Exceptional point in a simple textbook example

  title={Exceptional point in a simple textbook example},
  author={Francisco M. Fern'andez},
  journal={European Journal of Physics},
  • F. Fern'andez
  • Published 24 September 2017
  • Mathematics
  • European Journal of Physics
We propose to introduce the concept of exceptional points in intermediate courses on mathematics and classical mechanics by means of simple textbook examples. The first one is an ordinary second-order differential equation with constant coefficients. The second one is the well-known damped harmonic oscillator. From a strict mathematical viewpoint both are the same problem that enables one to connect the occurrence of linearly dependent exponential solutions with a defective matrix which cannot… 

Figures from this paper

Reply to the Comment on ‘Improving series convergence: the simple pendulum and beyond’
We correct an error pointed out by a reader in an example included in a previous article about a method for improving series convergence. After the correction, the conclusion of the previous article,
Signatures of Liouvillian Exceptional Points in a Quantum Thermal Machine
Viewing a quantum thermal machine as a non-Hermitian quantum system, we characterize in full generality its analytical time-dependent dynamics by deriving the spectrum of its non-Hermitian
Self-Stabilization of Light Sails by Damped Internal Degrees of Freedom
We consider the motion of a light sail that is accelerated by a powerful laser beam. We derive the equations of motion for two proof-of-concept sail designs with damped internal degrees of freedom.
Signatures of exceptional points in a quantum thermal machine
The concepts and tools from the theory of non-Hermitian quantum systems are used to investigate the dynamics of a quantum thermal machine. This approach allows us to characterize in full generality


Exceptional Points – Their Universal Occurrence and Their Physical Significance
Exceptional points are singularities that occur generically in the spectrum and eigenfunctions of operators (matrices) that depend on a parameter. For self-adjoint operators they always lie in the
The chirality of exceptional points
Abstract:Exceptional points are singularities of the spectrum and wave functions of a Hamiltonian which occur as functions of a complex interaction parameter. They are accessible in experiments with
Projective Hilbert space structures at exceptional points
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
Mathematical physics: Circling exceptional points
Going around an exceptional point in a full circle can be a non-adiabatic, asymmetric process. This surprising prediction is now confirmed by two separate experiments.
Avoided level crossing and exceptional points
The connection between level repulsions and the singularities associated with the analytically continued energy levels is investigated. The authors also conjecture that there are necessarily specific
Experimental observation of the topological structure of exceptional points.
A microwave cavity experiment where exceptional points (EPs), which are square root singularities of the eigenvalues as function of a complex interaction parameter, are encircled in the laboratory and one of the Eigenvectors undergoes a sign change which can be discerned in the field patterns.
Encircling an exceptional point.
The geometric phases that occur when an EP is encircled four times are measured and it is confirmed that for this system anEP is a branch point of fourth order.
Exceptional points in atomic spectra.
The resonances of the system are investigated and it is shown how exceptional points can be found by exploiting characteristic properties of the degeneracies, which are branch point singularities.
Repulsion of resonance states and exceptional points
  • Heiss
  • Physics
    Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics
  • 2000
It is shown that an encircling of an exceptional point induces a phase change of one wave function but not of the other, and it is argued that level anticrossing (crossing) must imply crossing of the corresponding widths of the resonance states.
Theoretical Mechanics:
  • A. F.
  • Education
  • 1936
AbstractWITH the publication of this volume on rigid dynamics, Prof. MacMillan has completed a task of first-rate importance. This work, together with its companion volumes on statics and particle