Eigenvectors for Fixed Boundary Conditions

We examined in sections 2.5, 2.6 and 2.7 the case in which periodical boundary conditions are adopted and, in that case, the transformations that take us from position space to momentum space are given by the finite Fourier transforms. When other types of boundary conditions are adopted in position space we will still have a momentum space, as well as transformations between it and position space, but these will no longer be the Fourier transforms, but other type of transform involving complete sets of orthogonal functions.

With the use of fixed boundary conditions, in which the values of the field at the border are given beforehand and kept fixed, the appropriate mode functions for the transformation to momentum space are no longer the usual complex phases, used with periodical boundary condition in the Fourier transformations. They are instead the real eigenfunctions $f_{\vec{k}}(\vec{n})$ of the Laplacian operator that satisfy the boundary conditions imposed on the box. In principle the field can be kept fixed at arbitrary and independent values at each one of the sites of the external border. This is the situation which is typically of interest in the classical theory, for example when we study techniques to determine the electric potential inside a box when its walls are kept at arbitrarily given values of the potential.

Another type of boundary condition of interest in the classical theory are those in which, instead of the field itself, it is the derivatives of the field that are kept at arbitrarily given values at the border. This is relevant, for example, in situations where the electric field rather than the electric potential is known at the border. The case in which the derivatives are zero at the border is also known by the name of “free” boundary conditions, since in this case we may simply eliminate form the model the values of the field at the border and the links that connect the interior to the border, which is equivalent to fixing at zero the value of the normal derivative at the border. This type of boundary condition is often used for models in statistical mechanics, which is mathematically quite similar to the quantum theory of fields, as a simpler alternative to periodic boundary conditions.

However, when we study in more detail the question of the boundary conditions in the quantum theory, later on, we will see that this type of situation is not of much interest in that case, in which the important aspects of the boundary conditions are of another nature. Since our study of fixed boundary conditions here is meant mostly at illustrating how to deal with them, we will limit ourselves here to the case of null boundary conditions, in which we make $\varphi=0$ at all the border sites. This type of boundary condition will be of some use in the development of the quantum theory, and it can be easily generalized, in all situations of interest, to the case in which $\varphi$ is kept constant at some non-zero single value over the whole border. On finite lattices, for boundary conditions $\varphi=0$, it can easily be shown (problems 2.8.1 and 2.8.2) that the eigenfunctions of the finite-difference Laplacian which vanish at the border are given by

  $$f^{N}_{\vec{k}}(\vec{n})= 2^{d/2} \sin\left(\frac{\pi k_{1}n_{1}}{N+1}\right)\ldots \sin\left(\frac{\pi k_{d}n_{d}}{N+1}\right),$$ (2.8.1)

where the integer coordinates $k_{\mu}=1,\ldots,N$, $\mu=1,\ldots,d$ are the coordinates that identify each one of the discrete eigenmodes of the Laplacian in momentum space.

Observe that in this case, unlike the case of periodical boundary conditions, we cannot make $k_{\mu}=0$ for any value of $\mu$, since the corresponding eigenfunction would be identically zero. Besides this, for any $\mu$ the change $k_{\mu}\rightarrow -k_{\mu}$ will only change the sign of the eigenfunction and therefore does not produce an independent eigenfunction. Due to this, we do not have here the freedom we had in the periodical case, of choosing the range of variation of $k_{\mu}$, which must be as given above. This implies that there is no zero mode, because the fixed boundary conditions would force the corresponding eigenfunction to be identically zero, over the complete extension of the lattice. In other words, with these boundary conditions the Laplacian has no normalizable (non-null) eigenvector with a zero eigenvalue.

