Self-adjoint extensions with Friedrichs lower bound

We produce a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. Applications of this abstract result to a few instructive examples are then discussed.


Motivation
We start with a familiar example. In the Hilbert space H = L 2 (0, 1) let us consider the densely defined, closed, and symmetric operator S is actually the operator closure of the negative Laplacian defined on C ∞ 0 (0, 1). Here and in the following D(R) denotes the domain of the operator R acting on H, and if R is symmetric we denote by It is also a general property of the Friedrichs extension the fact that S F S for any other S = S * ⊃ S, namely S F is the largest of all the self-adjoint extensions of S in the sense of operator form ordering.
As well known, as follows from (1.5), S F is diagonalizable over an orthonormal basis { √ 2 sin nπx | n ∈ N} of eigenfunctions, with (simple and pure point) spectrum This occurrence is well known: a lower semi-bounded symmetric operator may admit self-adjoint extensions other than the Friedrichs, with the same bottom of the Friedrichs spectrum. Actually this is not typical of symmetric operators with deficiency index 2 only, as was the case for S here. In Section 3 also examples with deficiency indices 1 will be recalled and discussed. By standard direct sum, these examples also cover the case of infinite deficiency indices. Now, while the possibility of non-Friedrichs self-adjoint extensions with the same Friedrichs lower bound is folk knowledge, we are not aware of an explicit operatortheoretic explanation of this phenomenon, nor of a characterisation in terms of transparent conditions which, once they are met, allow to construct all extensions with such a feature.
In this note we present a simple criterion and a constructive recipe to identify those self-adjoint extensions of a lower semi-bounded symmetric operator on Hilbert space which have the same lower bound as the Friedrichs extension. The abstract main results, Theorems 2.4, 2.5, and 2.6 below, are discussed in Section 2, and illustrative concrete examples where such results can be applied to are then presented in Section 3.

