Embedding Theorems for the Dunkl Harmonic Oscillator on the Line

Embedding results of Sobolev type are proved for the Dunkl harmonic oscillator on the line.


Introduction
The subindex ev/odd is added to any space of functions on R to indicate its subspace of even/odd functions; in particular, C ∞ = C ∞ ev ⊕C ∞ odd for C ∞ := C ∞ (R). The Dunkl operator T σ (σ > −1/2) on C ∞ is the perturbation of d dx defined by T σ = d dx on C ∞ ev and T σ = d dx + 2σ 1 x on C ∞ odd . The corresponding Dunkl harmonic oscillator is the perturbation L σ = −T 2 σ + s 2 x 2 of the harmonic oscillator H = − d 2 dx 2 +s 2 x 2 (s > 0). The conjugation E σ = |x| σ T σ |x| −σ on |x| σ C ∞ is equal to d dx − σx −1 on |x| σ C ∞ ev and d dx + σx −1 on |x| σ C ∞ odd ; note that |x| σ C ∞ ev/odd consists of even/odd functions, possibly not smooth or not even defined at 0. Up to the product by a constant, E σ was introduced by Yang [39]. In the form T σ , this operator was generalized to R n by Dunkl [12,13,14], giving rise to what is now called Dunkl theory (see the survey [31]); in particular, the Dunkl harmonic oscillator on R n was studied in [29,15,26,25]. See [27] for further generalizations on R. Sometimes the terms Yang-Dunkl operator and Yang-Dunkl harmonic oscillator are used in the case of R [27].
Let p k be the sequence of orthogonal polynomials for the measure e −sx 2 |x| 2σ dx, taken with norm one and positive leading coefficient. Up to normalization, these are the generalized Hermite polynomials [32,p. 380,Problem 25]; see also [9,11,16,10,29,30]. The corresponding generalized Hermite functions are φ k = p k e −sx 2 /2 . For each m ∈ N, let S m be the Banach space of functions φ ∈ C m (R) with sup x |x i φ (j) (x)| < ∞ for i + j ≤ m; the corresponding Fréchet space S = m S m is the Schwartz space on R. With domain S, L σ is essentially self-adjoint in L 2 (R, |x| 2σ dx), and the spectrum of its self-adjoint extension, L σ , consists of the eigenvalues (2k + 1 + 2σ)s (k ∈ N), with corresponding eigenfunctions φ k [29]. For each real m ≥ 0, let W m σ be the Hilbert space completion of S with respect to the scalar product φ, ψ W m σ := (1 + L σ ) m φ, ψ σ , where , σ denotes the scalar product of L 2 (R, |x| 2σ dx), obtaining a Fréchet space W ∞ σ = m W m σ . We show the following embedding theorems; the second one is of Sobolev type. Corollary 1.3. S = W ∞ σ as Fréchet spaces. In other words, Corollary 1.3 states that an element φ ∈ L 2 (R, |x| 2σ dx) is in S if and only if the "Fourier coefficients" φ, φ k σ are rapidly decreasing on k. This also means that S = m D(L m σ ) (S is the smooth core of L σ with the terminology of [6]) because the sequence of eigenvalues of L σ is in O(k) as k → ∞.
We introduce a version S m σ of every S m , whose definition involves T σ instead of d dx . They satisfy much simpler embeddings:  [35,36,37].
Next, we consider other perturbations of H on R + . Let S ev,U denote the space of restrictions of even Schwartz functions to some open set U , and set φ k,U = φ k | U .
is a primitive of f 1 . Then the following properties hold: (i) P , with domain h S ev,U , is essentially self-adjoint in L 2 (R + , e 2F1 dx); (ii) the spectrum of its self-adjoint extension, P, consists of the eigenvalues (4k + 1 + 2σ)s (k ∈ N) with multiplicity one and normalized eigenfunctions √ 2 hφ 2k,U ; and (iii) the smooth core of P is h S ev,U .
