ?/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U(1) circle in the attractive case and on [Formula: see text] in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunctions. In all cases, different extensions are incompatible one with the other. For an actual experiment with an attractive potential, the bound spectrum can be used to discriminate which extension is the correct one.

SUBMITTER: Abramovici G

PROVIDER: S-EPMC7190704 | biostudies-literature | 2020 Apr

REPOSITORIES: biostudies-literature

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Incompatible Coulomb hamiltonian extensions.

Abramovici G G

Scientific reports 20200429 1

We revisit the resolution of the one-dimensional Schrödinger hamiltonian with a Coulomb λ/|x| potential. We examine among its self-adjoint extensions those which are compatible with physical conservation laws. In the one-dimensional semi-infinite case, we show that they are classified on a U(1) circle in the attractive case and on [Formula: see text] in the repulsive one. In the one-dimensional infinite case, we find a specific and original classification by studying the continuity of eigenfunct ...[more]

PMID: 32350289

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Project description:IntroductionEarly phase dose-finding (EPDF) studies are critical for the development of new treatments, directly influencing whether compounds or interventions can be investigated in further trials to confirm their safety and efficacy. There exists guidance for clinical trial protocols and reporting of completed trials in the Standard Protocol Items: Recommendations for Interventional Trials (SPIRIT) 2013 and CONsolidated Standards Of Reporting Randomised Trials (CONSORT) 2010 statements. However, neither the original statements nor their extensions adequately cover the specific features of EPDF trials. The DEFINE (DosE-FIndiNg Extensions) study aims to enhance transparency, completeness, reproducibility and interpretation of EPDF trial protocols (SPIRIT-DEFINE) and their reports once completed (CONSORT-DEFINE), across all disease areas, building on the original SPIRIT 2013 and CONSORT 2010 statements.Methods and analysisA methodological review of published EPDF trials will be conducted to identify features and deficiencies in reporting and inform the initial generation of the candidate items. The early draft checklists will be enriched through a review of published and grey literature, real-world examples analysis, citation and reference searches and consultation with international experts, including regulators and journal editors. Development of CONSORT-DEFINE commenced in March 2021, followed by SPIRIT-DEFINE from January 2022. A modified Delphi process, involving worldwide, multidisciplinary and cross-sector key stakeholders, will be run to refine the checklists. An international consensus meeting in autumn 2022 will finalise the list of items to be included in both guidance extensions.Ethics and disseminationThis project was approved by ICR's Committee for Clinical Research. The Health Research Authority confirmed Research Ethics Approval is not required. The dissemination strategy aims to maximise guideline awareness and uptake, including but not limited to dissemination in stakeholder meetings, conferences, peer-reviewed publications and on the EQUATOR Network and DEFINE study websites.Registration detailsSPIRIT-DEFINE and CONSORT-DEFINE are registered with the EQUATOR Network.

Dataset Information

Incompatible Coulomb hamiltonian extensions.

Publications

Incompatible Coulomb hamiltonian extensions.

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