self adjoint equations and the symplectic metric

The form in which the initial value differential equation for a pair of first order equations is written can be called its standard form,

$$\frac{dZ(z)}{dz}$$ = M(z) Z(z), (341)

but for Sturn-Liouville applications, it is preferable to write it in canonical form, which is

$$\alpha(z) \frac{dZ(z)}{dz} + \{\beta(z) + \frac{1}{2} \frac{d\alpha(z)}{dz}\}Z(z) \lambda \gamma(z) Z(z).$$ = (342)

Here $\beta$ takes the place of M from the standard equation, but without any eigenvalue which M might have contained. The factor $\alpha$, only half of whose derivative is written explicitly (the other half being combined with M in $\beta$). allows for the possibility of attaching multipliers to the derivatives, which is sometimes either necessary or convenient. Finally, $\gamma$ positions the eigenvalue where it belongs in the matrix multiplier of Z.

If $\alpha$ is invertible, it is easy to interconvert the standard and canonical forms; if it is not, the order of the system of differential equations ought to be reduced. By inspection,

$$\alpha^{-1}(z)\{\lambda\gamma(z)-\beta(z)-\frac{1}{2}\frac{d\alpha(z)}{dz}\}.$$ M(z) = (343)

So the only outstanding question concerns the motivation for the strange way of writing the canonical equation.

There is a further innovation, which consists in introducing the adjoint of the canonical equation,

$$-\alpha^T(z) \frac{dW(z)}{dz} + \{\beta^T(z) - \frac{1}{2} \frac{d\alpha^T(z)}{dz}\}W(z) \lambda \gamma^T(z) W(z).$$ = (344)

If the combination of signs and transposes seems mystifying, think of the adjoint W as the transpose of the inverse of Z. More exactly, direct substitution and invoking the uniqueness of solutions relative to initial conditions,

$$(\alpha Z)^{-1 \ T}$$ Z^A = (345)

By regarding the left hand side of the canonical equation as the application of an operator ${\cal L}$ to Z, and likewise the left hand side of the adjoint equation as the application of ${\cal M}$ to W, the two can be written in the abbreviated form

$${\cal L}(Z) \lambda\gamma Z$$ = (346)

$${\cal M}(W) \lambda\gamma^T W.$$ = (347)

There are conditions under which an operator is just the same as its adjoint; by comparison they are

$$\alpha - \alpha^T$$ = (348)

$$\beta \beta^T$$ = (349)

$$\gamma \gamma^T.$$ = (350)

Once the definitions have all been established, Green's formula, which asserts that

$$\int_a^b\{\phi^T{\cal L}(\psi) - {\cal M}(\phi)^T\psi\}dz \phi^T\alpha\psi\vert _b - \phi^T\alpha\psi\vert _a,$$ = (351)

can be derived. In general terms, it is a consequence of ${\cal L}$ and ${\cal M}$ acting like derivatives applied to a product which evaluates into the product evaluated at its endpoints. The $\alpha$ sandwiched between the vectors follows from the detailed structure of these differential operators.

If the vectors $\psi$ and $\phi$ are eigenvectors,

$${\cal L}(\psi) \lambda \gamma \psi$$ = (352)

$${\cal M}(\phi) \mu \gamma^T \phi$$ = (353)

the left hand simplifies to give the Christoffel-Darboux formula,

$$(\lambda - \mu) \int_a^b\phi^T\gamma\psi dz \phi^T \alpha \psi \vert _a^b.$$ = (354)

Among other things, it justifies eigenfunction expansions with respect to the solutions of differential equations.

The matrix form of these results is more complicated than the version which is usually seen in textbooks, but it has the advantage of broad applicability. For example, for Schrödinger's equation,

$$\gamma \left[ \begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array} \right].$$ = (355)

Although $\gamma$ is singular, rather than disrupting the results, it makes the norm of functions depend on their values alone, and not at all on their derivatives. Spaces in which functions have this more complicated norm are called Sobolev spaces rather than Banach spaces. For the Dirac equation, $\gamma = {\bf 1}$, and both positive and negative energy components figure in the calculation of norms.

The selfadjoint $\alpha$ is antisymmetric - a multiple of ${\bf i}$ - turning the inner product $\phi^T \alpha \psi$ into a determinant; concretely, the Wronskian of the solutions $\psi$ and $\phi$, in the case of the Schrödinger equation.

The Christoffel-Darboux formula establishes the orthogonality of solutions belonging to different eigenvalues, but cannot establish a norm because of the zero factor resulting from equal eigenvalues. The norm might be gotten as a confluent case, taking limits as two eigenvalues approach, or the undefined norm might just be left in place in expansion formulas.

By using Hermitean conjugates instead of transposes in the relevant formulas, and treating the eigenvalue as a complex variable, the factor turns into $(\lambda^*-\mu)$ which leaves the imaginary part of the eigenvalue when the two eigenfunctions are the same. The real result can be gotten by taking the limit as the imaginary part vanishes.