This theorem follows by showing that the stated condition on f 1 and f 2 characterizes the cases where P can be obtained by the following process: first, restricting L σ to even functions, then restricting to U , and finally conjugating by h. The term of P with d dx can be removed by conjugation with the product of a positive function, obtaining the operator H + σ(σ − 1)x −2 ; in this way, we get all operators of the form H + cx −2 with c > −1/4.
The conditions of Theorem 1.4 are satisfied by P = H − 2c 1 x −1 d dx + c 2 x −2 (c 1 , c 2 ∈ R) on R + if and only if there is some a ∈ R such that a 2 +(2c 1 −1)a−c 2 = 0 and a + c 1 > −1/2; in this case, h = x a and e 2F1 = x 2c1 . For some c 1 , c 2 ∈ R, there are two values of a satisfying these conditions, obtaining two different self-adjoint operators defined by P in different Hilbert spaces. For instance, L σ may define a self-adjoint operator when σ ≤ −1/2.
This example is applied in [2] to prove a new type of Morse inequalities on strata of compact stratifications [33,20,34] with adapted metrics [23,24,5], where Witten's perturbation [38] is used for the minimal/maximal ideal boundary conditions of de Rham complex [7,8,6]. The version of Morse functions used in [2] is different from the version of Goresky-MacPherson [18]. More precisely, the operator P describes the radial direction of Witten's perturbed Laplacian in the local conic model of a stratification around each critical point. The two possible choices of a give rise to the minimal/maximal ideal boundary conditions.

Dunkl operator on the line. For any
for all 1 m ∈ N (see e.g. [19, Theorem 1.1.9]). Let us use the notation Consider matrix expressions of operators on C ∞ with respect to the decomposition C ∞ = C ∞ ev ⊕ C ∞ odd , as direct sum of subspaces of even and odd functions. For each function h, the notation h is also used for the operator of multiplication by h.
The perturbed factorial m! σ (m ∈ N) is inductively defined by 0! σ = 1, and Notice that m! σ > 0 if σ > −1/2. For k ≤ m, even when k! σ = 0, the quotient m! σ /k! σ can be understood as the product of the factors used in the definition of m! σ and not used in the definition of k! σ . For φ ∈ C ∞ and m ∈ N, by (1) and induction on m, we get 2.2. Dunkl harmonic oscillator on the line. On C ∞ , the harmonic oscillator, and the annihilation and creation operators are . Their perturbations L = −T 2 σ + s 2 x 2 , B = sx + T σ and B ′ = sx − T σ are called Dunkl harmonic oscillator, and Dunkl annihilation and creation operators. By (2), [L, Σ] = BΣ + ΣB = B ′ Σ + ΣB ′ = 0 .
For each m ∈ N, let S m be the space of functions φ ∈ C ∞ such that This expression defines a norm S m on S m , which becomes a Banach space. We have S m+1 ⊂ S m continuously 2 , and S = m S m , with the induced Fréchet topology, is the Schwartz space on R. Note that φ ′ S m ≤ φ S m+1 for all m.
We can restrict the above decomposition of C ∞ to every S m and S, obtaining S m = S m ev ⊕ S m odd and S = S ev ⊕ S odd . The matrix expressions of operators on S are taken with respect to this decomposition. For φ ∈ C ∞ ev , ψ = x −1 φ and i, j ∈ N, we get from (1) that for all x ∈ R. So ψ S m ≤ φ S m+1 for all m ∈ N, obtaining that S odd = x S ev and x −1 : C ∞ odd → C ∞ ev restricts to a continuous operator x −1 : S odd → S ev . Hence x : S ev → S odd is an isomorphism of Fréchet spaces, and T σ , B, B ′ and L define continuous operators on S.
