Revisited version of Weyl's limit point-limit circle criterion for essential self-adjointness

A new proof of the Weyl limit point-limit circle criterion is obtained, with systematic emphasis on Sobolev-space methods.


Introduction
Limit-Point Limit-Circle theory was first developed by the young Herman Weyl in the early 1900's in one of his first articles [1]. Since then, such methods (hereafter denoted by LP and LC, respectively) have become increasingly important thanks to their accurate predictions on the form of the potential in the applications, which can easily supply foundamental information about the solution of a great variety of singular second-order Sturm-Liouville problems. In the modern literature, the work in [2,3] provides an enlightening introduction to the link between these singular second-order problems and functional analysis, as well as to the applications to ordinary quantum mechanics.
Section 2 describes what is known from a Carathéodory theorem on ordinary differential equations; section 3 studies the Sobolev functional space for solutions of our singular second-order problems; Weyl's LP-LC criterion is studied with extensive and original use of Sobolev spaces in sections 4 and 5. Concluding remarks are made in section 6, while relevant details are given in the appendix. Throughout our paper, the reader is assumed to have some background on the LP-LC theory [4,5] and on operator theory [2,3]. In the LP-LC literature, whenever one deals with equation (2.1), it is necessary to specify the functional space to which the coefficients p(x), p ′ (x) and q(x) belong, in order to develop the theory and reach the desired results.

Extended Carathéodory's existence Theorem
For example, in [4,5,6] the coefficients p, p ′ and q belong to the set of real-valued continuous functions and hence the solution y must be globally of class C 2 on the interval I ⊆ R of interest. On the other hand, in [7,8], weaker conditions on p −1 , q are given -i.e.-they are L 1 loc while the solution y and its derivative y ′ are absolutely continuous (AC loc ) on the interval I of interest.
In order to clarify the hypotesis made on such functions and on the solutions, we will make use of some extended Carathéodory's existence theorems: Theorem 2.1. Let I ⊂ R be a closed interval and let f x, u(x) be G-regular on I (see appendix). Then there exists at least one absolutely continuous function u such that where x 0 is the average point of I.
We note that if f obeys the above theorem, there exists at least one absolutely continuous function u that satisfies the equation u ′ = f x, u(x) almost everywhere. 1) For every y ∈ V , f (x, y) is measurable on I and it is continuous in V .
Then there exists an absolutely continuous function u(x) such that u ′ (x) = f x, u(x) almost everywhere on I.
In [9] it is shown that the requirement 2) of Theorem 2.2 can be replaced by the following: 2') For every y ∈ C(I), f x, y(x) is summable in I, and upon taking y ∈ C(I), the functions x a f t, y(t) dt describe an absolutely-equicontinuous family on I obtaining a more general existence theorem.
Here we want to show that, if the assumptions of Theorem 2.1 are verified and we also take hypothesis 2') instead of 2), then Theorem 2.2 must also be true. From this the former will be a restricted case of the latter.
As already mentioned, for the definition of G-regularity we remind to the Appendix at the end of this paper. Here we will only give the main condition that ensures the occurrence of this property. For this purpose we need some further notions: Definition 1. Let I ⊂ R and let h 1 , ..., h m , k 1 , ..., k m ∈ L 1 (I, R). We define the subse- for some M ∈ L 1 (I, R) and it is measurable in the x variable for any fixed y and it is also continuous in the y variable, then there exists a G(h, k) such that f is G-regular on I.
Then f is said to be G-integrable on I.
As we can see from the definition of G-regularity in our Appendix, the G-integrability is necessary for the G-regularity. By using some content in [10] we can easily see that the G-integrability implies the absolute continuity of x x 0 f t, y(t) dt in the x variable where y is taken to be absolutely continuous as in (2.4).
From this, one finds that the G-regularity makes (2.2) absolutely continuous whenever a particular absolutely continuous function u(x) is chosen, and hence Since under our hypothesis f is taken G-integrable, it is also bounded from the Definition 2 and this suggests us that x x 0 f s, u(s) ds is an equi-absolutely continuous family of functions. From this we have already proved that Theorem 2.1 is a special case of Theorem 2.2 when the 2) hypothesis is replaced with 2').

Sobolev functional space for solutions
Now, by expressing (2.1) in the subsequent form of first-order differential system: it is easy to see, by applying Theorem (2.2) with the 2') hypothesis, that the required summability in the x variable forces the coefficients p(x) −1 , p ′ (x) and q(x) to belong to L 1 (I) while the solution y to (2.1) and its derivative y ′ are absolutely continuous functions. Now we will face the fact that an absolutely continuous function must belong to a Sobolev space -i.e.-W 1,1 defined, for example, in [11]: for all φ ∈ C 1 0 (a, b). We also recall the following 3) then f is said to be absolutely continuous.
It is well known that the weaker classical hypothesis that makes it possible to perform an integration by parts (Lebesgue fondamental integral theorem) such as is the absolute continuity of the function f . Taking into account the Definition 4, jointly with the properties obtained above, we see that every absolutely continuous function belongs to the W 1,1 ([a, b]) Sobolev space.

