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

Abstract

Let and . We prove that the Segal-Bargmann transform is a bounded operator from fractional Hermite-Sobolev spaces to fractional Fock-Sobolev spaces .

1. 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, is the Planck constant, is its potential energy, and is the wave function.

Let be the most basic Schrödinger operator in , 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 , we define the fractional Hermite operator of order . Let . The Hermite-Sobolev space of fractional order is the space of all tempered distributions for which the distribution is given by an function on .

Let be the complex -space and let be the ordinary volume measure on . If and are points in , we writeFor any the Fock space denotes the space of entire functions on such that the function is in . We define For the norm in is defined by

LetBoth and , as defined above, are densely defined linear operators on (unbounded though). We consider the radial derivative defined by Let be a real number and . The fractional Fock-Sobolev space of order is the space of all entire functions for which is given by an function.

The Segal-Bargmann transform is defined bywhere is the volume measure on . It is well-known that the Segal-Bargmann transform is a unitary isomorphism between and [2, 3].

We prove that the radial derivative has a parallel behavior to the Hermite operator . In particular, 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.

Theorem 1. Let and . Then the Segal-Bargmann transform is bounded.

2. 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 , the Hermite functions are defined by Here, 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, is an orthonormal basis for .

Let be the space of finite linear combinations of Hermite functionswhereThe space is dense in , and so, by the orthonormality of the Hermite functions,

For , we define the fractional Hermite operator of order . For , the Hermite series expansionconverges to uniformly in (and also in ), since , for all , and each , and we have (see [4])

Definition 2. Let and . One defines the fractional Hermite operator by

The fractional Hermite operators were introduced in [5].

Definition 3. Let and . The fractional Hermite-Sobolev space of order is the space of all tempered distributions for which the distribution is given by an function on . The fractional Hermite-Sobolev norm of order is defined accordingly,

The fractional Hermite-Sobolev spaces of order were introduced in [6].

3. Radial Derivative

We consider the radial derivative defined onbywhereWe haveThe following example tells us that . Thus is an unbounded operator on .

Example 4. LetThen , but .

Proof. Note thatwhere is the Riemann zeta function. However, we have

Lemma 5. is a positive, self-adjoint operator on .

Proof. Let be the set of all holomorphic polynomials on . We know that is dense in and is self-adjoint on . Hence is the domain of its unique self-adjoint extension.
Note thatThus is positive.

Lemma 6. has the discrete spectrum .

Proof. By (29), we have .
We defineThen is an orthonormal basis for . It is easy to see that is the set of all eigenvalues.
Let . First, we show that is injective and surjective.
Suppose that . ThenThis implies . Thus is injective.
For letbe the orthonormal decomposition of . We defineSinceis a Cauchy sequence in , the series in (33) converges in . Henceis a well-defined element of and it satisfies . This means that is surjective.
Moreover,where . Hence is bounded and so .

For let be the orthonormal decomposition of . Associated with the operator is a semigroup defined by the expansionWe can check that is the solution of the heat-type equation:It is easy to see thatThus is contractive.

Proposition 7. is a strongly continuous semigroup.

Proof. We note thatFor and we define by Thenwhere is a discrete measure defined byBy Lebesgue dominate convergence theorem, we have Hence is a strongly continuous semigroup.

Proposition 8. is the infinitesimal generator of . That is,

Proof. By using the previous discrete measure , it follows thatTaking limit on both sides and by Lebesgue dominate convergence theorem, Thus we get the result.

By Proposition 8, we have

4. Fractional Fock-Sobolev Spaces

Since has discrete spectrum , by using the spectral theorem, we define the fractional radial derivative for as follows.

Definition 9. Let . For letbe the orthonormal decomposition of . By the spectral theorem, is given by

Definition 10. Let be a real number and . The fractional Fock-Sobolev space of order is the space of all entire functions for which is given by an function. The fractional Fock-Sobolev norm of of order is defined accordingly,

We refer the reader to [7–10] for other Fock-Sobolev spaces.

5. -Boundedness of the Segal-Bargmann Transform

The Hermite operator is self-adjoint on the set of infinitely differentiable functions with compact support , and it can be factorized aswhere

Lemma 11. For each , one has

Proof. Let . By the integration by parts, we haveThis givesWe differentiateunder the integral sign to obtainThis givesBy (57) and (60), it follows that

Corollary 12. Consider

Proof. By Lemma 11, we have

Proposition 13. Let . Then

Proof. We defineThen is an orthonormal basis for and . For we haveand soSince is a unitary isomorphism, we have . Hence Thus we get the result.

We consider the mapping property of the Segal-Bargmann transform as a map from to for . Note that one-dimensional case is in [11].

Theorem 14. Consider

Proof. We have Note thatHenceand soThus we get the result.

The following Stein-Weiss interpolation theorem is well-known. See, for example, [3, 12].

Lemma 15. Let , and be positive weight functions on a measure space . If and , then with equal norms, where

Theorem 16. Let . There exists such that

Proof. The -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.

Theorem 17. Let and . Then the Segal-Bargmann transform is bounded.

Disclosure

An earlier version of this work was presented as an abstract at the International Conference on the 70th Anniversary of the Korean Mathematical Society, 2016.

Competing Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors would like to thank the referee for his/her valuable remarks and suggestions. The first author was supported by NRF of Korea (NRF-2016R1D1A1B03933740).