Let , σ and σ be the scalar product and norm of L 2 (R, |x| 2σ dx). Suppose from now on that σ > −1/2, obtaining that S is dense in L 2 (R, |x| 2σ dx). The following properties hold considerng these operators in L 2 (R, |x| 2σ dx) with domain S: −T σ is adjoint of T σ , B ′ is adjoint of B, and L is essentially self-adjoint. Let L, or L σ , denote the self-adjoint extension of L (with domain S). Its spectrum consists of the eigenvalues (2k + 1 + 2σ)s (k ∈ N). The corresponding normalized eigenfunctions φ k are inductively defined by Furthermore These properties follow from (4)- (7), like in the case of H.
2.3. Generalized Hermite polynomials. By (8), (9) and the definition of B ′ , we get φ k = p k e −sx 2 /2 , where p k is the sequence of polynomials inductively given by p 0 = s (2σ+1)/4 Γ(σ + 1/2) −1/2 and Up to normalization, p k and φ k are the generalized Hermite polynomials and functions [32, p. 380, Problem 25]. Each p k is of degree k, even/odd if k is even/odd, and with positive leading coefficient. Moreover T σ p 0 = 0 and From (12) and (13), we obtain the recursion formula By (14) and induction on k, we easily get the following when k is odd 3 : The following theorem contains a simplified version of the asymptotic estimates satisfied by φ k and ξ k = |x| σ φ k . They can be obtained by expressing p n in terms of the Laguerre polynomials [30,31], whose the asymptotic estimates are studied in [17,3,22,21]. The method of Bonan-Clark [4] can be also applied [1].

Perturbed Schwartz space
We introduce a perturbed version S m σ of each S m that will be appropriate to show our embedding results. Since S m σ must contain the functions φ k , Theorem 2.1 indicates that different definitions must be given for σ ≥ 0 and σ < 0.
When σ ≥ 0, for any φ ∈ C ∞ and m ∈ N, let This defines a norm S m σ on the linear space of functions φ ∈ C ∞ with φ S m σ < ∞, and let S m σ denote the corresponding Banach space completion. There are direct sum decompositions into subspaces of even and odd functions, S m σ = S m σ,ev ⊕ S m σ,odd .
When σ < 0, the even and odd functions are considered separately: let for φ ∈ C ∞ ev , and for φ ∈ C ∞ odd . These expressions define a norm S m σ on the linear spaces of func- The corresponding Banach space completions will be denoted by S m σ,ev/odd , and let S m σ = S m σ,ev ⊕ S m σ,odd . In any case, there are continuous inclusions S m+1 σ ⊂ S m σ , and a perturbed Schwartz space is defined as S σ = m S m σ , with the corresponding Fréchet topology, which decomposes as direct sum of the subspaces of even and odd functions, S σ = S σ,ev ⊕ S σ,odd ; in particular, S 0 = S. It easily follows that S σ consists of functions that are C ∞ on R \ {0} but a priori possibly not even defined at zero, Obviously, Σ defines a bounded operator on each S m σ . It is also easy to see that T σ defines a bounded operator S m+1 σ → S m σ for any m; notice that, when σ < 0, the role played by the parity of i + j fits well to prove this property. Similarly, x defines a bounded operator S m+1 σ → S m σ for any m because, by (2), So B and B ′ define bounded operators S m+1 σ → S m σ , and L a bounded operator S m+2 σ → S m σ . Thus T σ , x, Σ, B, B ′ and L define continuous operators on S σ . In order to prove Theorems 1.1 and 1.2, we introduce an intermediate weakly perturbed Schwartz space S w,σ . Like S σ , it is a Fréchet space of the form S w,σ = m S m w,σ , where each S m w,σ is the Banach space defined like S m σ by using d dx instead of T σ in the right hand sides of (16)- (18); in particular, S 0 w,σ = S 0 σ as Banach