Abstract results
Let H be a Hilbert space (over R or C, with scalar product ·, · anti-linear in the first entry, and with norm ) and let S be a densely defined, symmetric, semibounded operator on H with lower bound m(S). S in not necessarily closed. For clarity of the presentation we shall assume non-restrictively m(S) > 0. This implies that S −1 F is everywhere defined and bounded on H. It will be clear both from this abstract discussion and from the applications in Section 3 that the case of general finite m(S) can be covered by suitably shifting S to S − µ1 with µ < m(S). Unless such S is already essentially self-adjoint, it admits non-trivial self-adjoint extensions. In this case ker S * , the deficiency space for S, is non-trivial either. Standard extension schemes produce convenient classifications of the whole family of extensions. It can be shown within the modern theory of boundary triplets [2], or equivalently the classical 'universal' parametrization by Grubb [7], and in fact the very original extension theory by Kreȋn [9], Višik [12], and Birman [3], that the extensions of S can be labelled as follows.
Theorem 2.1. There is a one-to-one correspondence between the family of all selfadjoint extensions of S on H and the family of the self-adjoint operators on Hilbert subspaces of ker S * .
(i) If T is any such operator, in the correspondence T ↔ S T each self-adjoint extension S T of S is given by As a consequence, Theorem 2.1 collects results that are proved, e.g., in [8,Chapt. 13], [11,Chapt. 14], and [5,Sect. 3].
For convenience, let us denote by S(K) the collection of all self-adjoint operators defined in Hilbert subspaces of a given Hilbert space K: Theorem 2.1 states that the self-adjoint extensions of S are all of the form S T for some T ∈ S(ker S * ).
The Friedrichs extension of S can be expressed in terms of the classical decomposition formula (see, e.g., An ancillary result that tends to be somehow less highlighted, but which is most relevant for our discussion, is the following. Theorem 2.2. If, with respect to the notation of (2.1), S T is a self-adjoint extension of S, and if µ < m(S), then (2.7) As an immediate consequence, Theorem 2.2 reproduces the inequality m(T ) m(S T ) for any semi-bounded S T and shows, in particular, that positivity or strict positivity of the bottom of S T is equivalent to the same property for T , that is, To make this presentation self-contained, and for later convenience, let us deduce Theorem 2.2 from Theorem 2.1. To this aim, let us first single out a simple operatortheoretic property.
Proof of Theorem 2.2. For generic f ∈ D(S F ) and v ∈ D(T ), one has g := f + v ∈ D(S T ) and Since µ < m(S), and hence f, S F f − µ f 2 > 0, last inequality holds true if and only if for arbitrary f ∈ D(S F ) and v ∈ D(T ). By re-writing (*) as and by the fact that the above inequality is valid for arbitrary f ∈ D(S F ) and hence holds true also when the supremum over such f 's is taken, Lemma 2.3 then yields which completes the proof.
With these abstract results at hand, let us now turn to the identification of the non-Friedrichs extensions with the same Friedrichs lower bound.
It is worth observing that inequality (2.5) is not informative in this respect: indeed, owing to (2.5), a sufficient condition for the bottom of S T to equal the bottom of S F would be to impose m(S)m(T )/(m(S) + m(T )) m(S), but such inequality is only satisfied, in the form of an identity, when m(T ) = ∞, therefore the above sufficient condition only selects S T = S F , the Friedrichs extension.
We rather focus on (2.7) from Theorem 2.2. There, the operator S F − µ1 is invertible with everywhere bounded inverse on the whole H: indeed, µ < m(S) and then m(S F − µ1) > 0. Instead, S F − m(S)1 fails to be invertible on H, because its bottom is by construction equal to zero.
The informal idea now is that even if (S F − m(S)1) −1 cannot be defined as a bounded operator on the whole H, yet it makes sense on ran(S F − m(S)1), and if it happens that the latter space has a non-trivial intersection with ker S * , then there are non-zero vectors v ∈ ran(S F − m(S)1) ∩ ker S * on which v, (S F − m(S)1) −1 v is unambiguously defined and hence the right-hand side of the second expression in (2.7) is meaningful also when µ = m(S). Moreover, if on such v's one can define an operator T ∈ S(ker S * ) satisfying (2.7) when µ = m(S), then by suitably exploiting the limit µ ↑ m(S) this should give a characterisation of the condition m(S T ) m(S), which is equivalent to m(S T ) = m(S), as S F S T , the Friedrichs extension is the largest of all self-adjoint extensions of S.
By elaborating on such idea we finally come to our main results, Theorems 2.4, 2.5, and 2.6 below.
Clearly, underlying (2.7) is the quadratic form language, so the actual operator to possibly invert in some subspace of ker S * is rather (S F − m(S)1) 1/2 , a positive self-adjoint operator with zero lower bound.
