A Dirac delta operator

If $T$ is a (densely defined) self-adjoint operator acting on a complex Hilbert space $\mathcal{H}$ and $I$ stands for the identity operator, we introduce the delta function operator $\lambda \mapsto \delta \left(\lambda I-T\right) $ at $T$. When $T$ is a bounded operator, then $\delta \left(\lambda I-T\right) $ is an operator-valued distribution. If $T$ is unbounded, $\delta \left(\lambda I-T\right) $ is a more general object that still retains some properties of distributions. We derive various operative formulas involving $\delta \left(\lambda I-T\right) $ and give several applications of its usage.

1. The delta function δ (λI − T ) The scalar delta 'function' λ → δ (λ − a) along with its derivatives were introduced by Paul Dirac in [1], and later in [2,Section 15], although its definition can be traced back to Heaviside. The rigorous treatment of this object in the context of distribution theory is due to Laurent Schwartz [6,12]. In this paper we extend the definition of δ (λ − a) from real numbers to self-adjoint operators on a Hilbert space H. We denote by D (R) = lim − → D ([−n, n]) the linear space of infinitely differentiable complex-valued functions of compact support, equipped with the inductive limit topology. As usual in physics we shall assume that the scalar product in H is anti-linear for the first variable.
If T is a densely defined self-adjoint operator 1 on H and I stands for the identity operator, we define the delta function operator λ → δ (λI − T ) at T by for each f ∈ C (R), i. e., for each real-valued continuous function f (λ). Here dλ is the Lebesgue measure of R, but the right-hand side of (1.1) is not a true integral. If T is a bounded operator, we shall see at once that δ (λI − T ) must be regarded as a vector-valued distribution, i. e., as a continuous linear map from the space D (R) into the locally convex space L (H) of the bounded linear operators (endomorphisms) on H equipped with the strong operator topology [10,11], whose action on f ∈ D (R) we denote as an integral. If T is unbounded we shall see that δ (λI − T ) still retains some useful distributional-like properties. The previous equation means Let us recall that if T is a (densely defined) self-adjoint operator, there is a unique spectral family {E λ : λ ∈ R} 1 In what follows σ (T ) will denote the spectrum of T . Recall that the residual spectrum of a self-adjoint operator T is empty, so that σ (T ) = σp (T ) ∪ σc (T ), where σp (T ) denotes the point spectrum (the eigenvalues) and σc (T ) the continuous spectrum of T . of self-adjoint operators defined on the whole of H that satisfy (i) E λ ≤ E µ and E λ E µ = E λ for λ ≤ µ, (ii) lim ǫ→0 + E λ+ǫ x = E λ x, and (iii) lim λ→−∞ E λ x = 0 and In this case, the spectral theorem (cf. [8,Section 107]) and the Borel-measurable functional calculus provide a selfadjoint operator f (T ) defined by for each Borel-measurable function f (λ), whose domain 3) tell us that f (λ) need not be defined on R \ σ (T ). Thanks to (1.3) the definition of δ (λI − T ) may be extended to Borel-measurable functions by declaring that the equation (1.1) holds for (x, y) ∈ D (f (T )) × H and each Borel function f . But, by reasons that will become clear later, we shall restrict ourselves to those Borel functions which are continuous at each point of σ p (T ). Moreover, working with the real and complex parts, no difficulty arises if the function f involved in the equation (1.1) is complex-valued (except that f (T ) is no longer a self-adjoint operator whenever Im f = 0). Thus, unless otherwise stated, we shall assume that both in (1.1) and (1.3) the function f is complex-valued. Note that the complex Stieltjes measure d E λ x, y need not be dλcontinuous. In what follows we shall denote by B p (R) the linear space over C consisting of all complex-valued Borel-measurable functions of one real variable which are continuous on σ p (T ). If it turns out that f n (T ) → f (T ) in the strong operator topology [3,10.2.8 Corollary]. Therefore, in this case δ (λI − T ) is an L (H)-valued distribution.
As all integrals considered so far are over σ (T ), we have On the other hand, if µ ∈ σ p (T ) and y is an eigenvector corresponding to the eigenvalue µ, clearly for every λ ∈ R. In the particular case when T a is the linear operator defined on H by T a x = ax for a fixed a ∈ R, then T a is a self-adjoint linear operator with σ (T a ) = σ p (T a ) = {a}. In this case δ (λI − T a ) x = δ (λ − a) x for every x ∈ H, i. e., δ (λI − T a ) = δ (λ − a) I.
Since equality f (T ) † y, x = y, f (T ) x holds for all x, y ∈ D (f (T )) and each f ∈ B p (R), we may infer that holds (in a 'distributional' sense) for all x, y ∈ D (T ). This suggests that in certain sense δ (λI − T ) may be regarded (possibly for almost all λ ∈ R) as a Hermitian operator on D (T ).
Let us also point out that as equation (1.1) holds for all f ∈ D (R), in a distributional sense we have So, from (1.6) and (1.7) we get Y ′ (λI − T ) = δ (λI − T ).
The same equality holds if T is unbounded but f ∈ D (R).
If T is a self-adjoint operator and f ∈ B p (R), then where the latter equality is the definition of |f (T )| 2 . So, we have the following result.