Supposing that

$$\sum_{i=0}^\infty c_i \psi^i(z),$$ f(z) = (356)

and setting aside questions of convergence,

$$\int_a^b\psi_i^T(z)f(z)dz c_i \int_a^b\psi_i^T(z)\psi_i(z)dz,$$ = (357)

so that

$$\frac{\int_a^b\psi_i^T(\sigma)f(\sigma)d\sigma} {\int_a^b\psi_i^T(\sigma)\psi_i(\sigma)d\sigma},$$ c_i = (358)

The Stieltjes integral comes into play when we find that the eigenfunctions are quite numerous and the actual separation between eigenvalues is very small but nevertheless the eigenvalues are packed irregularly. It is also desirable to separate the parts of the coefficient which are due to integrating to function to be represented, and the small factor occasioned by the large norm.

Therefore, keep the integral of the function as a coefficient

$$\xi_i \int_a^b\psi_i^T(\sigma)f(\sigma)d\sigma,$$ = (359)

and introduce the distribution function $\rho(\lambda)$ which is a step function vanishing at $-\infty$, with increments at the eigenvalues $\lambda_i$ in the amount of the reciprocals of those denominators, which are actually the square of the norm of the eigenfunction.

Altogether,

$$\sum_{i=0}^\infty \xi_i \psi_i(z)(\rho_{i+1}-\rho_i),$$ f(z) = (360)

$$\int_{-\infty}^{\infty}\xi(\lambda)\psi(\lambda,z)d\rho(\lambda)$$ = (361)

There still remains the selection of an appropriate right hand boundary condition, and the choice of a uniform normalization for all the eigenfunctions. Help in making the selections can be had from looking at Parseval's equality,

$$\int_a^b\vert f(x)\vert^2 dx \sum_{i=0}^\infty\vert c_i\vert^2,$$ = (362)

which is a generalization of the Pythagorean theorem to function space, and which can be written in the more symbolic form

$$\sum_{i=0}^\infty\vert(\psi_i,f)\vert^2.$$ (f,f) = (363)

By the Christoffel-Darboux formula, and using the complex version of Green's formulas rather than the real version,

$$\frac{[f,f](b)-[f,f](a)}{\lambda-\lambda^*} \sum_{i=0}^\infty\frac{\vert[f,\psi_i(b)]-[f,\psi_i(a)]\vert^2} {\vert\lambda - \lambda_i\vert^2}$$ = (364)

Having turned integrals in function space into boundary sums (an interval has just two boundary points, a and b), it would be convenient to eliminate the dependence on b. One mechanism is to look at a solution $f = \phi + m \psi$ which is a combination of the two standard solutions, and to suppose that f satisfies a real boundary condition at b, resulting in [f,f](b)=0.

Whatever that boundary condition, it should be used for the $\psi$'s as well; in other words,

$$[f,\psi_i](b)$$ = 0 (365)

Having removed the influence of the right boundary point by working with real boundary values in the complex domain, there remains the left boundary to assign some standard form. Using real values there too, and recalling the two linearly independent solutions $\psi$ and $\phi$, altogether,

$$[\psi, \psi_i](a)$$ = 0 (366)

with

$$[f,\psi_i](a) = [\phi, \psi_i](a) = r_i.$$

This quantity r_i, which is the increment in the Stieltjes integral, is the initial amplitude of a real, normalized solution of the differential equation over the finite interval $a,\ b$. That is another way to get the step in the spectral distribution function, because Parseval's equality now reads

$$\frac{m-m^*}{\lambda-\lambda^*} \sum_{i=0}^\infty\frac{r_i^2}{\vert\lambda-\lambda_i\vert^2}$$ = (367)

$$\int_{-\infty}^\infty\frac{d\rho(\mu)}{\vert\lambda - \mu\vert^2}$$ = (368)

$$\int_{-\infty}^\infty\frac{\rho'(\mu)d\mu}{\vert\lambda - \mu\vert^2}.$$ = (369)

The last line is admissable for points in the continuous spectrum of the differential operator, but the Stieltjes form must be retained for the point spectrum. If $\rho'$ exists, it is called the spectral density.

The description of the spectral density calls for some care in its presentation. In works on solid state theory especially, the spectral density is often considered to be ``the number of eigenvalues per unit of frequency interval,'' which is only a part of the story. This works well for plane waves, or functions which are more or less homogeneous throughout their extent, but a much more important consideration is the relationship between near amplitude and far amplitude. That determines the weight any given function requires to build up a given wave packet, and it is that weight which is properly the spectral density.

A good way to appreciate the difference is to return to the Dirac Harmonic Oscillator previously discussed. The spectrum is continuous, and there is no particular reason to think of the number of eigenvalues per unit interval because they are pretty much uniformly distributed. But most of them dissipate the presence of their particle by their large relative amplitude at infinity. Only eigenvalues in small, selected, intervals contribute to the presence of a particle near at hand, and it is this emphasis which assigns them a high spectral density.

  

Figure 26: The spectral density function for the Dirac Harmonic Oscillator is the section of this surface bisecting the figure. Actually, there is a spectral density matrix [5], of which this section only reveals the even, or (0,0), element of that matrix.

Figure 26, although it graphs probability density as a function of both energy and distance, has been normalized to unit amplitude at infinity, so the values over zero distance portray the Titchmarsh-Weyl m-function, or in other words, the spectral density.

Whereas it might be overly ambitious to write

$$m(\lambda) \int_{-\infty}^\infty\frac{\rho'(\mu)d\mu}{\lambda - \mu}$$ = (370)

and compare it to Cauchy's integral formula, the relation certainly holds for the imaginary part of the equation, and invites considering $\rho'$ as the boundary value of an analytic function which could be extrapolated throughout a half-plane, at least [19].