In this respect, as S F is self-adjoint on H, and so is S F − m(S)1 with lower bound zero, then upon decomposing the negative powers (S F − m(S)1) −δ , δ > 0, are naturally defined as self-adjoint operators on the Hilbert subspace ran(S F − m(S)1), or also on the whole H upon extension by zero on ker(S F − m(S)1).
In the first statement we characterise the occurrence of non-Friedrichs extensions with the same Friedrichs lower bound.
Theorem 2.4. Let S be a densely defined and symmetric operator on a given Hilbert space H with lower bound m(S) > 0. Necessary and sufficient condition for S to admit self-adjoint extensions other then the Friedrichs extensions and with the same lower bound m(S) is that In the applications both ker S * and ran(S F −m(S)1) 1/2 are in general spaces that one can qualify rather explicitly. Thus, condition (2.9) is practically manageable and qualify the operator-theoretic mechanism for non-Friedrichs extensions with the Friedrichs lower bound. In Section 3 we shall give examples of that.
Our next result concerns the actual recipe to construct such extensions, when (2.9) is matched, thus in practice how to identify the corresponding extension parameters T in S(ker S * ). We shall use the customary notation of square brackets for the domain D[q] of a quadratic form q on H and for the evaluation q[v] on elements of its domain; as usual, we shall denote by q[v 1 , v 2 ] the evaluation of the corresponding sesquilinear form defined by polarisation.
Theorem 2.5. Same assumptions as in Theorem 2.4, and assume further that condition (2.9) is satisfied.
(i) The expression defines a symmetric, closed, and strictly positive quadratic form q.
(ii) Let T q be the operator on the Hilbert subspace D[q] uniquely associated with q. Then T q ∈ S(ker S * ).
(iii) For any T ∈ S(ker S * ) with T T q , the corresponding self-adjoint extension S T of S (Theorem 2.1) has the property (iv) Any self-adjoint extension S T of S satisfying (2.11) corresponds to an extension parameter T ∈ S(ker S * ) with T T q .
In view of the general classification of Theorem 2.1, the above results admit a natural corollary that it is worth stating as a separate theorem. It is convenient to introduce the meaningful terminology 'top extensions' for all those S T 's with m(S T ) = m(S) (in particular, S F is a top extension), and 'least-top extension' for the extension S LT := S Tq . Theorem 2.6. Same assumptions as in Theorem 2.4. Each top extension S top of S satisfies in the sense of operator form ordering. Each such extension is of the form S top = S T for some T ∈ S(ker S * ) with T T q , where T q is qualified in Theorem 2.5(ii), and they are all ordered with T in the sense of (2.4).
Proof of Theorem 2.4, necessity part. Let S T be a self-adjoint extension of S, labelled by some T ∈ S(ker S * ), with the property m(S T ) = m(S F ) and S T = S F . Let (µ n ) n∈N be an increasing sequence of real numbers such that µ n < m(S) ∀n and µ n → m(S) as n → ∞. Since S T µ n 1, In fact, for each v the sequence of square norms (S F − µ n 1) − 1 where ν v is the scalar spectral measure of the self-adjoint operator S F relative to the vector v. Therefore, As the latter conclusion is tantamount as (2.13) Proof. The inclusion for D(T ) was already proved. Next, as a follow-up of the reasoning of the previous proof, let us observe that for each v ∈ D(T ) one has Indeed, by dominated convergence. Therefore, one can take the limit n → ∞ in the inequality thus obtaining the second line of (2.13).
Proof of Theorem 2.5 and of Theorem 2.4, sufficiency part.
(i) The fact that (2.10) defines a symmetric quadratic form with strictly positive lower bound is obvious. As for q being closed, let us show that if and q[v n − v] → 0. This is indeed equivalent to saying that q is closed (see, e.g., [11,Prop. 10.1]). Now, the above assumption on ( A first conclusion, since ker S * is closed in H and hence v ∈ ker S * as well, is that v ∈ ran((S F − m(S)1) Part (i) of Theorem 2.5 is thus proved.
(ii) As q is densely defined in the Hilbert subspace D[q], and it is symmetric, closed, and semi-bounded from below, then q uniquely identifies a self-adjoint operator T q on D[q] defined by T q v := z v (see, e.g., [11,Theorem 10.7]). Since D[q] ⊂ ker S * and ker S * is closed in H, then D[q] ⊂ ker S * , thus proving that T q ∈ S(ker S * ). This establishes part (ii) of Theorem 2.5.
(iii) Let T ∈ S(ker S * ) with T T q . This means that D(T ) ⊂ D(T q ) and for every v ∈ D(T ). Consider now an arbitrary µ < m(S). With the very same argument used in the proof of the necessity part of Theorem 2.5 one sees that for all v ∈ D(T ). Owing to Theorem 2.2, the self-adjoint extension S T of S parametrised by the considered T is such that S T µ1. By the arbitrariness of µ, one concludes that S (iv) Last, let S T be a self-adjoint extension of S with m(S T ) = m(S). The necessity statement of Theorem 2.4 implies that the intersection ran(S F −m(S)1) 1 2 ∩ ker S * is non-trivial, so one can define the form q and the operator T q ∈ S(ker S * ) as in parts (i) and (ii) of Theorem 2.5. Owing to Corollary 2.7, This means precisely that T T q .

