Bi-orthogonal quantum systems

ORAL

Abstract

Models of PT symmetric quantum mechanics provide examples of bi-orthogonal quantum systems. The latter incorporporate all the structure of PT symmetric models, and allow for generalizations, especially in situations where the PT construction of the dual space fails. This happens for periodic solutions that occur at spectral singularities. For these periodic solutions, the dual functions are associated polynomials that obey inhomogeneous equations. The formalism is illustrated by a few exact results for some elementary potential models. In some non-trivial cases, equivalent hermitian theories are obtained and shown to be very simple: They are just free (chiral) particles. Field theory extensions are briefly considered. Supersymmetric bi-orthogonal systems are also discussed.

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Authors

  • R.F. Kelly

    SVT Associates, Department of Material Science and Engineering, Department of Chemistry, University of Florida, Florida International University, WebAssign, North Carolina State University, Broughton High School, Dept.~of Chemistry, Univ.~of Florida, Dept.~of Physics, Univ.~of Florida, National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA, Center for Superconductivity Research, Dept. of Physics, University of Maryland, College Park, MD, 20742, USA, Dept. of Physics, University of Florida, 32611, USA, Experimentalphysik VI, Center for Electronic Correlations and Magnetism, Institute of Physics, Augsburg, Germany, Physics \& Astronomy, UNC-CH, Chapel Hill, NC, University of North Carolina, Auburn University, University of Virginia, Tech. Univ. Eindhoven, University of Florida, Los Alamos National Labs, University of New Mexico, Advanced Materials Research Institute, University of New Orleans, New Orleans, LA, Department of Physics, University of Florida, UF, NHMFL, FSU / NHMFL, FSU, University of Arkansas, Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA, Dept. of Physics, University of Florida, Gainesville, FL 32611-8440, USA, Dept. Chemistry Florida State Univeristy, University of Brewen, Tohoku University, Okayama University, Dept of Chemistry, Florida State University, Dept. of Chemistry, Florida State University, National High Magnetic Field Laboratory, Tallahassee, FL, Laboratoire Lois Neel, Grenoble, France, Dept. of Chemistry, Texas A\&M University, Tsinghua Univ., INEL, JINR, Vanderbilt Univ./LBNL, Vanderbilt Univ., SVT Associates, Inc., Department of Chemical Engineering, University of Florida, Department of Materials Science and Engineering, University of Florida, Department of Electrical Engineering, National Central University, Taiwan, University of Miami, North Carolina Central University, University of Missouri Rolla, AB Millimetre, France, Thomas Keating Ltd., UK, Dept. of Physics, Univ. of Florida, Department of Material Science and Engineering University of Florida, Department of Chemistry University of Florida, Department of Chemical Eng. University of Florida, Naval Research Lab, Washington, DC, University of Rajshahi, LENIN All Russian Electrotechnical Institute, Moscow, Russia, Independent Researcher, Argentina

  • R.F. Kelly

    SVT Associates, Department of Material Science and Engineering, Department of Chemistry, University of Florida, Florida International University, WebAssign, North Carolina State University, Broughton High School, Dept.~of Chemistry, Univ.~of Florida, Dept.~of Physics, Univ.~of Florida, National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA, Center for Superconductivity Research, Dept. of Physics, University of Maryland, College Park, MD, 20742, USA, Dept. of Physics, University of Florida, 32611, USA, Experimentalphysik VI, Center for Electronic Correlations and Magnetism, Institute of Physics, Augsburg, Germany, Physics \& Astronomy, UNC-CH, Chapel Hill, NC, University of North Carolina, Auburn University, University of Virginia, Tech. Univ. Eindhoven, University of Florida, Los Alamos National Labs, University of New Mexico, Advanced Materials Research Institute, University of New Orleans, New Orleans, LA, Department of Physics, University of Florida, UF, NHMFL, FSU / NHMFL, FSU, University of Arkansas, Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA, Dept. of Physics, University of Florida, Gainesville, FL 32611-8440, USA, Dept. Chemistry Florida State Univeristy, University of Brewen, Tohoku University, Okayama University, Dept of Chemistry, Florida State University, Dept. of Chemistry, Florida State University, National High Magnetic Field Laboratory, Tallahassee, FL, Laboratoire Lois Neel, Grenoble, France, Dept. of Chemistry, Texas A\&M University, Tsinghua Univ., INEL, JINR, Vanderbilt Univ./LBNL, Vanderbilt Univ., SVT Associates, Inc., Department of Chemical Engineering, University of Florida, Department of Materials Science and Engineering, University of Florida, Department of Electrical Engineering, National Central University, Taiwan, University of Miami, North Carolina Central University, University of Missouri Rolla, AB Millimetre, France, Thomas Keating Ltd., UK, Dept. of Physics, Univ. of Florida, Department of Material Science and Engineering University of Florida, Department of Chemistry University of Florida, Department of Chemical Eng. University of Florida, Naval Research Lab, Washington, DC, University of Rajshahi, LENIN All Russian Electrotechnical Institute, Moscow, Russia, Independent Researcher, Argentina

