On the Inverse to the Harmonic Oscillator

Abstract

Abstract. Let b d be the Weyl symbol of the inverse to the harmonic oscillator on R d . We prove that b d and its derivatives satisfy convenient bounds of Gevrey and Gelfand-Shilov type, and obtain explicit expressions for b d . In the even-dimensional case we characterize b d in terms of elementary functions.In the analysis we use properties of radial symmetry and a combination of different techniques involving classical a priori estimates, commutator identities, power series and asymptotic expansions. IntroductionA fundamental operator in quantum physics and classical analysis is the harmonic oscillatorIn physics the operator H appears in the stationary Schrödinger equation for a particle under the action of a quadratic potential. In classical analysis, H is also known as the Hermite operator, and possesses several convenient properties. For example, the operator H is strictly positive in L 2 (R d ) with discrete spectrum, and the eigenfunctions are the Hermite functions, see for example [7,15,19].By means of the Hermite functions one can express also the kernel of the inverse H −1 . On the other hand, coherently with the point of view of the quantum physics, H −1 can be written as Weyl pseudo-differential operator(see for example the general calculus in [9,10,12,16]

On the Weyl symbol of the resolvent of the harmonic oscillator

“…The results of our paper are stronger than those of [2]. First, we consider the more general case of the resolvent (H − z) −1 , whereas [2] is restricted to z = 0.…”

“…The results of our paper are stronger than those of [2]. First, we consider the more general case of the resolvent (H − z) −1 , whereas [2] is restricted to z = 0. Second, our explicit representation in terms of Bessel-type function and in terms of an integral representation is absent in [2].…”

“…It also proves that its derivatives satisfy some estimates. The authors of [2] call them Gelfand-Shilov bounds.…”

“…First, we consider the more general case of the resolvent (H − z) −1 , whereas [2] is restricted to z = 0. Second, our explicit representation in terms of Bessel-type function and in terms of an integral representation is absent in [2]. Third, our bounds on the derivatives easily imply those proven in [2].…”

Pseudo-Differential Calculus in Anisotropic Gelfand–Shilov Setting

“…In [10] we considered the more general case s = σ > 0, which is interesting in particular in connection with Shubintype pseudo-differential operators, cf. [5,9]. Although the extension of the complete calculus developed in [3,4] in this case is out of reach due to the lack of compactly supported functions in S σ s (R d ) and Σ σ s (R d ), nevertheless some interesting results can be achieved also in this case by using different tools than the usual micro-local techniques, namely a method based on the use of modulation spaces and of the short time Fourier transform.…”