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 ...

Abstract. We study a non-Hermitian PT −symmetric generalization of an Nparticle, two-mode Bose-Hubbard system, modeling for example a Bose-Einstein condensate in a double well potential coupled to a continuum via a sink in one of the wells and a source in the other. The effect of the interplay between the particle interaction and the non-Hermiticity on characteristic features of the spectrum is analyzed drawing special attention to the occurrence and unfolding of exceptional points (EPs). We find that for vanishing particle interaction there are only two EPs of order N + 1 which under perturbation unfold either into [(N + 1)/2] eigenvalue pairs (and in case of N + 1 odd, into an additional zero-eigenvalue) or into eigenvalue triplets (third-order eigenvalue rings) and (N + 1) mod 3 single eigenvalues, depending on the direction of the perturbation in parameter space. This behavior is described analytically using perturbational techniques. More general EP unfoldings into eigenvalue rings up to (N + 1)th order are indicated.

Vigilance, anxiety, epileptic activity, and muscle tone can be modulated by drugs acting at the benzo- (Fig. la) was constructed containing a 6.4-kb genomic region including exons 7, 9, and 10 of the y2 subunit gene isolated from a 129SV mouse genomic library. A 1.2-kb genomic Pvu II-Nco I fragment including exon 8 (coding for amino acids 306-375 of the y2 polypeptide) was replaced with the phosphoglycerate kinase (PGK)-neo cassette (11), and a tk expression cassette (12) was added at the 3' end of the y2 sequence. Splicing from exon 7 to exon 9 would result in a stop of the translational reading frame and prohibit expression of sequences downstream of exon 7. Before electroporation into E14 ES cells (13), the plasmid was linearized at a polylinker site adjacent to the 5' end of the 7y2 genomic sequence. E14 ES cells were cultured on irradiated G418-resistant feeder cells obtained from CD1-M-TKneo2 mouse embryos [BRL, Fullinsdorf (Basel)] in GMEM (Glasgow modification of Eagle's medium; Flow Laboratories) containing 10% total calf serum and leukemia inhibitory factor (103 units/ml, Life Technologies). The cells were transfected and screened for homologous recombinants (14) by using PCR and the primers y2.19 (5'-CATCT CCATC GCTAA GAATG TTCGG derived from 7y2 sequences upstream of the targeting vector and Y2.20 (5'-ATGCT CCAGA CTGCC TTGGG AAAAG C-3') derived from PGK promoter sequences (11). Chimeric mice were generated (15) and mated to C57BL/6 females, and the offspring were genotyped by PtR amplification of tail DNAs. Reactions specific for the disrupted y2 allele [(0) Abbreviations: BZ, benzodiazepine; GABA, y-aminobutyric acid; DRG, dorsal root ganglion (ganglia); ES, embryonic stem; E, embryonic day; P, postnatal day.§To whom reprint requests should be addressed. 7749The publication costs of this article were defrayed in part by page charge payment. This article must therefore be hereby marked "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

We demonstrate the presence of parity-time (PT) symmetry for the non-Hermitian two-state Hamiltonian of a dissipative microwave billiard in the vicinity of an exceptional point (EP). The shape of the billiard depends on two parameters. The Hamiltonian is determined from the measured resonance spectrum on a fine grid in the parameter plane. After applying a purely imaginary diagonal shift to the Hamiltonian, its eigenvalues are either real or complex conjugate on a curve, which passes through the EP. An appropriate basis choice reveals its PT symmetry. Spontaneous symmetry breaking occurs at the EP.

A non-Hermitian complex symmetric 2 × 2 matrix toy model is used to study projective Hilbert space structures in the vicinity of exceptional points (EPs). The bi-orthogonal eigenvectors of a diagonalizable matrix are Puiseuxexpanded in terms of the root vectors at the EP. It is shown that the apparent contradiction between the two incompatible normalization conditions with finite and singular behavior in the EP-limit can be resolved by projectively extending the original Hilbert space. The complementary normalization conditions correspond then to two different affine charts of this enlarged projective Hilbert space. Geometric phase and phase jump behavior are analyzed and the usefulness of the phase rigidity as measure for the distance to EP configurations is demonstrated. Finally, EP-related aspects of PT −symmetrically extended Quantum Mechanics are discussed and a conjecture concerning the quantum brachistochrone problem is formulated.