Proposition 2.
If T is self-adjoint and f ∈ B p (R), then for every x, y ∈ D (f (T )).
Proof. We adapt a classic argument. Indeed, for every x, y ∈ D (f (T )) we have Since E µ E λ = E µ whenever µ ≤ λ, and E λ y, x does not depend on µ, by splitting the integral we get where clearly the first integral is f (T ) y, E λ x . Plugging d f (T ) y, E λ x into (1.8), we are done.
Corollary 3. Under the same conditions of the previous theorem, the equality holds for every x ∈ D (f (T )).
Proof. This is a straightforward consequence of preceding corollary and the Lebesgue dominated convergence theorem.
This proposition holds in particular if f n → f in D (R). Hence, even in the unbounded case, δ (λI − T ) behaves as a vector-valued distribution-like object.
is continuous on R and makes sense if we replace λ by a self-adjoint operator T , the value of the integral does not depend on the integration ordering.
Proof. Since g (·, µ) ∈ B p (R) for every µ ∈ R., one has On the other hand, by the definition of δ (λI − T ) we have for (x, y) ∈ D (T ) × H. So, the proposition follows.
On the other hand, it is clear that So, the right-hand side of (1.10) coincides with If we denote by L (H) the linear space of all linear endomorphisms on H, the next theorem summarize some previous results.
, whose action on f ∈ B p (R) we denote by holds for every λ ∈ R, and the action f (T ) of δ (λI − T ) on f ∈ S (R) is given by Indeed, if f ∈ S (R) we have Functionally, the action of δ T on f ∈ S (R) by means of equation (2.2) becomes Consequently, we have with the order of the integration as stated.
Remark 10. Consider the one-parameter unitary group {U (t) : t ∈ R} generated by the self-adjoint operator T , that is, U (t) = exp (−itT ) for every t ∈ R. If F denotes the Fourier transform, equation (2.2) can be written as So, equation (2.3) reads as In what follows we shall compute the spectral family {E λ : λ ∈ R} for some useful self-adjoint operators of Quantum Mechanics by means of the delta δ (λI − T ).
x is an eigenvector corresponding to λ, then and the right-hand integral makes no sense (see [4] for a useful discussion). If λ / ∈ σ p (T ) we define If λ belongs to σ p (T ), then (µ → Y (λ − µ)) / ∈ B p (R). In order to define E λ we enlarge a little the interval of integration by considering the integral The limit is well-defined since lim ǫ→0 + E λ+ǫ = E λ pointwise on H. In the particular case when λ belongs to σ d (T ), the discrete part of σ p (T ), λ is isolated in σ p (T ).
Example 11. The spectral family of the (up to a sign) one-dimensional Quantum Mechanics momentum operator of the free particle P = iD, where Dϕ = ϕ ′ , acting on the Hilbert space H = L 2 (R) is given by for every regular compactly supported ϕ ∈ D (P ).
Proof. As is well-known P is a self-adjoint operator with D (P ) = H 2,1 (R) and σ c (P ) = R. Since for a regular enough ϕ ∈ D (P ), by Corollary 9 we have Note that the integral of the right-hand side does exist because ϕ has compact support. According to the definition of E λ for the continuous spectrum and keeping in mind the order of integration as indicated in Corollary 9, one has bearing in mind the distributional relation we get where the last integral must be understood in Cauchy's principal value sense.
Example 12. The spectral family of the one-dimensional Quantum Mechanics kinetic energy term of the free particle, corresponding to the Laplace operator T = −D 2 on H = L 2 (R), where D 2 ϕ = ϕ ′′ , is given by for λ > 0 and E λ = 0 whenever λ < 0, where ϕ is a regular function with compact support belonging to D (T ).
Proof. In this case T is a self-adjoint operator with σ (T ) = [0, +∞). Since T = (iD) 2 , according to (1.10) we have regarded as a functional on S (R) through dλ-integration over [0, +∞). Plugging into the previous expression and keeping in mind the correct order of integration, we see that for every f ∈ S (R). By the definition of E λ if λ > 0 and the fact that δ (µI − T ) = 0 whenever µ < 0, we have Working out the penultimate integral with µ instead of λ and f (µ) = Y (λ − µ), we obtain for λ > 0. So, by setting u = √ µ we get where the integrals are understood in Cauchy's principal value sense.
Example 13. Spectral family of the (up to a sign) onedimensional Quantum Mechanics momentum operator S for a bounded particle on As is well-known this is a self-adjoint operator with discrete spectrum σ (S) = Z whose eigenfunction system {ϕ n : n ∈ Z}, with ϕ n (x) = (2π) −1/2 e −inx , are the solutions of the eigenvalue problem iϕ ′ = λϕ with ϕ (−π) = ϕ (π). So, for ϕ ∈ D (S) we have ϕ L2 = n∈Z c n ϕ n with for every n ∈ Z. Since σ (S) = σ d (S), recalling the definition of the operator E λ for λ ∈ σ d (S), clearly we have for every λ ∈ R. So, the fact that E λ is a bounded operator yields Remark 14. Since in the previous example S is bounded on H = L 2 [−π, π], the delta operator δ (λI − S) should be regarded as a continuous endomorphism as well. In this case δ (λI − S) ϕ = n∈Z c n δ (λ − n) ϕ n .
Example 15. The one-dimensional Quantum Mechanics position operator on L 2 (R). This operator is defined on H = L 2 (R) by (Qϕ) (x) = xϕ (x) for every x ∈ R. Clearly σ c (Q) = R and ϕ ∈ D (Q) if (x → x ϕ (x)) ∈ L 2 (R). Moreover, it is clear that Hence, in this case we can write Let us compute the spectral family and the projection operator onto the eigenspace ker (M + I). Clearly for every λ ∈ R. If λ 1 = −1, the orthogonal projection P λ 1 onto ker (I + M ) is Proof. If x ∈ H, we can write x = ∞ i=1 x, u i u i . Since (λI − K) −1 is a bounded operator whenever λ / ∈ σ (K), we have so we obtain the classic series For the solution of the equation (I − zK) x = y with z ∈ C we get the Schmidt series On the other hand, since δ (λI − K) acts on H as a continuous endomorphism, equation holds for every x ∈ H. If T is an unbounded self-adjoint operator then D (T ) = H and D (T n ) becomes smaller as n grows. So, the following result, makes sense only if the operator T is bounded.
Theorem 18. In general, if T is a bounded self-adjoint operator, one has which is the Taylor series of δ (λI − T ) at λI.
Proof. Developing the operator function exp (itT ), which is well-defined by the spectral theorem, we get so that, formally interchanging the sum and the integral, we may write Using the fact that for every n ∈ N, we obtain (2.9).

