MR2045938 Heiss, W. D. Exceptional points of non-Hermitian 
operators. J. Phys. A 37 (2004), no. 6, 2455--2464.

The paper reviews the appealing physics which is hidden behind 
the Kato's concept of the exceptional points (EP) and which 
became ``a physical reality" in the light of recent non-quantum 
experiments. In the introduction one appreciates the amazing 
variability of the areas of physics where EP's proved to play a 
role. On the more mathematical side I miss a remark on the truly 
seminal older work by C. Bender and T. T. Wu (e.g., Phys. Rev. 
184 (1969) 1231) who made the EP's exceptionally popular via 
their occurrence in the quantum harmonic oscillator H with the 
quartic perturbation gV at the complex couplings g. 

The author emphasizes an ease of a removal of the Hermiticity 
assumption from the matrices H and/or V in the EP context of 
classical physics. Of course, one could complement his argument 
by the older works on the cubic imaginary V in quantum context 
(cf., e.g., the review by G. Alvarez in J. Phys. A: Math. Gen. 27 
(1995) 4589 giving a few other references). Reference [23] which 
presented a breakthrough in this direction would deserve a 
completion by a sample of the subsequent intensive activity in 
the field. At random I just recollect the Mostafazadeh's detailed 
mathematical outlines of the EP-related mathematics (also known 
as a Jordan-block matrix structure deserving, in principle, a 
further detour to the classical works by M. G. Krein etc). I 
would even dare to add also my own - and probably the first 
exactly solvable - explicit EP differential Schroedinger equation 
example [cf. M. Z., Phys. Lett. A 259 (1999) 220].

The most appealing parts of the present review lie in the 
simplicity and transparency of all the detailed non-quantum 
examples: two coupled damped pendula in section 2, a generic 
two-dimensional matrix model in section 3 as well as the 
phenomenologically oriented insightful completion of the 
description of the repelling-level phenomena in section 4.