The classical Sturm-Liouville theory consists of finding the eigensolutions (and eigenvalues) for second order ordinary differential equations on a finite interval with no singularities of the potential inside. An important extension (in addition to coping with many dimensions) is to determine the spectral properties of the second order differential operator in the singular case, for instance at the origin for the Coulomb problem and in general for taking the limit to an infinite interval. This extension is rigorously given by Weyl's theory yielding an alternative approach to Schrödinger type equations. The spectrum may then be both descrete and continuous including appropriate spectral densities and the full mathematical apparatus to deal with both quantized states and scattering theory. Although S-L theory is well known as an introduction to quantum mechanics the further interplay between the spectral properties in the singular case is a more demanding and important enterprize.

Is this theory not restricted to 1D (a single real variable)? Quantum theory has to cope with many dimensions so that the direct relevance of Sturm-Liouville theory is necessarily limited.

S-L theory is well developed and need not any steps away to be applied to quantum theory. As to strings, I do not believe this way to be correct for many reasons. You are right writing that S-L theory is crucial for quantization. Actually we can realize quantization within the framework of the Einstein – Cartan – Shroedinger program, as it is shown here:

http://vixra.org/abs/1311.0058

This way we are able to resolve many actual problems of physics (nature of the Planck constant, problem of cosmological constant, unification of Gravity with Quant. Theory, problem of singularities, etc.). And main point for this is S-L theory.

@Ulrich, since by solving a PDE with separation techniques it is always possible to split any 3d problem in three 1d problems, I think yes, S-L is sufficient.

Http://www.ms.uky.edu/~rwalker/research/sturmliouvilletheory.pdf

The classical Sturm-Liouville theory consists of finding the eigensolutions (and eigenvalues) for second order ordinary differential equations on a finite interval with no singularities of the potential inside. An important extension (in addition to coping with many dimensions) is to determine the spectral properties of the second order differential operator in the singular case, for instance at the origin for the Coulomb problem and in general for taking the limit to an infinite interval. This extension is rigorously given by Weyl's theory yielding an alternative approach to Schrödinger type equations. The spectrum may then be both descrete and continuous including appropriate spectral densities and the full mathematical apparatus to deal with both quantized states and scattering theory. Although S-L theory is well known as an introduction to quantum mechanics the further interplay between the spectral properties in the singular case is a more demanding and important enterprize.

@Demetris

It is not always possible to separate a Schrodinger equation. An example is an hydrogen atom in an external electric field.

Dear Oscar,

You are right in principle, but your example is wrong. The hydrogen atom in an electric field is exactly separable (nonrelativistic problem) in paraboloc coordinates., see e.g.

Hehenberger et al. Phys. Rev. A 10, 1494 (1974). This paper which also appears among my publications also belabours my earlier comments regarding S-L theory.

Since I am not allowed to upload it I can only send pdf's privately.

Although, improvement has been achieved after 1837, the main idea of classical quantization is based on a simple S-L problem: Quantization arises when we have bounded problems. This is what we have forgotten and we are trying to find 'quantum anything' even when we have unbounded problems.

@Oscar, I have not studied in so details Schrodinger equation but I think by using Moon & Spencer 's book 'Field Theory Handbook' probably we can find one of the 11 systems to separate it, probably Erkki Brändas has already calculated the solution.

@Erkki

My bad. Sorry.

Oh Demetris, how it is possible for you to think these days that boundary conditions are the important point? The unified principle is: wave functions have to be normalized. This means in particular that in unbounded space it is not sufficient to satisfy the boundary condition lim |psi(x)|=0 for |x| tending to infinity.

@Erkki, can you give me an example of the "...general for taking the limit to an infinite interval"?

Demetris, If you send an email to me at [email protected] I will send you examples.

Ulrich, as you know Schrödinger described in his famous treatese, posing quantization as an eigenvalue problem, how to acquisce physically suitable solutions of the boundary value problem of the Schrödinger differential equation. He imposed the requirements of continuity, single valued- and finiteness. The quandry or complication, though well understood and worked out today yet sometimes misread, is the relation between these demands and the ones of square integrability, i.e. that the solution of interest should belong to a suitably defined Hilbert space. To rigorously develop useful generaliztions one needs to either pose precise boundary conditions for the physically interesting solutions or to find the relevant topology for the actual spaces.

Perhaps, I don’t know the history of Sturm-Liouville theory in detail, but I also clearly see that “Quantization arises when we have bounded problems”. It is natural.

