Thursday, February 24, 2011

Varieties defined by Boundaries

One of the uprising methods to study the scattering properties of set of points are spectral zeta functions. Applications have been found on distribution of prime numbers, energy states of quantum mechanical systems, eigenvalues of operators and because of this, several connection results and conjectures have been made among these apparently unrelated areas.

In general, it is often useful when study the properties of the distribution of a set of points $S$, to study the behavior of a the zeta function which is associated in some know way with the set of points under analysis. Maybe the most famous example of this fact is the study of the distribution of prime numbers by means of the Riemann Zeta function.

The main connection among this different points of view often is, or at least has being conjectured to be, the starting set of points can be realized as the spectrum of differential operators acting on the $L^2$ space of functions defined in some special kind of space, for example manifolds or graphs.

When talking about differential operators, it is capital to specify boundary conditions in order to have a well defined problem, whenever the notion of boundary makes sense.

Recently, most of the study of zeta functions has been made by defining it in an implicit way, that is, without knowing explicitly the values eigenvalues of such differential operators, and this can be made by applying a result that resembles the spectral theorem in functional analysis

$$\int_\gamma \lambda^{-2s} \frac{d}{d\lambda}\log(f(\lambda)) d\lambda$$

which is basically an application of Cauchy's Residue Theorem.

The role that plays this function $f$ that appears in the residue theorem is of great importance for analyzing properties of the zeta function. This function is a meromorphic function whose zeros are located $S$, and often can be found by imposing the boundary conditions.

Using this interpretation, we can say that the set $S$ is an analytic variety defined by $f$. This analytic variety is the intersection of the varieties defined by imposing one boundary condition at a time, i.e. we have that

$\displaystyle S=\bigcap_{\beta\in B} Z (f_\beta)$

where $B$ is the set of boundaries and $f_\beta$ is a function obtained when imposing the $\beta$ boundary condition, and $Z(g)$ is the zero set for the function $g$.

Each of the boundary conditions can define an analytic variety by means of $f_\beta$, and hence, the intersection of these varieties give the spectrum of the desired differential operator. Because of this reasoning, it is natural to think that the zeta function can be decomposed as the zeta functions of the corresponding boundary conditions that define the problem.

Studying this decomposition would be of great importance for finding the general behavior of a spectral zeta function when changing boundary conditions and see what is the net affect that these play in the overall result.

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