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The condition of self-adjointness ensures that the eigenvalues of a Hamiltonian are real and bounded below. Replacing this condition by the weaker condition of P T symmetry, one obtains new infinite classes of complex Hamiltonians whose spectra are also real and positive. These P T symmetric theories may be viewed as analytic continuations of conventional… (More)

- Carl M . Bender, Dorje C. Brody, Hugh F Jones, Bernhard K. Meister
- Physical review letters
- 2007

Given an initial quantum state |psi(I)> and a final quantum state |psi(F)>, there exist Hamiltonians H under which |psi(I)> evolves into |psi(F)>. Consider the following quantum brachistochrone problem: subject to the constraint that the difference between the largest and smallest eigenvalues of H is held fixed, which H achieves this transformation in the… (More)

- Carl M . Bender, Dorje C. Brody, Hugh F Jones
- Physical review letters
- 2002

Requiring that a Hamiltonian be Hermitian is overly restrictive. A consistent physical theory of quantum mechanics can be built on a complex Hamiltonian that is not Hermitian but satisfies the less restrictive and more physical condition of space-time reflection symmetry (PT symmetry). One might expect a non-Hermitian Hamiltonian to lead to a violation of… (More)

If one defines the distance between two points as the Manhattan distance (the sum of the horizontal distance along streets and the vertical distance along avenues) then one can define a city as being optimal if the average distance between pairs of points is a minimum. In this paper a nonlinear differential equation for the boundary curve of such a city is… (More)

This paper shows that there is a correspondence between quasi-exactly solvable models in quantum mechanics and sets of orthogonal polynomials {Pn}. The quantum-mechanical wave function is the generating function for the Pn(E), which are polynomials in the energy E. The condition of quasi-exact solvability is reflected in the vanishing of the norm of all… (More)

Carl M. Bender, Dorje C. Brody, and Bernhard K. Meister 1Department of Physics, Washington University, St. Louis MO 63130, USA 2Blackett Laboratory, Imperial College, London SW7 2BZ, UK and Department of Applied Mathematics and Theoretical Physics, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK 3Goldman Sachs, Peterborough Court, 133 Fleet… (More)

The HamiltonianH = p2+x4+iAx, whereA is a real parameter, is investigated. The spectrum of H is discrete and entirely real and positive for |A| < 3.169. As |A| increases past this point, adjacent pairs of energy levels coalesce and then become complex, starting with the lowest-lying energy levels. For large energies, the values ofA at which this merging… (More)

It is shown that if a Hamiltonian H is Hermitian, then there always exists an operator P having the following properties: (i) P is linear and Hermitian; (ii) P commutes with H ; (iii) P = 1; (iv) the nth eigenstate of H is also an eigenstate of P with eigenvalue (−1). Given these properties, it is appropriate to refer to P as the parity operator and to say… (More)

Abstract A non-Hermitian Hamiltonian that has an unbroken PT symmetry can be converted by means of a similarity transformation to a physically equivalent Hermitian Hamiltonian. This raises the following question: In which form of the quantum theory, the non-Hermitian or the Hermitian one, is it easier to perform calculations? This paper compares both forms… (More)

- Carl M Bendera, Carl M . Bender, E. Ben-Naim
- 2008

The nonlinear integral equation P(x) = ∫ β α dyw(y)P(y)P(x + y) is investigated. It is shown that for a given function w(x) the equation admits an infinite set of polynomial solutions Pn(x). For polynomial solutions, this nonlinear integral equation reduces to a finite set of coupled linear algebraic equations for the coefficients of the polynomials.… (More)