The resolvent operator and δ (λI − T )
Recall that the spectrum σ (T ) of a (densely defined) self-adjoint operator on a complex Hilbert space H is a closed subset of C contained in R (see for instance [9, 3.2]). If z ∈ C \ σ (T ), i. e., if z is a regular point of T , and R (z, T ) = (zI − T ) −1 denotes the resolvent operator of T at z (see [7,Definition 8.2]), the function λ → (z − λ) −1 is continuous on σ (T ). The resolvent is well-defined over H, so it is a bounded normal operator. If z ∈ R \ σ (T ) then R (z, T ) is even self-adjoint. From (1.1) it follows that with z ∈ C \ σ (T ) and using the fact that then, according to (2.5), for Im z > 0 we have From here, it follows that if Im z > 0, where L is the Laplace transform. This is the Hille-Yosida theorem which relates the resolvent with the one-parameter group of unitary transformations {U (t) : t ∈ R} generated by the self-adjoint operator T . If T is a bounded self-adjoint operator, γ is a closed Jordan contour that encloses σ (T ) and f (z) is holomorphic inside the connected region surrounded by the path γ, the Dunford integral formula asserts that In [13] is pointed out that (2πi) −1 R (z, T ) can be considered as the indicatrix of a vector-valued distribution with values in L (H). Dunford integral formula is easily obtained by using the δ (λI − T ) operator since, if we apply the Proposition 5 with g (λ, Example 19. Derivation of the orthogonal projection operator onto ker (M + I) of the Hermitian matrix M of the Example 16 by the resolvent technique. We must compute Clearly, we have we reproduce the result we got earlier.