Applications
Let us discuss now a few instructive examples of application of Theorems 2.4-2.5.

Schrödinger quantum particle on an interval.
Let us revisit in more systematic terms the example presented in Sect. 1. The operator S has deficiency index equal to 2, and explicitly (3.1) ker S * = span{1, x} .

By standard ODE methods one finds that the general solution is
thus the minimal norm solution is the one with B = 0. This proves that the completing the proof of (3.4) and (3.5).

As the intersection space (3.4) is non-trivial, Theorem 2.4 ensures that S admits non-Friedrichs self-adjoint extensions with the same Friedrichs lower bound. This is consistent with what discussed in the introduction: m(S F ) = m(S A ), Friedrichs and anti-periodic extension have the same lower bound.
It is instructive to apply the constructive recipe of Theorem 2.5 so as to identify all such extensions. With the notation therein, thus T q is an operator of multiplication by some real number t q , (having used (3.5) in the third step and 1 − 2x 2 2 = 1 3 in the last step), then necessarily t q = 12.
Theorem 2.5, in parts (iii) and (iv), then states that the self-adjoint extensions S T of S with m(S T ) = m(S F ) are those labelled by self-adjoint operators T with T T q . Such T 's, apart from the one parametrising the Friedrichs extension, are therefore such that T is the multiplication by some t 12 . Keeping into account, as is immediate to check, that (3.9) W := V ⊥ ∩ ker S * = span{1} , the extension S T for each T satisfying (3.8) has domain given by formula (2.1) of Theorem 2.1, that is, (3.10) The action of the everywhere defined and bounded operator S −1 F on the subspace ker S * = span{1, x} is easily computed by solving a boundary value problem completely analogous to the one considered in the proof of Lemma 3.1. The result (as found, e.g., in [5, Eq. (91)]) is Thus, Formula (3.12), for each fixed t 12, identifies those self-adjoint extensions of S different from the Friedrichs extension, but with the same lower bound. In order to identify the boundary condition of self-adjointness satisfied by a generic element g ∈ D(S T ) for each extension of type (3.12), we compute the boundary values (3.14) It was indeed convenient to cast (3.13) in the form (3.14) because the latter can be more easily matched with the general conditions of self-adjointness of the extensions of S, as we shall now do. We refer to the following very standard result, obtained for example by exploiting Theorem 2.1 for all possible extension parameters (see, e.g., [11,Example 14.10]), or equivalently by means of the alternative extension scheme a la von Neumann applied to S (see, e.g., [6, Sect. 6.2.3.1]).
Proposition 3.2. The family of self-adjoint extensions on L 2 (0, 1) of the operator S defined in (1.1) consists of restrictions of S * , and hence of operators of the form − d 2 dx 2 , to domains of H 2 (0, 1)-functions g satisfying boundary conditions of one of the following four classes: (1) , where c ∈ C and b 1 , b 2 ∈ R and qualify each extension. By direct comparison between (3.14) and (3.15)-(3.18) we see that (3.14) can only be of type (3.16) with We have thus proved the following. For completeness, here is how the direct check would have proceeded. Let us limit the analysis to the eigenvalue problem for a generic self-adjoint extension of type (3.16) with the choice (3.19), namely for fixed t ∈ R. g must be of the form g(x) = A cos √ λx + B sin √ λx, A, B ∈ C, and for sure the pairs (g, λ) with (3.22) g(x) = sin((2n + 1)πx) , λ = (2n + 1) 2 π 2 , n ∈ N 0 solve (3.21), showing that all such extensions have the eigenvalues (2n + 1) 2 π 2 , n ∈ N 0 , in common. The remaining (i.e., non-sin-only) solutions to (3.21) are obtained imposing B = 0, and it is then simple to conclude that the admissible λ's are the (t-dependent) roots of (3.23) (and understanding the above trigonometric functions as hyperbolic functions when λ < 0). As F (λ) increases with λ in all intervals in which it is defined, and F (π 2 ) = 12, one deduces that only for t 12 the admissible λ's selected by (3.23) satisfy λ π 2 (see Figure 3.1). The spectrum thus determined from (3.22) and (3.23) indeed confirms, by direct inspection, what found in Prop. 3.4 by means of our Theorem 2.5.