The eigenfunctions $f^{N}_{\vec{k}}(\vec{n})$ on finite lattices satisfy the orthogonality and completeness relations

$$\sum_{\vec{n}}f^{N}_{\vec{k}}(\vec{n})f^{N}_{\vec{k}'}(\vec{n}) = (N+1)^{d}\delta^{d}(\vec{k},\vec{k}'),$$ (2.8.2)

$$\sum_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})f^{N}_{\vec{k}}(\vec{n}') = (N+1)^{d}\delta^{d}(\vec{n},\vec{n}').$$ (2.8.3)

where in this case the Kronecker delta functions and sums are defined by

$$\begin{eqnarray*} \sum_{\vec{n}}& = & \sum_{n_{1}=1}^{N}\ldots\sum_{n_{d}=1}^{N}... ...},\vec{n}')& = & \delta(n_{1},n'_{1})\ldots\delta(n_{d},n'_{d}). \end{eqnarray*}$$

It is not difficult to demonstrate these orthogonality and completeness relations (problem 2.8.4) by writing the eigenfunctions in terms of complex exponentials, with the use of

$$\sin\left(\frac{\pi k_{\mu}n_{\mu}}{N+1}\right)=\frac{1}{2\i... ...\mu}}{N+1}} -e^{-\imath\frac{\pi k_{\mu}n_{\mu}}{N+1}}\right],$$

where there is no sum over $\mu$, and then using the formula for the sum of a geometrical progression, generalized to the complex context, as we already did in section 2.5. Note that, since the sine functions are zero for $n=0$, $n=N+1$, $k=0$ and $k=N+1$, it is possible to extend the sums, both those over $\vec{n}$ and those over $\vec{k}$, from the interval $[1,\ldots,N]$ to the interval $[0,\ldots,N+1]$. It is important to emphasize that, just as in the case of periodical boundary conditions, these orthogonality and completeness relations are exact on each finite lattice.

For fixed null boundary conditions the transformation of the field from position space to momentum space and its inverse are written as

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & \frac{1}{(N+1)^{d}} \sum_{\v... ...um_{\vec{k}}f^{N}_{\vec{k}}(\vec{n})\widetilde\varphi (\vec{k}). \end{eqnarray*}$$

It is also not difficult to show (problem 2.8.1) that the eigenvalues of the finite-difference Laplacian for the base of functions defined in equation (2.8.1) are given by

  $$\rho_{f}^{2}=4\left\{\sin^{2}\left[\frac{\displaystyle \pi k_{1}}... ...^{2}\left[\frac{\displaystyle \pi k_{d}}{\displaystyle 2(N+1)}\right]\right\}.$$ (2.8.4)

Observe however that, unlike what happened in the periodical case, the functions given in (2.8.1) are not eigenfunctions of the finite-difference operator $\Delta_{\mu}$ (problem 2.8.3). This is related to the fact that the finite-difference operator acts over the lattice functions as a generator of translations and, since the boundary conditions are fixed, it is not possible to execute such translations without violating the boundary conditions. As we shall see in future volumes, just as in the case of periodical boundary conditions, also in this case all quantities of physical interest on finite lattices will be functions of $\vec{k}$ only through the combination $\rho_{f}^{2}(\vec{k})$.

In the limit $N\rightarrow \infty$ each one of these quantities approaches the corresponding continuum-limit quantity. Assuming as always the existence of an external dimensional scale in which the side of the box is given by $L$, in this limit the eigenfunctions of the Laplacian are given by

  $$f_{\vec{k}}(\vec{x})=2^{d/2}\sin\left(\frac{\pi k_{1}x_{1}}{L}\right)\ldots\sin\left(\frac{\pi k_{d}x_{d}}{L}\right),$$ (2.8.5)

where $x_{\mu}=n_{\mu}a$, $a=L/(N+1)$, $n_{\mu}=0,\ldots,N+1$ define the continuum coordinates $x_{\mu}$ within the box, which in the limit are defined in the interval $[0,L]$. These functions satisfy the orthogonality and completeness relations (problem 2.8.5)

$$\int_{0}^{L}{\rm d}x_{1}\ldots\int_{0}^{L}{\rm d}x_{d} \;f_{\vec{k}}(\vec{x})f_{\vec{k}'}(\vec{x}) = \delta^{d}(\vec{k},\vec{k}'),$$ (2.8.6)

$$\sum_{\vec{k}}f_{\vec{k}}(\vec{x})f_{\vec{k}}(\vec{x}') = V\delta^{d}(\vec{x}-\vec{x}'),$$ (2.8.7)

where the integer coordinates $k_{\mu}$ extend now from $1$ to $\infty$ and $\delta^{d}(\vec{x}-\vec{x}')$ is the Dirac delta function in $d$ dimensions. The eigenvalues of the Laplacian corresponding to these eigenfunctions are given (problem 2.8.5) by the limits of the quantities $\rho_{f}^{2}/a^{2}$. In the limit the values of the dimensionfull momenta are given by $p_{\mu}=\pi k_{\mu}/L$ and the eigenvalues of the Laplacian are

  $$p^{2}=\frac{\pi^{2}}{L^{2}}\left(k_{1}^{2}+\ldots+k_{d}^{2}\right).$$ (2.8.8)