spaces. Let S m w,σ denote the norm of S m w,σ . As before, S w,σ consists of functions which are C ∞ on R\{0} but a priori possibly not even defined at zero, S w,σ ∩C ∞ is dense in S w,σ , there is a canonical decomposition S w,σ = S w,σ,ev ⊕ S w,σ,odd , and d dx and x define bounded operators on S m+1 w,σ → S m w,σ . Thus d dx and x define continuous operators on S w,σ . Proof. Let φ ∈ S. For all i and j, we have |x| Proof. Let φ ∈ S w,σ . For all i and j, for |x| ≥ 1. It remains to prove an inequality of this type for |x| ≤ 1, which will be a consequence of the following assertion. and k ≥ n, such that, for all φ ∈ C ∞ , Assuming that Claim 1 is true, the proof can be completed as follows. Let φ ∈ S w,σ and set n = ⌈σ⌉. For |x| ≤ 1, according to Claim 1, |φ(x)| is bounded by Let m, i, j ∈ N with i + j ≤ m. By applying the above inequality to the function x i φ (j) , and expressing each derivative (x i φ (j) ) (r) as a linear combination of functions of the form x p φ (q) with p+q ≤ i+j+r, it follows that there is some C ≥ 1, depending only on σ and m, such that for |x| ≤ 1. By (19) and (20), φ S m ≤ C φ S mσ w,σ because m σ = m + M n . Now, let us prove Claim 1. By induction on n and using integration by parts, it is easy to prove that This shows directly Claim 1 for n ∈ {0, 1}. Proceeding by induction, let n ≥ 2 and assume that Claim 1 holds for n − 1. By (21), it is enough to find appropriate expressions of x r φ (r) (x) for 0 < r < n. For that purpose, apply Claim 1 for n − 1 to each function φ (r) , and multiply the resulting equality by x r to get where a, b, k, ℓ, u and v run in finite subsets of N with b, ℓ, v ≤ M n−1 and k ≥ n−1; thus r + k ≥ n and r + b, r + ℓ, r + v ≤ n − 1 + M n−1 = M n − 1. Therefore it only remains to rise the exponent of t by a unit in the integrals of the last sum. Once more, integration by parts makes the job: Lemma 3.3. If σ < 0, then S m+1 ⊂ S m w,σ continuously. Proof. This is proved by induction on m. For φ ∈ C ∞ ev and |x| ≥ 1, we have (1). Now, assume that m ≥ 1 and the result holds for m − 1. Let i, j ∈ N such that i + j ≤ m, and let φ ∈ S ev ∪ S odd . Independently of the parity of φ and i + j, we Suppose that φ ∈ S ev . If i = 0 and j is odd, then φ (j) ∈ S odd . Thus there is some ψ ∈ S ev such that φ (j) = xψ, obtaining |x| σ |φ (j) (x)| ≤ |ψ(x)| for 0 < |x| ≤ 1. If i + j is odd and i > 0, then Finally, assume φ ∈ S odd , and let ψ = x −1 φ ∈ S ev . If i is even and j = 0, then |x| σ |x i φ(x)| ≤ |x i ψ(x)| for 0 < |x| ≤ 1. If i + j is even and j > 0, then Therefore, by (1) and the induction hypothesis, there are some C ′ , C ′′ > 0, independent of φ, such that Lemma 3.4. If σ < 0, then S m+1 w,σ ⊂ S m continuously. Proof. Let i, j ∈ N such that i + j ≤ m. Since for any φ ∈ C ∞ , we get φ S m ≤ φ S m+1 w,σ . Corollary 3.5. S = S w,σ as Fréchet spaces.    Proof. This follows by induction on m. It is true for m = 0 because S 0 w,σ = S 0 σ as Banach spaces. Now, let m ≥ 1, and assume that the result holds for m − 1. For . But, by the induction hypothesis and since M m,ev = M m−1,odd + 1, there are some C, C ′ > 0, independent of φ, such that For φ ∈ C ∞ odd , let ψ = x −1 φ, and take i, j and x as above. Then . But, by the induction hypothesis, Corollary 3.6, and since there are some C, C ′ > 0, independent of φ, such that Corollary 3.9. S w,σ ⊂ S σ continuously.

Perturbed Sobolev spaces
Observe that S σ ⊂ L 2 (R, |x| 2σ dx). Like in the case where S is considered as domain, it is easy to check that, in L 2 (R, |x| 2σ dx), with domain S σ , B is adjoint of B ′ and L is symmetric.