Schrödinger quantum particle in R 3 with point interaction.
This is an example with deficiency index equal to 1. With respect to the Hilbert space H = L 2 (R 3 ) we consider the operator S is densely defined and symmetric, with m( S) = 0. The self-adjoint extensions of S are Hamiltonians for a quantum particle in three dimensions subject to a point interaction supported at x = 0, and they are very well studied and understood.  (ii) For each α ∈ R, The negative eigenvalue −(4πα) 2 , when it exists, is simple and the corresponding eigenfunction is |x| −1 e 4πα|x| .
In order to recover such a conclusion from the abstract setting of Sect. 2, let us consider (3.30) S := S + 1 .
Clearly, m(S) = 1. The self-adjoint extensions of S and of S then only differ by a trivial shift. As we intend to analyse the extensions of S within the extension scheme of Theorem 2.1, rather than using von Neumann's extension theorem as in [1], let us follow closely the discussion made in [10,Sect. 3], were indeed the Kreȋn-Višik-Birman scheme was employed. We shell denote, respectively, by andˇthe Fourier and inverse Fourier trans- In particular, It is possible to prove the following. (ii) The Friedrichs extension of S is the operator (3.33) (iii) All other self-adjoint extensions of S are of the form S t for some t ∈ R, where D(S t ) = g ∈ L 2 (R 3 ) g = f + (p 2 + 1) −2 tξ + (p 2 + 1) −1 ξ f ∈ D(S) , ξ ∈ C S t g = (p 2 + 1) f + (p 2 + 1) −2 tξ .

(3.34)
This is precisely formula (2.1) of Theorem 2.1 specialised to the case where ker S * is one-dimensional and T is therefore the operator of multiplication by the real number t. (iv) One has (3.35) Clearly, S F − m(S)1 = S F − 1 = S F , the self-adjoint (negative) Laplacian on L 2 (R 3 ). Therefore, unlike the example discussed in Subsect. 3.1, Proof. The fact that G 1 ∈ ker S * is stated in Theorem 3.6(i). As 1 |p|(p 2 + 1) ∈ L 2 (R 3 , dp) and (S F − 1) V can be at most one-dimensional, thus (3.37) is proved, and so is (3.38) as well.
Owing to Lemma 3.7, Theorem 2.4 is applicable: S admits non-Friedrichs extensions with Friedrichs lower bound, and so does therefore S, consistently with what previously observed in (3.29). Furthermore, with the notation of Theorem 2.5, thus T q is an operator of multiplication by some real number t q , (having used (3.38) in the second identity), then necessarily t q = 2.
Theorem 2.5, in parts (iii) and (iv), then states that the self-adjoint extensions S T of S with m(S T ) = m(S F ) are those labelled by self-adjoint operators T with T T q . Such T 's, apart from the one parametrising the Friedrichs extension, are therefore such that D(T ) = V = span{G 1 } T is the multiplication by some t 2 . For what argued in Theorem 3.6(iii), such extensions are precisely the operators S t that one reads out from formula (3.34) with t 2. In turn, the correspondence formula (3.35) leads to the conclusion that the self-adjoint extensions of S with Friedrichs lower bound are precisely those −∆ α 's with α 0.

Radial problem in hydrogenoid-like Hamiltonians.
It is worth mentioning another example with unit deficiency index, even without working out here the steps through which Theorems 2.4 and 2.5 are applied, which are in fact completely analogous to the computations of Sect. 3.1 and 3.2.
For given ν ∈ R, let us now consider a densely defined and symmetric operator on the Hilbert space H = L 2 (R + ) with lower bound m(S ν ) = 0. One typical emergence of S ν in mathematical physics is as the minimally defined zero-momentum radial operator in the construction of a quantum hydrogenoid Hamiltonian with an additional point interaction at the center of the Coulomb potential: S ν is indeed well known and thoroughly studied, and we refer to [4, Sect. 1.4] and references therein for an updated historical overview.
Hardy's inequality implies that S ν is lower semi-bounded, and in particular obviously (3.43) m(S ν ) = 0 ∀ν 0 (repulsive Coulomb interaction). A standard limit-point limit-circle argument shows that S ν has unit deficiency index. Its self-adjoint extensions are studied in the literature by means of various extension schemes, including recently in [4] by means of the general Theorem 2.1 above. −g + ν r g ∈ L 2 (R + ) and g 1 = 4πα g 0 S (α) ν g = −g + ν r g ,  and of its self-adjoint extensions was worked in [4, Sect. 2], which we refer to for the details. The special value of the shift (3.51) was chosen in [4] in order to be able to solve the ODE S * ν u = 0 by means of special functions, this way characterising explicitly the deficiency space ker S * ν . The Friedrichs extension S ν,F of S ν was also characterised in [4,Sect. 2]. This provides all the ingredients to investigate the intersection (2.9) and apply Theorems 2.4-2.5 so as to reproduce (3.50).