In an overlap (or union?) of the fields of mathematical physics, of
spectral theory and of stochastic analysis (let us abbreviate it as
PSS), the interest in non-self-adjoint operators is quickly growing
during the recent years. Together with the authors, let me redirect
all readers which would like to know the reasons to the Brian
Davies' 2007 book cited as [8]. Adding, for the basic reference,
also the citation of SPECTRA AND PSEUDOSPECTRA by Lloyd N. Trefethen
and Mark Embree (Princeton University Press, Princeton and Oxford,
2005) with its ``sixty short sections'' such that ``nobody will read
them all'', with readers still advised to ``look at all the
figures''.

In the present review (prepared, originally, for the PSS conference
in Goslar in 2011) there are no figures. The methods are taken,
basically, from the authors' two previous works [27] and [25] on the
distribution of eigenvalues of non-selfadjoint operators. The former
approach (via complex analysis, CA) relies heavily on Borichev,
Golinskii and Kupin  as predecessors ([5], cf. Theorem 3.3.1). The
eigenvalues are interpreted as zeros of a holomorphic function
$d_\alpha$ called perturbation determinant. Under the accepted
restrictions it is important that the (extended) resolvent set of
the unperturbed operator is conformaly equivalent to the unit disk.

Without further specific assumptions the alternative,
operator-theoretic approach using the concept of numerical range
looks more appropriate for the underlying technical study of the
rate of accumulation of the isolated eigenvalues towards a point in
essential spectrum. The reason is that, due to assumptions, the
closure of the numerical range and the spectrum of the unperturbed
operator coincide. The key initial reference is made to several
variants of the classical estimate (5.1.1) which is due to Kato.

In applications to concrete operators, each of the two approaches in
question is shown to have specific advantages over the other. For
the sake of definiteness, the choice of the operators is often
restricted, typically, to the order$-p$ Schatten-class perturbations
of self-adjoint bounded operators and/or to the relatively-Schatten
perturbations of non-negative operators. In both approaches the
eigenvalue moments are put in sums which are, typically, bounded in
terms of the Schatten norm of the non-self-adjoint perturbation. In
general, adding assumptions is usually observed to strengthen the
relative merits of the initially less efficient CA approach.

Ultimately, in a way motivated by the role of the Lieb-Thirring
inequalities (cf. (7.2.4) in self-adjoint case) in studies of the
stability of quantum world [35], specific applications of the
abstract results to certain non-self-adjoint Jacobi and
Schr\"odinger operators are considered. Theorem 7.1.7 may be
perceived as one of the paper's most typical and most relevant
``applied-theory'' results. It appears efficient, in particular,
when a sequence of eigenvalues converges to an endpoint of the
essential spectrum. The authors also add a list of most interesting
open questions claiming, e.g., that the ``validity or falsehood of
estimate (7.2.12)'' may be perceived to be ``one of the major open
problems'' at present.

In a broader phenomenological context of growing popularity and of
an active current use of non-self-adjoint Hamiltonians with real
spectra in quantum physics (cf., e.g., the Mostafazadeh's extensive
review paper ``Pseudo-Hermitian Quantum Mechanics'' in Int. J. Geom.
Meth. Mod. Phys. 7 (2010) 1191-1306 (arXiv: 0810.5643)), I would
like to add a remark that the turn of attention to the Lieb-Thirring
inequalities in non-self-adjoint setup might in fact be also
perceived as an ill-devised strategy. Typically, in the ``most
natural generalization of the self-adjoint Lieb-Thirring
inequalities to the non-self-adjoint setting'', Frank, Laptev, Lieb
and Seiringer [16] were forced to consider just the {\em non-real}
energy eigenvalues ({\em a ``contradictio in adjecto''?}) in the
sectors avoiding the positive half-line.


MR3077277 (Sent 2014-01-21) Demuth, Michael; Hansmann, Marcel;
Katriel, Guy Eigenvalues of non-selfadjoint operators: a comparison
of two approaches. Mathematical physics, spectral theory and
stochastic analysis, 107--163, Oper. Theory Adv. Appl., 232,
Birkhäuser/Springer Basel AG, Basel, 2013. 47A10 (47A75 47B36)