Weyl's LP-LC criterion
Let us consider the following special case of Sturm-Liouville equation on (a, b) taking This is an eigenvalue equation whose differential operator is defined on L 2 (a, b). The aim of the following Weyl's Criterion is to provide the condition on the operator (4.2) in order to ensure its self-adjointness in terms of the LP-LC property.
In this way, such a theorem provides a magnificent link between operator theory on Hilbert spaces and LP-LC theory [2,3]: Then the closureL has got deficiency indices: From the extended version of Carathéodory existence theorem we know that, if a solution to equation (4.1) exists in a compact I, then it must be absolutely continuous together with his first derivative and thus it belongs to W 1,1 (I).
We can introduce the following Sobolev space that will be the basic living place for our solutions: Definition 6.
Of course, we are only interested in functions which are absolutely continuous with their first derivative.
We note that the operator (4.2) must act on a Hilbert space -i.e.-L 2 (I) and therefore, from (4.1), the function d 2 y dx 2 ∈ L 2 (I). The subsequent theorem [9] shows that the hypothesis on the second derivative of our solutions to belong to L 2 is sufficient to guarantee us the local absolute continuity of the solutions and their first derivative: loc (I) and take for some y 0 ∈ I the following expression: Then, v(x) is continuous in I and By applying recursively the above theorem one finds that, under the hypothesis of square summability of its second derivative, y and its first derivative are locally absolutelly continuous on I whatever I is. It is also clear that, in the case of a compact real interval, y and y ′ are absolutely continuous functions of W 1,1 (I) and hence y belongs to the (4.3) set.
Now, if I is not bounded or half-bounded, we are dealing with functions belonging to W 2,1 loc (I) and taking the square summability required for operator (4.2), the basic functional space to which our solution belongs is W 2,1 loc (I) ∩ L 2 (I).
Now we can summarise our results in the following theorem: Of course, if I is compact then W 2,1 loc (I) ≡ W 2,1 (I) while if the square summability is required, then y ′′ must belong to L 2 (I) and thus we obtain the local absolute continuity of y and y ′ and the Carathéodory's existence theorem is fullfilled.

Proof of Weyl's criterion
We can now proceed with the proof of Theorem 4.1 by following the logical steps that can be found in [3]. Our method will make use of Theorem 4.2 jointly with all the information obtained in the previous section.
Proof of statement (i): If the operator L is LC at both ends of the interval I, then every solution to the equation Ly = ly, ∀l ∈ C for which ℑl = 0, belongs to L 2 (I). This means that there exist two linearly independent solutions to each of the equations Ly = iy and Ly = −iy, and therefore the deficiency indices are n + (L) = n − (L) = 2.

