Generalized Bargmann inequalities

Pan Lian

In this paper, we establish the Heisenberg’s uncertainty inequality associated with the Caputo derivative of order q ∈ (1, ∞) in the generalized Bargmann–Fock space. We also determine exactly when the equality occurs in the uncertainty inequality. It is done by estimating the growth of the eigenvalues of the commutator [Dq,zq]. We prove that the sequence of eigenvalues of [Dq,zq] tends to ∞ when the fractional order q belongs to (1, ∞). However, the sequence converges to zero when q belongs to (0, 1), which shows different behavior. Hence, only a weak uncertainty inequality is obtained for the latter case.

Luigi Schiavulli

Stephen Bruce Sontz

The Segal-Bargmann transform A is a unitary isomorphism of the quantum mechanical configuration space L2(Rn,ν) onto the Segal-Bargmann, or phase, space HL2(Cn,μ) of entire functions square integrable with respect to μ. Here μ and ν are Gaussian measures, and for z∈Cn and measurable f:Rn→CAf(z) :=∫Rn dν(t)f(t)exp(−z2/2+√z⋅t) if the integral converges. Af is defined for all f∈Lp(Rn,ν) if 1<p⩽∞, but A does not preserve Lp structure if p≠2. Specifically, it is proven here that A is a compact map from Lp(Rn,ν) to Lq(Cn,μ) if 1⩽q<2 and p>1+q/2 (which implies p>q) and in this case ‖A‖p→q⩽((1+q/2−p′q/2)(1−q/2))−n/2q where p′ is the dual of p. Moreover, a calculation with test functions shows that ‖A‖p→q=∞ if q>2 or p<1+q/2 (which is implied by p<q). Using an improved operator norm estimate (analogous to the Hausdorff-Young inequality) we prove Hirschman type entropy inequalities for f∈L2(Rn,ν), (p−1−2−1)S(f )⩽(q−1−2−1)S(Af )+nCp,q‖f‖22, where 1⩽q<2, p>1+q/2, Cp,q is given explicitly as a function of p and q and S(g) is the entropy of g. Examples given here show A is not entropy preserving. So these inequalities provide constraints on the change in entropy. Similarly, we get energy-entropy, or log-Sobolev, inequalities a1S(Af )+a2S(f )⩽a3‖f‖22+a4〈f,Nf〉, for f in a dense subspace of L2(Rn,ν). Here 〈f,Nf〉 is the Dirichlet form of the energy operator N in the configuration space and the ai can have various signs and are also given explicitly. The principal results of this article are generalized Heisenberg uncertainty relations, namely they are constraints on Af, the phase space representation, given information about f, the configuration space representation.

Claudio Pita-Ruiz, Stephen Bruce Sontz

We consider a μ-deformation of the Segal-Bargmann transform, which is a unitary map from a μ-deformed quantum configuration space onto a μ-deformed quantum phase space (the μ-deformed Segal-Bargmann space). Both of these Hilbert spaces have canonical orthonormal bases. We obtain explicit formulas for the Shannon entropy of some of the elements of these bases. We also consider two reverse log-Sobolev inequalities in the μ-deformed Segal-Bargmann space, which have been proved in a previous work, and show that a certain known coefficient in them is the best possible.

Sanket Tikare, Svetlin G. Georgiev