4.
The δ (λI − T ) operator as a limit Since in the sense of distributions 1 2πi as ǫ → 0 + . Hence, if g n is defined by the left-hand side µ-parametric integral with ǫ = 1/n, then g n → f pointwise on R. So, if f ∈ D (R) and T is bounded (hence with σ (T ) compact), as can be easily checked {g n } ∞ n=1 is a uniformly bounded sequence of continuous functions, with sup n∈N g n ∞ ≤ f ∞ , that converges pointwise on R to f . Thus, by [3,10.2.8 Corollary] one has g n (T ) → f (T ) in the strong operator topology, that is as ǫ → 0 + in the strong operator topology of L (H). Therefore, if T is bounded and f ∈ D (T ) then coincides with δ (λI − T ) as an L (H)-valued distribution.

Unitary equivalence of δ (λI − T )
Theorem 20. If T is a self-adjoint operator defined on the whole of H, there exist a finite measure µ on the Borel sets of the compact space σ (T ) and a linear isometry U from L 2 (σ (T ) , µ) onto H such that where (Qϕ) (x) = x ϕ (x) is the position operator.
Proof. According to [5] there exist a finite measure µ on the Borel sets of the compact space σ (T ) and a linear isometry U from L 2 (σ (T ) , µ) onto H such that for every ϕ ∈ L 2 (σ (T ) , µ). So, since U −1 T U is a selfadjoint operator on L 2 (σ (T ) , µ), we have as stated.

Commutation relations
For position Q and momentum P of a one-dimensional particle, one has H = L 2 (R) and [Q, P ] = i I. Therefore If the limit as ǫ → 0 + the bracketed function is equal to 0 if µ ∈ R \ [a, b], equal to π if a < µ < b and equal to π/2 if µ ∈ {a, b}. So, if a, b / ∈ σ p (T ) so that χ (a,b) and χ [a,b] both belong to B p (R), setting then g n (µ) → f (µ) for every µ ∈ R and sup n∈N g n ∞ ≤ 1 which, according to Proposition 4, implies that g n (T ) x → f (T ) x for every x ∈ D (T ). In other words holds pointwise on the domain D (T ) of T . Hence, by virtue of (7.1) we get which is Stone's formula.