The toolbox of an experienced condensed matter theorist
is divided into two major compartments, each one crowded with calculational
techniques. One of the two compartments bears the label *universal tools*,
the other one *precision instruments*.

The universal-tools compartment contains an assortment
of *general methods* for the calculation of observable quantities
of interest in condensed matter physics. Among them are general methods
for a particular purpose. General methods for the calculation of dynamic
correlation functions which are applicable to arbitrarily selected model
systems, for example, belong in that compartment. Also found there are
multi-purpose methods with a wide range of applicability. Universal tools
must have a certain robustness against conditions that may invalidate their
applicability. However, they are not meant to yield exact results on such
a wide territory of applications.

The general calculational or computational techniques that are commonly used in condensed matter theory may be categorized as follows:

Methods with *extrinsic *limitations, such as computer
simulations, Green's function methods, the recursion method, or finite-size
studies. In these methods, the limitations are set by the amount of calculational
effort or computational power invested in them.

Methods with *intrinsic* limitations, such as mean
field theory, linear spin-wave theory (harmonic approximation), the random-phase
approximation (within the framework of Green's function methods), the *n*-pole
approximation (within the framework of the recursion method), have built-in
limitations that cannot be overcome within their own respective scope.

The precision instruments, stored in the other compartment
of the theorist's toolbox, are a collection of *special methods*.
They have been designed for the exact solution of particular problems.
The small but precious collection of exactly solved models in statistical
mechanics and solid state physics is their main source of origin.

Once a challenging problem has been solved by a special method, it is by no means guaranteed that the solution can be reproduced by a general method. Nevertheless, it is usually instructive and illuminating to test the performance of universal tools on problems that have previously been solved by specially designed precision instruments. Many a general method has its root in a special method designed for the exact solution of a specific problem. What makes it a general method are applications to problems of a similar nature, where it is subject to intrinsic or extrinsic limitations. For example, mean field theory may be regarded as a special method for the solution of certain model systems with long-range interaction, and spin-wave theory derives its legitimacy from those special situations in which anharmonicities can be ignored in magnetic excitations.

For a thorough study of a broadly defined topic we need
universal tools *and* precision instruments, i.e. general methods
with their wide range of applicability *and* special methods that
provide deeper insight for particular circumstances. We must combine *systematics
in breadth* with *systematics in depth* in order to gain the best
possible understanding of the topic under scrutiny. The use in isolation
of (i) general methods with severe intrinsic limitations or (ii) special
methods applicable only under highly non-generic circumstances is likely
to invite misleading conclusions. Systematics in both directions is key
to an understanding of the least accessible territory.

Exact solutions are usually out of reach except for particular circumstances and by special methods. The particular circumstances are always describable in terms of a simplification of the problem. There are basically two types of simplifications that may bring an exact solution to within reach:

Simplifications due to a *special type of interaction*
between the degrees of freedom. The free-particle limit or special types
of infinite-range interaction are obvious examples.

Simplifications due to a *special state *of a model
system which otherwise exhibits generic behavior. A typical example is
the ordered ground state of the Heisenberg ferromagnet.

Do these particular, simplifying circumstances translate
into an improved performance of applicable general methods as well? The
answer depends on the specifics of the general method under consideration.
(i) Green's function methods pose the problem of approximating the infinite
hierarchy of equations of motion in a controlled and systematic way. That
is notoriously difficult even for weakly coupled degrees of freedom. However,
in the noninteracting limit, that hierarchy reduces to a closed set of
equations, from which the exact solution can readily be extracted. Simplifications
due to a special state of the system do not, in general, result in a more
tractable hierarchy of equations of motion. The reason is that the state
of the system remains unspecified in the hierarchy of equations of motion.
(ii) In the recursion method, the properties of generic systems manifest
themselves in highly complex patterns exhibited by the sequences of continued-fraction
coefficients, as we shall see. For noninteracting degrees of freedom, the
amount of simplification in those sequences is comparable to that in the
Green's function approach. However, the recursion method is decidedly better
equipped to handle situations in which the simplification is due to a special
state of the system. The reason is that the specification of the state
has its impact on every continued-fraction coefficient as it is evaluated
in the recursive calculational procedure.