In the continuum the transformation of the field from position space to momentum space and its inverse are written as

$$\begin{eqnarray*} \widetilde\varphi (\vec{k}) & = & \frac{1}{V}\int_{V}{\rm d}^{... ...um_{\vec{k}}f^{N}_{\vec{k}}(\vec{x})\widetilde\varphi (\vec{k}). \end{eqnarray*}$$

Since the role played by the transformations to momentum space is always the same and they are always associated to decompositions of the functions of position in some basis of orthogonal functions, we often will, for simplicity of exposition, commit the abuse of language of referring to the transformation to momentum space as Fourier transforms, whatever the boundary conditions in use may actually be.

Problems

1. Show that, with fixed boundary conditions in which the field is zero at the border, the functions $f^{N}_{\vec{k}}(\vec{n})$ given in equation (2.8.1) are eigenfunctions of the finite-difference Laplacian. Derive also the expression for the corresponding eigenvalues $\rho_{f}^{2}$.

2. Show that the exponential functions $\exp[\imath 2\pi(\vec{k}\cdot\vec{n})/N]$, that we used in the case of periodic boundary conditions, are not eigenfunctions of the Laplacian with fixed boundary conditions where the field is zero at the border. In order to see this, examine in detail the situation at the sites next to the border.

3. Show that, with fixed boundary conditions where the field is zero at the border, the functions $f^{N}_{\vec{k}}(\vec{n})$ given in equation (2.8.1) are not eigenfunctions of the finite-difference operator $\Delta_{\mu}$. Examine in detail the situation at the sites next to the border.

4. Demonstrate the orthogonality and completeness relations for fixed boundary conditions given in equation (2.8.2), decomposing the sine functions that appear in the eigenfunctions $f^{N}_{\vec{k}}(\vec{n})$ into complex exponentials and using the formula for the sum of a geometrical progression, which can be generalized to the complex context, as we already saw in section 2.5.

5. Show that in the continuum limit inside a finite box the eigenfunctions of the Laplacian are given by the functions $f_{\vec{k}}(\vec{x})$ defined in equation (2.8.5). Starting from the orthogonality and completeness relations on finite lattices given in equation (2.8.2) demonstrate that the corresponding relations in the continuum limit are those given in equation (2.8.6). Show also that in this limit the eigenvalues of the Laplacian are those given in equation (2.8.8).

6. Show that the functions defined on a one-dimensional lattice with $N$ sites and fixed boundary conditions, given by

   
   

   $$f_{\kappa}(n)=\left\{ \begin{array}{ll} \kappa=0: & 1, \\ \k... ...}{N+1}\right),\mbox{~~where~~}k=1,\ldots,N, \end{array}\right.$$
   
   

   where $n=0,\ldots,N+1$ and $\kappa=0,\ldots,2N$, are all orthogonal to one another, so long as the sums over position space are defined by

   
   

   $$\frac{1}{2}f_{\kappa}(0)f_{\kappa'}(0) +\sum_{n=1}^{N}f_{\ka... ...(n)f_{\kappa'}(n) +\frac{1}{2}f_{\kappa}(N+1)f_{\kappa'}(N+1),$$
   
   

   reflecting the fact that the sites at the border are associated to integration elements with half the volume of the integration elements associated to the internal sites.

7. ($\star$) Find, if at all possible, a subset of $N$ of the functions given in problem 2.8.6 that is complete for the representation of functions $\varphi(n)$ of the $N$ sites in the interior of the lattice, given two independent fixed values $\varphi(0)$ and $\varphi(N+1)$ at the two ends of the lattice^2.2.