Non-Hermitian quantum mechanics (NHQM) is an important alternative to the standard (Hermitian) formalism of quantum mechanics, enabling the solution of otherwise difficult problems. The first book to present this theory, it is useful to advanced graduate students and researchers in physics, chemistry and engineering. NHQM provides powerful numerical and analytical tools for the study of resonance phenomena - perhaps one of the most striking events in nature. It is especially useful for problems whose solutions cause extreme difficulties within the structure of a conventional Hermitian framework. NHQM has applications in a variety of fields, including optics, where the refractive index is complex; quantum field theory, where the parity-time (PT) symmetry properties of the Hamiltonian are investigated; and atomic and molecular physics and electrical engineering, where complex potentials are introduced to simplify numerical calculations.

No abstract

The visualization of an exceptional point in a PT symmetric directional coupler(DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a PT symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a PT symmetric DC to measure the radius of convergence for non PT symmetric structures. For such systems the physical meaning of the rather mathematical term: radius of convergence, is exemplified.In the past several years, following the seminal paper by Bender and Boettcher [1], non-hermitian PTsymmetric Hamiltonians have caught a lot of attention (see [2] and references therein). Under certain conditions PT -symmetric Hamiltonians can have a completely real spectrum and thus can serve, under the appropriate inner products, as the Hamiltonians for unitary quantum systems [3].Recently, the realization of PT -symmetric "Hamiltonians" has been studied using optical waveguides with complex refractive indices [4,5]. The equivalence of the Maxwell and Schrödinger equations in certain regimes provides a physical system in which the properties of PT -symmetric operators can be studied and exemplified.An extremely interesting property of PT -symmetric operators stems from the anti-linearity of the time symmetry operator. Consider a PT -symmetric operatorĤ, i.e., [PT ,Ĥ] = 0. Due to the non-linearity of T one cannot in general choose simultaneous eigenfunctions of the operators PT andĤ. However, if an eigenvalue of thê H is real then it's corresponding eigenfunction is a also an eigenfunction of the PT operator. This property has come to be known as exact/spontaneously-broken PTsymmetry. Exact PT -symmetry refers to the case when every eigenfunction of the PT symmetric operator is also an eigenfunction of the PT operator. In any other case the PT -symmetry is said to be broken.Usually, the transition between exact and spontaneously-broken PT symmetry can be controlled by a parameter in the Hamiltonian. This parameter serves as a measure of the non-hermiticity. An important class of PT -symmetric Hamiltonians are of the form: H(λ) = H 0 + iλV . Where H 0 (and V ) are real and symmetric(anti-symmetric) with respect to parity so that [PT ,Ĥ] = 0. When λ = 0 the Hamiltonian is hermitian and the entire spectrum is real. The spectrum remains real even when λ = 0 as long as λ < λ c . At this critical value and beyond, pairs of eigenvalues collide and become complex, see for example [6]. Bender et al. [7] showed that the reality of the spectrum is explained by the real secular equations one can write for PT -symmetric matrices. These secular equations will depend on the non-hermiticity parameter and, consequently, yield either real or complex solutions. Delabaere et al. [8] showed for the one-parameter family of complex cubic oscillators that pairs of eigenvalues cross each other at Bender and Wu branch points. Dorey et al. [9] after proving ...

Physical systems with loss or gain feature resonant modes that are decaying or growing exponentially with time. Whenever two such modes coalesce both in their resonant frequency and their rate of decay or growth, a so-called "exceptional point" occurs, around which many fascinating phenomena have recently been reported to arise [1][2][3][4][5][6] . Particularly intriguing behavior is predicted to appear when encircling an exceptional point sufficiently slowly 7,8 , like a state-flip or the accumulation of a geometric phase 9,10 . Experiments dedicated to this issue could already successfully explore the topological structure of exceptional points [11][12][13] , but a full dynamical encircling and the breakdown of adiabaticity inevitably associated with it 14-21 remained out of reach of any measurement so far. Here we

No abstract