Let now y = ui(x),i = 1, 2, 3, ..., be the series of Sturm-Liouville proper functions; then the series of functions , i = 1, 2, 3, . . . , forms a complete orthogonal system for the domain; i.e. in the first place, if ui(x) and uk(x) are the proper functions belonging to the values Ei and Ek , then
(3)
(Integrals without limits are to be taken over the domain, throughout this paper.) The expression “complete” signifies that an originally arbitrary continuous function is condemned to vanish identically, by the mere postulation that it must be orthogonal with respect to all the functions . (More shortly: “There exists no further orthogonal function for the system.”) We can and will always regard the proper functions ui(x) in all general discussions as “normalised”, i.e. we imagine the constant factor, which is still arbitrary in each of them on account of the homogeneity of (2), to be defined in such a way that the integral (3) takes the value unity for i = k. Finally we again remind the reader that the proper values of (2) are certainly all real.
Let now the proper values Ei and functions ui(x) be known. Let us, from now on, direct our attention specially to a definite proper value, Ek say, and the corresponding function uk(x), and ask how these alter, when we do not alter the problem in any way other than by adding to the left-hand side of (2) a small “perturbing term”, which we will initially write in the form
(4)
In this λ is a small quantity (the perturbation parameter), and r(x) is an arbitrary continuous function of x. It is therefore simply a matter of a slight alteration of the coefficient q in the differential expression (1). From the continuity properties of the proper quantities, mentioned in the introduction, we now know that the altered Sturm-Liouville problem
(2′)
must have, in any case for a sufficiently small λ, proper quantities in the near neighbourhood of Ek and uk , which we may write, by way of trial, as
(5)
On substituting in equation (2′), remembering that uk satisfies (2), neglecting λ2 and cutting away a factor λ we get
(6)
For the defining of the perturbation vk of the proper function, we thus obtain, as a comparison of (2) and (6) shows, a non-homogeneous equation, which belongs precisely to that homogeneous equation which is satisfied by our unperturbed proper function uk (for in (6) the special proper value Ek stands in place of E). On the right-hand side of this non-homogeneous equation occurs, in addition to known quantities, the still unknown perturbation εk of the proper value.
This occurrence of εk serves for the calculation of this quantity before the calculation of vk. It is known that the non-homogeneous equation—and this is the starting-point of the whole perturbation theory—for a proper value of the homogeneous equation possesses a solution when, and only when, its right-hand side is orthogonaleo to the allied proper function (to all the allied functions, in the case of multiple proper values). (The physical interpretation of this mathematical theorem, for the vibrations of a string, is that if the force is in resonance with a proper vibration it must be distributed in a very special way over the string, namely, so that it does no work in the vibration in question; otherwise the amplitude grows beyond all limits and a stationary condition is impossible.)
The right-hand side of (6) must therefore be orthogonal to uk, i.e.
(7)
or
(7′)
or, if we imagine ui already normalised, then, more simply,
(7”)
This simple formula expresses the perturbation of the proper value (of first order) in terms of the perturbing function r(x) and the unperturbed proper function uk(x). If we consider that the proper value of our problem signifies mechanical energy or is analogous to it, and that the proper function uk is comparable to “motion with energy Ek”, then we see in (7”) the complete parallel to the well-known theorem in the perturbation theory of classical mechanics, viz. the perturbation of the energy, to a first approximation, is equal to the perturbing function, averaged over the unperturbed motion. (It may be remarked in passing that it is as a rule sensible, or at least aesthetic, to throw into bold relief the factor ρ(x) in the integrands of all integrals taken over the entire domain. If we do this, then, in integral (7”), we must speak of and not r(x) as the perturbing function, and make a corresponding change in the expression (4). Since the point is quite unimportant, however, we will stick to the notation already chosen.)
We have yet to define vk(x), the perturbation of the proper function , from (6). We solveep the non-homogeneous equation by putting for vk a series of proper functions, viz.
(8)
and by developing the right-hand side, divided by ρ(x), likewise in a series of proper functions, thus
(9)
where
(10)
The last equality follows from (7). If we substitute from (8) and (9) in (6) we get
(11)
Since now ui satisfies equation (2) with E = Ei , it follows that
(12)
By equating coefficients on left and right, all the γki ’s, except γkk , are defined. Thus
(13)
while γkk, as may be understood, remains completely undefined. This indefiniteness corresponds to the fact that the postulation of normalisation is still available for us for the perturbed proper function. If we make use of (8) in (5) and claim for the same normalisation as for uk(x) (quantities of the order of λ2 being neglected), then it is evident that γkk = 0. Using (13) we now obtain for the perturbed proper function
(14)
(The dash on the sigma denotes that the term i = k has not to be taken.) And the allied perturbed proper value is, from the above,
(15)
By substituting in (2ʹ) we may convince ourselves that (14) and (15) do really satisfy the proper value problem to the proposed degree of approximation. This verification is necessary since the development, assumed in (5), in integral powers of the perturbation parameter is no necessary consequence of continuity.