Another aspect of Sturm-Liouville theory is the asymptotics of the number

of discrete eigenvalues and it's connection with the geometry of domain (spectral geometry). This theory is gone much further in XX-XXI century and is still developing even today.

@Erkki

Is Schroedinger equation separable in crossed fields?

@Oscar

No the Schrödinger equation is not separable for hydrogen in a magnetic field, see e.g. chapter 7 Hydrogenic Systems in Electric and Magnetic Fields by J. C. Gay in The Spectrum of Atomic Hydrogen. ADVANCES, World Scientific Singapore (1988) p. 365

Dear Erkki, thank you for providing me your work.

Here are my remarks about the 1974 paper.

1)First of all I'd like to notice that indeed complex numbers are rotations in the 2d plane since exp(i*theta)*r*exp(i*phi)=r*exp(i*(theta+phi)), so the Eq. 3 of 74' paper is interesting. Have you any idea about the original thought of Weyl (whom by the way we know more for Weyl's tensor)?

2)The problem is always bounded, although the use of complex plane leads to the limit case b-->infinity.

3)When you are solving numerically Eq.42 of 74' paper you really have a linear system of ODEs, so again we have linear science here (look another favorite topic for me for details)

4) You have used parabolic coordinates (both cases). Have you tried some of the other 9 coordinate systems that are available for separating Helmoltz equation?

5)Finally, which is your favorit explanation of lambda=E+i*epsilon?

@Demetris

Weyl introduced the complex iε in the eigenvalue parameter to allow taking the two limits, i.e. (i) interval —> ∞ (or coordinate approaching the singularity) and then (ii) ε—>0. These limiting procedures are part of Weyl’s limit point-limit circle theory. The “limit point” guarantees a unique square integrable solution, which gives the differential equation a unique solution with unique boundary conditions at the singularities (e.g. 0, ∞). The numerical integration yields straightforwardly the Green’s function (essentially the m-function) and the spectral function. Actually we tested both the conventional parabolic and squared parabolic coordinates. The only other possibility would be to consider the ellipsoidal coordinate system, see e.g the Advances review, but this is so because every other orthogonal coordinate system separating the Helmholz equation, can be obtained by taking appropriate limits in the curvilinear system. In fact in most applications, when the asymptotic solution to the differential quation is known, one can avoid the ε−limit (putting ε=0). Finally note that the complex rotation of the coordinates are only imparted to find resonances (corresponding to Gamow waves) on the second Riemann sheet. In the Stark problem all hydrogen states become resonances and the continuum goes from -∞ to +∞. Fortunately the Stark problem is Weyl’s limit point!

@Erkki, and what if we weren't so lucky? What is the intuition behind the intoduction of E+i*e? That's the interesting point plus the rotation matrix defined with cos(a), sin(a).

Dear Demetris, Here are some answers:

1. ….and what if we weren't so lucky?

In case of Weyl’s limit circle case all solutions (going into the singularity) becomes square integrable. This happens in fact for the hydrogen atom problem separated in spherical polar coordinates, at the origin. However, the standard choice of the solution being zero at the origin assures selfadjointness of the three-dimensional Laplacian. In general, unless you can find a mathematical argument for the unique boundary condition at the origin or infinity (singularity) the corresponding operator does not have a self-adjoint extension.

2. What is the intuition behind the introduction of E+iε?

The reason is to guarantee that the resolvent (or m-function) stays finite.

3. …..plus the rotation matrix defined with cos(a), sin(a)

The numerical integration must start at a point inside the interval, where there are no singularities. Choosing the point just outside the singularity (here the origin) imparts the problem of determining ”a” such that psi corresponds to the regular (square integrable) solution at the origin.

General comment:

For more details on points 1-3, see J. Math. Phys. 26, 2648 1985. I also inserted, in my previous answer, a sentence on complex rotation, which is only relevant for the studies of the resonance problem, see Phys. Rev. A 37, 3777 (1988), where the Coulomb problem is worked out in quite some detail.

The essential core formalism of sturm Liouville, with inner product with respect to a weight function, is naturally extensible either to higher dimensions or infinite intervals or both. As noted above proper bounaries lead to the phenomena of discrete levels (quantization in QM) and orthogonality with respect to some weight.(using the hermitian operators) Schrodinger is after all a special case.

There are some unconventional versions of QM quite adaptable to this kind of formalism. It turns out that matter conservation, usually proclaimed consequence of Schrodinger, is actually a much more general consequence within this framework.(redefined probability density)

I think there are examples in SL where some states are bounded, even though they should not be in Schrodinger.