Lemma 4.1. S σ is a core 4 of L. 4 Recall that a core of a closed densely defined operator T between Hilbert spaces is any subspace of its domain D(T ) which is dense with the graph norm.
Proof. Let R denote the restriction of L to S σ . Then L ⊂ R ⊂ R * ⊂ L * = L in L 2 (R, |x| 2σ dx) because S ⊂ S σ by Corollaries 3.5 and 3.9.
For each m ∈ R, let W m σ be the Hilbert space completion of S with respect to the scalar product , W m σ defined by φ, ψ W m σ = (1 + L) m φ, ψ σ . The corresponding norm will be denoted by W m σ , whose equivalence class is independent of the parameter s used to define L. In particular, W 0 σ = L 2 (R, |x| 2σ dx). As usual,  (4) and (5) (the details of the proof are omitted because this observation will not be used). Thus L, Σ, B and B ′ define bounded operators on W ∞ σ . Also on the spaces W m σ , the parity of (generalized) functions is preserved by L and Σ, and reversed by B and B ′ . Observe that B ′ is not adjoint of B in W m σ for m = 0. The motivation of our tour through perturbed Schwartz spaces is the following embedding results; the second one is a version of the Sobolev embedding theorem.  Corollary 4.4. S σ = W ∞ σ as Fréchet spaces. For each non-commutative polynomial p (of two variables, X and Y ), let p ′ denote the non-commutative polynomial obtained by reversing the order of the variables in p; e.g., if p(X, Y ) = X 2 Y 3 X, then p ′ (X, Y ) = XY 3 X 2 . It will be said that p is symmetric if p(X, Y ) = p ′ (Y, X). Note that p ′ (Y, X)p(X, Y ) is symmetric for any p. Given any non-commutative polynomial p, the continuous operators p(B, B ′ ) and p ′ (B ′ , B) on S σ are adjoint of each other in L 2 (R, |x| 2σ dx); thus p(B, B ′ ) is a symmetric operator if p is symmetric.   If m is odd, write L ⌊m/2⌋ = f m (B, B ′ ) as above for some symmetric homogeneous non-commutative polynomial f m of degree m − 1. Then, by (4), Thus Claim 3 follows with Proof of Proposition 4.2. By the definitions of B and B ′ , and (16)- (18), for each homogeneous non-commutative polynomial p of three variables with degree d ≤ m + 1, there exists some C p > 0 such that, for all φ ∈ S σ : if σ < 0, |x| ≤ 1, and φ and d have the same parity, then ; and, otherwise, With the notation of Lemma 4.5, let d a denote the degree of each q a , and let q a (x, B, B ′ ) = x q a (B, B ′ ). If σ ≥ 0, then for φ ∈ S σ,ev , and by 2 da even For c = (c k ) and c ′ = (c ′ k ) in R N , and m ∈ R, the expressions c Cm = sup k |c k |(1 + k) m and c, c ′ m , with the induced Fréchet topologies. It is said that c is even/odd if c k = 0 for odd/even k. There are direct sum decompositions into subspaces of even and odd sequences, C m = C m,ev ⊕ C m,odd and ℓ 2 m = ℓ 2 m,ev ⊕ ℓ 2 m,odd (m ∈ R ∪ {∞}).
Proof. It is easy to see that c Cm ≤ c ℓ 2 2m and c ℓ 2 m ≤ c C m ′ ( k (1+k) m−2m ′ ) 1/2 for any c ∈ C ∞ , where the last series is convergent because m − 2m ′ < −1. According to Section 2.2, the "Fourier coefficients" mapping φ → ( φ k , φ σ ) defines a quasi-isometry W m σ → ℓ 2 m for all finite m, and therefore an isomorphism W ∞ σ → C ∞ of Fréchet espaces. This map is compatible with the decompositions into even and odd subspaces.
where the last series is convergent because m − m ′ < −1. Proof. Since 2m ′ > m+6, there are m 1 , m 2 ∈ R so that m ′ −m 2 > 2, 2m 2 −m 1 > 1 and m 1 − m > 1. Then, by Propositions 4.2 and 4.3, Lemmas 4.6 and 4.9, and using the "Fourier coefficients" mapping, we get the composition of bounded maps, σ,ev ֒→ S m σ,ev , which extends x −1 : S odd → S ev by (15).  Proof. We proceed by induction on m. The case m = 0 was already indicated in the proof of Lemma 3.8. Now, let m ≥ 1 and assume that the result holds for m− 1.