  • R.F. Kelly

    SVT Associates, Department of Material Science and Engineering, Department of Chemistry, University of Florida, Florida International University, WebAssign, North Carolina State University, Broughton High School, Dept.~of Chemistry, Univ.~of Florida, Dept.~of Physics, Univ.~of Florida, National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA, Center for Superconductivity Research, Dept. of Physics, University of Maryland, College Park, MD, 20742, USA, Dept. of Physics, University of Florida, 32611, USA, Experimentalphysik VI, Center for Electronic Correlations and Magnetism, Institute of Physics, Augsburg, Germany, Physics \& Astronomy, UNC-CH, Chapel Hill, NC, University of North Carolina, Auburn University, University of Virginia, Tech. Univ. Eindhoven, University of Florida, Los Alamos National Labs, University of New Mexico, Advanced Materials Research Institute, University of New Orleans, New Orleans, LA, Department of Physics, University of Florida, UF, NHMFL, FSU / NHMFL, FSU, University of Arkansas, Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA, Dept. of Physics, University of Florida, Gainesville, FL 32611-8440, USA, Dept. Chemistry Florida State Univeristy, University of Brewen, Tohoku University, Okayama University, Dept of Chemistry, Florida State University, Dept. of Chemistry, Florida State University, National High Magnetic Field Laboratory, Tallahassee, FL, Laboratoire Lois Neel, Grenoble, France, Dept. of Chemistry, Texas A\&M University, Tsinghua Univ., INEL, JINR, Vanderbilt Univ./LBNL, Vanderbilt Univ., SVT Associates, Inc., Department of Chemical Engineering, University of Florida, Department of Materials Science and Engineering, University of Florida, Department of Electrical Engineering, National Central University, Taiwan, University of Miami, North Carolina Central University, University of Missouri Rolla, AB Millimetre, France, Thomas Keating Ltd., UK, Dept. of Physics, Univ. of Florida, Department of Material Science and Engineering University of Florida, Department of Chemistry University of Florida, Department of Chemical Eng. University of Florida, Naval Research Lab, Washington, DC, University of Rajshahi, LENIN All Russian Electrotechnical Institute, Moscow, Russia, Independent Researcher, Argentina

  • R.F. Kelly

    SVT Associates, Department of Material Science and Engineering, Department of Chemistry, University of Florida, Florida International University, WebAssign, North Carolina State University, Broughton High School, Dept.~of Chemistry, Univ.~of Florida, Dept.~of Physics, Univ.~of Florida, National High Magnetic Field Laboratory, Florida State University, Tallahassee, FL 32306, USA, Center for Superconductivity Research, Dept. of Physics, University of Maryland, College Park, MD, 20742, USA, Dept. of Physics, University of Florida, 32611, USA, Experimentalphysik VI, Center for Electronic Correlations and Magnetism, Institute of Physics, Augsburg, Germany, Physics \& Astronomy, UNC-CH, Chapel Hill, NC, University of North Carolina, Auburn University, University of Virginia, Tech. Univ. Eindhoven, University of Florida, Los Alamos National Labs, University of New Mexico, Advanced Materials Research Institute, University of New Orleans, New Orleans, LA, Department of Physics, University of Florida, UF, NHMFL, FSU / NHMFL, FSU, University of Arkansas, Dept. of Physics and Astronomy, University of Delaware, Newark, DE 19716, USA, Dept. of Physics, University of Florida, Gainesville, FL 32611-8440, USA, Dept. Chemistry Florida State Univeristy, University of Brewen, Tohoku University, Okayama University, Dept of Chemistry, Florida State University, Dept. of Chemistry, Florida State University, National High Magnetic Field Laboratory, Tallahassee, FL, Laboratoire Lois Neel, Grenoble, France, Dept. of Chemistry, Texas A\&M University, Tsinghua Univ., INEL, JINR, Vanderbilt Univ./LBNL, Vanderbilt Univ., SVT Associates, Inc., Department of Chemical Engineering, University of Florida, Department of Materials Science and Engineering, University of Florida, Department of Electrical Engineering, National Central University, Taiwan, University of Miami, North Carolina Central University, University of Missouri Rolla, AB Millimetre, France, Thomas Keating Ltd., UK, Dept. of Physics, Univ. of Florida, Department of Material Science and Engineering University of Florida, Department of Chemistry University of Florida, Department of Chemical Eng. University of Florida, Naval Research Lab, Washington, DC, University of Rajshahi, LENIN All Russian Electrotechnical Institute, Moscow, Russia, Independent Researcher, Argentina