Proof of statement (ii):
Suppose that L is LP at a and LC at b.
Let us consider a restriction L 0 of the operator L acting on the subsequent linear domain: where [c, d] ⊂ I. From Theorem 4.2 and the comment below, we have that D(L 0 ) ⊂ It is easily seen that operator L 0 is symmetric and that the domain of its adjoint is because the equations have at most two linearly independent solutions in L 2 (c, d) and hence n + (L 0 ) ≤ 2 and n − (L 0 ) ≤ 2. We must rule out the case n + (L 0 ) = n − (L 0 ) = 0 because it is the self-adjoint one and this is not the case because D(L 0 ) ⊂ D(L * 0 ). Now we can show that each of equations (5.3) has only one solution in L 2 (c, d).
It is indeed well known that the adjoint domain for a linear operator on a Hilbert space admits the following decomposition: where K + (A) and K − (A) are the deficiency spaces of the operator under consideration.
Let us define the operator P 2 = − d 2 dx 2 on the domain (5.1) and let us denote it by P 2 0 . Of course P 2 0 is symmetric and its adjoint has domain (5.2), thus we can certainly say that . By solving the equations with the condition imposed by (5.2), we obtain two one-dimensional deficiency spaces of the form where ζ 1 , ζ 2 , ρ 1 and ρ 2 are locally square integrable on the real line. Equations (5.6) show is a two-dimensional linear space and hence the same holds for K + (L 0 ) ⊕K − (L 0 ). Now, taking into account the fact that [4,5] if, for some complex l 0 ∈ C all solutions to L 0 y = l 0 y are square integrable, than for every complex l ∈ C every solution to L 0 y = ly is square integrable as well, we must rule out the cases dim(K + (L 0 )) = 2, dim(K − (L 0 )) = 0 and dim(K + (L 0 )) = 0, dim(K − (L 0 )) = 2. From this, the only case left is dim(K + (L 0 )) = dim(K − (L 0 )) = 1 and therefore n + (L 0 ) = n − (L 0 ) = 1. This also means that there exists a non-vanishing functionũ that does not belong to Ran(L 0 − iI).
At this stage, let I − = (a, d] and let L 1 be a second restriction of L defined on We note that in the case in which the end-point a is at finite distance from the origin, the basic space in (5.7) can alwais be taken to be W 2,1 (I − ) instead of W 2,1 (I − ) ∩ L 2 (I − ).
All our reasoning on L 0 can be repeated on L 1 , leading us to the same conclusions: L 1 is symmetric on its domain and has got the same deficiency indices of L 0 . By using some arguments that can be found in [2], we can state that there exists at most one self-adjoint In order to do this, we first need some arguments on the Wronskian function where D(W) is defined according to We can now proceed with the proof of the (iii) statement, in which we will make use of the results obtained above.
Suppose that the end-point a is regular while b is LP. It is easily seen that L has got self-adjoint extensions because it is symmetric on C ∞ 0 (I) and it has deficiency indices n + (L) = n − (L) = 1 like the operator L 1 defined in the proof of the (ii) statement. Among all conceivable self-adjoint extensions, we want to choose that one for which there exists some (α, β) ∈ R 2 \(0, 0) such that αφ(a) + βφ ′ (a) = 0 (5.15) and call itL. In order to do this we will use a Theorem in [12], can be stated as follows [3,12]: If A is a closed Hermitian operator, there exists a 1 − 1 correspondence between closed symmetric extensions B of A, and partial isometries V , with initial space H I (V ) ⊂ K + domains of the closed symmetric extensions for a closed symmetric operator, by using partial isometries between the deficiency spaces K + (L) and K − (L). Of course L is closable, thus we can make use of von Neumann's Theorem.
Let us define the following unitary operator between the deficiency spaces that acts like a complex conjugation: Certainly U is an isometry, hence it is bijective.
From von Neumann's Theorem we know that the self-adjoint extension related to the unitary operator (5.16) has got the following domain: It is straightforward that the function in D(L) satifies the relation (5.15) for same α and β.
Eventually, if we show that W(x;φ, ψ) = 0 ∀φ, ψ ∈ D(L * ) (5.18) and final space H F (V ) ⊂ K − . This correspondence is expressed by or by This completely shows that (5.18) holds, and the desired proof is completed.