For φ ∈ C ∞ ev , i + j ≤ m with j > 0 and x ∈ R, we have But, by the induction hypothesis and because M m,ev = M m−1,odd +1, there are some C, For φ ∈ C ∞ odd , let ψ = x −1 φ, and take i, j and x as above. We have But, by the induction hypothesis, Corollary 4.10, and since M m,odd ≥ M m−1,ev + 1 and 2M m,odd > M m−1,ev + 6, there are some C, C ′ > 0, independent of φ, such that  Proof. This is a consequence of Corollaries 3.10 and 4.14 Corollaries 3.10 and 4.14 and Propositions 4.2 and 4.3 give Theorems 1.1 and 1.2.

Perturbations of H on R +
Since the function |x| 2σ is even, the decomposition S = S ev ⊕ S odd extends to an orthogonal decomposition L 2 (R, |x| 2σ dx) = L 2 ev (R, |x| 2σ dx) ⊕ L 2 odd (R, |x| 2σ dx) . Let L ev/odd and L ev/odd , or L σ,ev/odd and L σ,ev/odd , denote the corresponding components of L and L. L ev/odd is essentially self-adjoint in L 2 ev/odd (R, |x| 2σ dx), and its self-adjoint extension is L ev/odd , which satisfies an obvious version of Corollary 1. 3.
Fix an open subset U ⊂ R + of full Lebesgue measure. Let S ev/odd,U ⊂ C ∞ (U ) denote the linear subspace of restrictions to U of the functions in S ev/odd . The restriction to U defines a linear isomorphism S ev/odd ∼ = S ev/odd,U , and a unitary isomorphism L 2 ev/odd (R, |x| 2σ dx) ∼ = L 2 (R + , 2x 2σ dx). Via these isomorphisms, L ev/odd corresponds to an operator L ev/odd,U on S ev/odd,U , and L ev/odd corresponds to a self-adjoint operator L ev/odd,+ in L 2 (R + , x 2σ dx); the more explicit notation L σ,ev/odd,U and L σ,ev/odd,+ may be used. Let φ k,U = φ k | U , whose norm in L 2 (R + , x 2σ dx) is 1/ √ 2. Going one step further, for any positive function h ∈ C 2 (U ), the multiplication by h defines a unitary isomorphism h : L 2 (R + , x 2σ dx) → L 2 (R + , x 2σ h −2 dx). Thus hL ev,U h −1 , with domain h S ev,U , is essentially self-adjoint in L 2 (R + , x 2σ h −2 dx), and its self-adjoint extension is hL ev,+ h −1 . Via these unitary isomorphisms, we get an obvious version of Corollary 1.3 for hL ev,+ h −1 . By using it easily follows that hL ev,U h −1 has the form of P in Theorem 1.4. Then Theorem 1.4 is a consequence of the following. Proof. By (25), So P = hL σ,ev,U h −1 if and only if h −1 h ′ = σx −1 − f 1 and f 2 = h −1 h ′′ + 2h −1 h ′ f 1 , which are easily seen to be equivalent to the conditions of Theorem 1.4.
Remark 1. By (25), we get an operator of the same type if h and d dx is interchanged in the operator P of Theorem 1.4. Remark 2. By using (25) with h = x −1 on R + , it is easy to check that L σ,odd,R+ = xL 1+σ,ev,R+ x −1 on S odd,R+ = x S ev,R+ . So no new operators are obtained with this process by using L σ,odd instead of L σ,ev .