The procedure, here explained in fair detail for the simplest case, is capable of generalisation in many ways. In the first place, we can of course consider the perturbation in a quite similar manner for the second, and then the third order in λ, etc., in each case obtaining first the next approximation to the proper value, and then the corresponding approximation for the proper function. In certain circumstances it may be advisable—just as in the perturbation theory of mechanics—to regard the perturbation function itself as a power series in λ, whose terms come into play one by one in the separate stages. These questions are discussed exhaustively by Herr E. Fues in work which is now appearing in connection with the application to the theory of band spectra.
In the second place, in quite similar fashion, we can consider also a perturbation of the term in y′ of the differential operator (1) just as we have considered above the term −qy. The case is important, for the Zeeman effect leads without doubt to a perturbation of this kind—though admittedly in an equation with several independent variables. Thus the equation loses its self-adjoint form by the perturbation—not an essential matter in the case of a single variable. In a partial differential equation, however, this loss may result in the perturbed proper values no longer being real, though the perturbing term is real; and naturally also conversely, an imaginary perturbing term may have a real, physically intelligible perturbation as its consequence.
We may also go further and consider a perturbation of the term in y”. Indeed it is quite possible, in general, to add an arbitrary “infinitely small” lineareq and homogeneous differential operator, even of higher order than the second, as the perturbing term and to calculate the perturbations in the same manner as above. In these cases, however, we would use with advantage the fact that the second and higher derivatives of the proper functions may be expressed by means of the differential equation itself, in terms of the zero and first derivatives, so that this general case may be reduced, in a certain sense, to the two special cases, first considered—pert
urbation of the terms in y and y′ .
Finally, it is obvious that the extension to equations of order higher than the second is possible.
Undoubtedly, however, the most important generalisation is that to several independent variables, i.e. to partial differential equations. For this really is the problem in the general case, and only in exceptional cases will it be possible to split up the disturbed partial differential equation, by the introduction of suitable variables, into separate differential equations, each only with one variable.
§ 2. SEVERAL INDEPENDENT VARIABLES (PARTIAL DIFFERENTIAL EQUATION)
We will represent the several independent variables in the formulae symbolically by the one sign x, and briefly write ∫ dx (instead of ∫ ···∫ dx1dx2 . . .) for an integral extending over the multiply-dimensioned domain. A notation of this type is already in use in the theory of integral equations, and has the advantage, here as there, that the structure of the formulae is not altered by the increased number of variables as such, but only by essentially new occurrences, which may be related to it.
Let therefore L[y] now signify a self-adjoint partial linear differential expression of the second order, whose explicit form we do not require to specify; and further let ρ(x) again be a positive function of the independent variables, which does not vanish in general. The postulation “self-adjoint” is now no longer unimportant, as the property cannot now be generally gained by multiplication by a suitably chosen f(x), as was the case with one variable. In the particular differential expression of wave mechanics, however, this is still the case, as it arises from a variation principle.
According to these definitions or conventions, we can regard equation (2) of §1, as the formulation of the Sturm-Liouville proper value problem in the case of several variables also. Everything said there about the proper values and functions, their orthogonality, normalisation, etc., as also the whole perturbation theory there developed—in short, the whole of § 1—remains valid without change, when all the proper values are simple, if we use the abbreviated symbolism just agreed upon above. And only one thing does not remain valid, namely, that they must be simple.
(2)
Nevertheless, from the pure mathematical standpoint, the case when the roots are all distinct is to be regarded as the general case for several variables also, and multiplicity regarded as a special occurrence, which, it is admitted, is the rule in applications, on account of the specially simple and symmetrical structure of the differential expressions L[y] (and the “boundary conditions”) which appear. Multiplicity of the proper values corresponds to degeneracy in the theory of conditioned periodic systems and is therefore especially interesting for quantum theory.