Concluding remarks: central potentials and other issues
The original contribution of our paper is a systematic investigation of Weyl's LP-LC criterion that, while relying upon the work of Weyl [1] and the outstanding modern treatment of [2,3], makes use whenever possible of modern tools such as Sobolev spaces and G-regularity.
The world of atomic physics offers a wide range of applications of ordinary quantum mechanics. This is not an exact theory, because the use of relativity would make it necessary to use the spectral theory of pseudo-differential operators in order to develop the quantum theory of bound states [13]. Thus, one still resorts with profit to ordinary quantum mechanics, from which one can learn valuable lessons. For example, if a physical system ruled by a central potential V (r) is considered in R n (the choice n = 3 is frequent but not mandatory), one finds an effective potential where, on denoting by l the orbital angular momentum quantum number, one finds [14] ρ nl = (n − 1)(n − 3) As one learns from [2,3], the LP condition at the origin is achieved if In the particular case of a free particle, V (r) = 0 and (6.3) leads to Interestingly, this condition is violated just once, i.e. by s-wave stationary states (for which l = 0) in 3 dimensions. The same holds if V (r) is a Coulomb-type potential, because then the centrifugal term on the left-hand side of (6.3) dominates on the Coulomb term as r approaches 0. Here we want to show that indeed, the operator − d 2 dx 2 has got more that one self-adjoint extension, and these correspond to the Dirichlet and Neumann conditions at the origin. We can proceed in the following way: 1) First we consider a particular class of domains -i.e.-D {ǫ,µ} , that let our operator be closed and symmetric and from this, using some arguments contained in [2], we are ensuring the existence of self-adjoint extensions for such closed and symmetric restrictions.
2) We use the von Neumann's theorem [2,3,12]  Let us define the following two-parameter domains of symmetry for − d 2 dx 2 : It is easy to see that such domains are closed and on them our operator is symmetric, and from the fact that n + = n − = 1, there exist self-adjoint extensions for each fixed admissible pair (ǫ, µ).
In order to use von Neumann's Theorem, we need the expression of the deficiency spaces, and we easily find that We see that such spaces are one-dimensional linear spaces and, from von Neumann's Theorem we know that all possible symmetric extensions for each of D {ǫ,µ} , are in bijection with the isometries between the deficiency spaces. From the fact that the deficiency spaces are one-dimensional, the isometries required can only be phase factors of the form e iθ(x,c + ,c − ) and therein, following the statement of the Theorem, we must have and since it follows that c − c + = 1, we set c − c + = e ic with c ∈ [0, 2π[, and this shows that From the von Neumann's criterion [2,3,12] we can give the explicit form to the domains of symmetric extensions that we will call D {ǫ,µ} (c): where φ ∈ D {ǫ,µ} , z ∈ C. Taking into account the symmetry relation where ψ ∈ D {η,µ} (c) and ξ ∈ D * {η,µ} (c), we obtain the following equation: that defines the two following kinds of adjoint domains: with c = π 2 , and with c = π, in which we have ruled out π 2 and π values that lead to singular ratios and ξ ′ (0) ξ(0) , respectively. First of all, it is interesting to note that (6.11) and (6.12) are independent of the (ǫ, µ) pair. In this way we can certainly say that (6.11) and (6.12) cover all possible domains for the adjoints of the closed and symmetric extensions for any of the possible closed and symmetric realizations of − d 2 dx 2 over the real half-line. Now, by using the self-adjointness relations D {ǫ,µ} (c) = D * 1 (c) and D {ǫ,µ} (c) = D * 2 (c) we easily get the following self-adjointness domains: that correspond to the Dirichlet and Neumann condition at the origin.
The last thing that we want to note is that the sets (6.13) and (6.14) are are both closed and open.
For example, by using the following sequence in D 1 (π) ∩ D 2 π 2 : which converges in L 2 but not in the intersection of D 1 (π) and D 2 π 2 , we realize that (6.13) and (6.14) are open sets. On the other hand, if we choose the following: 2 ) , hence it belongs to the complement of each D 1 (π) and D 2 π 2 for every n ∈ N but its limit belongs D 1 (π) ∩ D 2 π 2 . This shows that the complement of (6.13) and (6.14) is an open set and therefore (6.13) and (6.14) must be closed sets.
The fact that they are closed sets also results from von Neumann's Theorem. In this way, the sequence (6.16) confirms the validity of such a Theorem. Eventually, we have obtained that the operator − d 2 dx 2 is Self-Adjoint only on domains (6.13) and (6.14), which are simultaneously closed and open. Similar techniques have been applied, over the years, to a wide range of topics. For example, the work in [15] studied essential self-ajointness in 1-loop quantum cosmology, whereas the work in [16] has suggested that a profound link might exist between the formalism for asymptotically flat space-times and the limit-point condition for singular Sturm-Liouville problems in ordinary quantum mechanics. Last, but not least, the parameter λ nl ≡ l + (n−2) 2 in Eq. (6.2) is neatly related to the parameter L used in large-N quantum mechanics [17], i.e. λ nl = L 2 − 1. (6.17) Moreover, since the Schrödinger stationary states are even functions of λ nl , this suggests exploiting the complex-λ nl plane in the analysis of scattering problems [18]. If n is kept arbitrary, this means complexifying a linear combination of l and n [14], including the particular case where l remains real while the dimension n is complexified.
Thus, there is encouraging evidence that Sobolev-space methods and yet other concepts of functional and complex analysis may provide the appropriate tool for investigating classical and quantum physics as well as correspondences among such frameworks.
Acknowledgments V F B is grateful to G E for support and useful advice. G E is grateful to Dipartimento di Fisica "Ettore Pancini" for hospitality and support.

Appendix: G-regularity
For the definition of G-regularity we need to endow the R n space with the norm ||y|| = n i=1 |y i |.
Here we make use of the topology induced by the uniform convergence in order to give to the C(I, R n ) vector space a Banach structure. Let us call it K.
We will say that the sequence u n ∈ K −→ u ∈ K if and only if, for every ǫ ∈ R + , there exists some ν ∈ N such that, for all n ≥ ν the following majorization is verified: Since K is also a metric space, we can use in (A2) the following notation for the distance between u n and u: At this stage we have only to define the following particular sequence on the I segment: x n = x − d n sign(x) (A4) and we note that x n −→ x on I only if u n −→ u on K.
We are now in a position to define the concept of G-regularity, while we refer to Definition 2 for the G-integrability.
Definition 7. Let f be G-integrable on the interval I whose middle point is x 0 , and let g ∈ G be defined in equation (2.3). Let the sequence u n (x) = x x 0 g n (s)ds ∈ G be such that u n tends uniformly to u(x) = then we say that f is G-regular on I.
For more insights on the G-regularity property of functions, we refer the reader to [10].