Boundedness of the Segal-Bargmann Transform on Fractional Hermite-Sobolev Spaces

Copyright © 2017 Hong Rae Cho et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Let s ∈ R and 2 ≤ p ≤ ∞. We prove that the Segal-Bargmann transformB is a bounded operator from fractional Hermite-Sobolev spacesWs,p H (Rn) to fractional Fock-Sobolev spaces Fs,p R .


Introduction
In quantum mechanics, the Schrödinger equation is a partial differential equation that describes how the quantum state of some physical system changes with time. The most famous example is the nonrelativistic Schrödinger equation for a single particle moving in a potential: where is the particle's mass, ℏ is the Planck constant, is its potential energy, and Ψ is the wave function. Let be the most basic Schrödinger operator in R , ≥ 1, the Hermite operator (or the harmonic oscillator): Then the Schrödinger equation can be written by This is an important model in quantum mechanics (see, e.g., [1]). For ∈ R, we define the fractional Hermite operator = (−Δ + | |) of order . Let 0 < ≤ ∞. The Hermite-Sobolev space , (R ) of fractional order is the space of all tempered distributions for which the distribution /2 is given by an function on R .
Let C be the complex -space and let be the ordinary volume measure on C . If = ( 1 , . . . , ) and = ( 1 , . . . , ) are points in C , we write For any 0 < ≤ ∞ the Fock space denotes the space of entire functions on C such that the function For = ∞ the norm in ∞ is defined by Let Journal of Function Spaces Both and * , as defined above, are densely defined linear operators on (unbounded though). We consider the radial derivative R defined by Let be a real number and 0 < ≤ ∞. The fractional Fock-Sobolev space , R of order is the space of all entire functions for which R /2 is given by an function. The Segal-Bargmann transform B is defined by where ( ) is the volume measure on R . It is well-known that the Segal-Bargmann transform is a unitary isomorphism between 2 (R ) and 2 [2,3].
We prove that the radial derivative R has a parallel behavior to the Hermite operator . In particular, R is densely defined, positive, self-adjoint and has the discrete spectrum; it generates a diffusion semigroup. Moreover, we show that the Segal-Bargmann transform intertwines fractional Hermite-Sobolev spaces with fractional Fock-Sobolev spaces as follows.

Fractional Hermite-Sobolev Spaces
In one dimension, the Hermite polynomials are defined by and by normalization we obtain the Hermite functions Note that In higher dimensions, for each multi-index = ( 1 , . . . , ) ∈ N 0 , the Hermite functions ℎ are defined by Here, N 0 = N ∪ {0} is the set of nonnegative integer. By (12), we know that these are the eigenfunctions of the Hermite operator defined in (2). In fact, Moreover, {ℎ : ∈ N 0 } is an orthonormal basis for 2 (R ).
Let H be the space of finite linear combinations of Hermite functions where The space H is dense in 2 (R ), and so, by the orthonormality of the Hermite functions, For ∈ R, we define the fractional Hermite operator = (−Δ + | |) of order . For ∈ S(R ), the Hermite series expansion converges to uniformly in R (and also in 2 (R )), since ‖ℎ ‖ ∞ (R ) ≤ , for all ∈ N 0 , and each ∈ N, and we have (see [4]) Definition 2. Let ∈ R and ∈ S(R ). One defines the fractional Hermite operator by The fractional Hermite operators were introduced in [5].
Definition 3. Let ∈ R and 0 < ≤ ∞. The fractional Hermite-Sobolev space , (R ) of order is the space of all tempered distributions for which the distribution /2 is given by an function on R . The fractional Hermite-Sobolev norm of order is defined accordingly, The fractional Hermite-Sobolev spaces , (R ) of order were introduced in [6].

Radial Derivative
We consider the radial derivative R defined on (24) We have The following example tells us that Dom(R) ⊊ 2 . Thus R is an unbounded operator on 2 .
Proof. Note that where (⋅) is the Riemann zeta function. However, we have Lemma 5. R is a positive, self-adjoint operator on Dom(R).
Proof. Let P(C ) be the set of all holomorphic polynomials on C . We know that P(C ) is dense in 2 and R is selfadjoint on P(C ). Hence Dom(R) is the domain of its unique self-adjoint extension.
Note that Thus R is positive.
be the orthonormal decomposition of . Associated with the operator R is a semigroup { } ≥0 defined by the expansion

Journal of Function Spaces
We can check that ( , ) fl ( ) is the solution of the heattype equation: It is easy to see that Thus is contractive.

Proposition 7. { } ≥0 is a strongly continuous semigroup.
Proof. We note that For ∈ N 0 and ⊂ N 0 we define ( ) by where ] is a discrete measure defined by By Lebesgue dominate convergence theorem, we have Hence { } ≥0 is a strongly continuous semigroup.
Proof. By using the previous discrete measure ], it follows that Taking limit on both sides and by Lebesgue dominate convergence theorem, Thus we get the result.
By Proposition 8, we have

Fractional Fock-Sobolev Spaces
Since R has discrete spectrum {2| |+ : ∈ N 0 }, by using the spectral theorem, we define the fractional radial derivative R for ∈ R as follows.
be the orthonormal decomposition of . By the spectral theorem, R is given by Definition 10. Let be a real number and 0 < ≤ ∞. The fractional Fock-Sobolev space , R of order is the space of all entire functions for which R /2 is given by an function. The fractional Fock-Sobolev norm of of order is defined accordingly, We refer the reader to [7][8][9][10] for other Fock-Sobolev spaces.

-Boundedness of the Segal-Bargmann Transform
The Hermite operator is self-adjoint on the set of infinitely differentiable functions with compact support ∞ (R ), and it can be factorized as (55) Proof. Let ∈ ∞ (R ). By the integration by parts, we have This gives We differentiate under the integral sign to obtain This gives By (57) and (60), it follows that

Corollary 12. Consider
Proof. By Lemma 11, we have Proposition 13. Let ∈ R. Then Proof. We define Then { : ∈ N 0 } is an orthonormal basis for 2 and B(ℎ ) = . For ∈ S(R ) we have and so Since B is a unitary isomorphism, we have ⟨ , ℎ ⟩ = ⟨B( ), ⟩. Hence Thus we get the result.
We consider the mapping property of the Segal-Bargmann transform B as a map from (R ) to for ∈ [2, ∞]. Note that one-dimensional case is in [11].

Theorem 14. Consider
Proof. We have Note that Hence (73) Thus we get the result.
Proof. The 2 -boundedness is followed by the unitary isomorphism of the Segal-Bargmann transform. In Theorem 14, we proved the ∞ -boundedness of the Segal-Bargmann transform. By Lemma 15, we have the required result.
By Proposition 13 and Theorem 16, we have the following result.