A proper value Ek is called α-fold, when equation (2), for E = Ek, possesses not one but exactly α linearly independent solutions which satisfy the boundary conditions. We will denote these by
(16)
Then it is true that each of these α proper functions is orthogonal to each of the other proper functions belonging to another proper value (the factor ρ(x) being included; cf. (3)). On the contrary, these α functions are not in general orthogonal to one another, if we merely postulate that they are α linearly independent proper functions for the proper value Ek , and nothing more. For then we can equally well replace them by α arbitrary, linearly independent, linear aggregates (with constant coefficients) of themselves. We may express this otherwise, thus. The series of functions (16) is initially indefinite to the extent of a linear transformation (with constant coefficients), involving a non-vanishing determinant, and such a transformation destroys, in general, the mutual orthogonality.
But through such a transformation this mutual orthogonality can always be brought about, and indeed in an infinite number of ways; the latter property arising because orthogonal transformation does not destroy the mutual orthogonality. We are now accustomed to include this simply in normalisation, that orthogonality is secured for all proper functions, even for those which belong to the same proper value. We will assume that our uki ’s are already normalised in this way, and of course for each proper value. Then we must have
. (17)
Each of the finite series of proper functions uki , obtained for constant k and varying i, is then only still indefinite to this extent, that it is subject to an orthogonal transformation.
We will now discuss, first in words, without using formulae, the consequences which follow when a perturbing term is added to the differential equation (2). The addition of the perturbing term will, in general, remove the above-mentioned symmetry of the differential equation, to which the multiplicity of the proper values (or of certain of them) is due. Since, however, the proper values and functions are continuously dependent on the coefficients of the differential equation, a small perturbation causes a group of α proper values, which lie close to one another and to Ek , to enter in place of the α-fold proper value Ek. The latter is split up. Of course, if the symmetry is not wholly destroyed by the perturbation, it may happen that the splitting up is not complete and that several proper values (still partly multiple) of, in summa, equal multiplicity merely appear in the place of Ek (“partial removal of degeneracy”).
As for the perturbed proper functions, those a members which belong to the α values arising from Ek must evidently also on account of continuity lie infinitely near the unperturbed functions belonging to Ek, viz. uki; i = 1, 2, 3 . . . α. Yet we must remember that the last-named series of functions, as we have established above, is indefinite to the extent of an arbitrary orthogonal transformation. One of the infinitely numerous definitions, which may be applied to the series of functions, uki; i = 1, 2, 3 . . . α, will lie infinitely near the series of perturbed functions; and if the value Ek is completely split up, it will be a quite definite one! For to the separate simple proper values, into which the value is split up, there belong proper functions which are quite uniquely defined.
This unique particular specification of the unperturbed proper functions (which may fittingly be designated as the “approximations of zero order” for the perturbed functions), which is defined by the nature of the perturbation, will naturally not generally coincide with that definition of the unperturbed functions which we chanced to adopt to begin with. Each group of the latter, belonging to a definite α-fold proper value Ek, will have first to be submitted to an orthogonal substitution, defined by the kind of perturbation, before it can serve as the starting-point, the “zero approximation”, for a more exact definition of the perturbed proper functions. The defining of these orthogonal substitutions—one for each multiple proper value—is the only essentially new point that arises because of the increased number of variables, or from the appearance of multiple proper values. The defining of these substitutions forms the exact counterpart to the finding of an approximate separation system for the perturbed motion in the theory of conditioned periodic systems. As we will see immediately, the definition of the substitutions can always be given in a theoretically simple way. It requires, for each α-fold proper value, merely the principal axes transformation of a quadratic form of α (and thus of a finite number of) variables.
When the substitution has once been accomplished, the calculation of the approximations of the first order runs almost word for word as in § 1. The sole difference is that the dash on the sigma in equation (14) must mean that in the summation all the proper functions belonging to the value Ek , i.e. all the terms whose denominators would vanish, must be left out. It may be remarked in passing that it is not at all necessary, in the calculation of first approximations, to have completed the orthogonal substitutions referred to for all multiple proper values, but it is sufficient to have done so for the value Ek, in whose splitting up we are interested. For the approximations of higher order, we admittedly require them all. In all other respects, however, these higher approximations are from the beginning carried out exactly as for simple proper values.
Of course it may happen, as was mentioned above, that the val
ue Ek, either generally or at the initial stages of the approximation, is not completely split up, and that multiplicities (“degeneracies”) still remain. This is expressed by the fact that to the substitutions already frequently mentioned there still clings a certain indefiniteness, which either always remains, or is removed step by step in the later approximations.
Let us now represent these ideas by formulae, and consider as before the perturbation caused by (4), § 1,
(4)
i.e. we imagine the proper value problem belonging to (2) solved, and now consider the exactly corresponding problem (2′),
(2′)
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