Sharp inequalities for coherent states and their optimizers
-
Rupert L. Frank
Abstract
We are interested in sharp functional inequalities for the coherent state transform related to the Wehrl conjecture and its generalizations. This conjecture was settled by Lieb in the case of the Heisenberg group, Lieb and Solovej for SU(2), and Kulikov for SU(1, 1) and the affine group. In this article, we give alternative proofs and characterize, for the first time, the optimizers in the general case. We also extend the recent Faber-Krahn-type inequality for Heisenberg coherent states, due to Nicola and Tilli, to the SU(2) and SU(1, 1) cases. Finally, we prove a family of reverse Hölder inequalities for polynomials, conjectured by Bodmann.
1 Introduction and main results
Coherent states appear in various areas of pure and applied mathematics, including mathematical physics, signal and image processing, and semiclassical and microlocal analysis. Some background can be found, for instance, in [33,39,40]. Here, we are interested in sharp functional inequalities for coherent state transforms.
To motivate the questions we are interested in, let us recall Wehrl’s conjecture [44] and its resolution by Lieb [23]. Following Schrödinger, Bargmann, Segal, Glauber, Klauder, and others, we consider a certain family of normalized Gaussian functions
where
known as the Husimi function, the covariant symbol, or the lower symbol. Thus, to a quantum state
showed that it is positive, and conjectured that its minimum value occurs when
is maximal for
In [10], Carlen gave an alternative proof of Lieb’s result, both for
In [24], Lieb and Solovej extended the earlier results and showed what they called the generalized Wehrl conjecture. Namely, for any convex function
Our first main result in this article gives an alternative proof of the theorem of Lieb and Solovej and includes a new characterization of the cases of equality.
Theorem 1
Let
and the supremum is attained for
Note that the value of the double integral with
Under a slightly stronger assumption on
Corollary 2
Let
and the supremum is attained for
Remark 3
The statement and proof of Theorem 1 and Corollary 2 extend, with minor changes, to the case of higher dimensions. We omit the details.
Coherent states are often closely related to representations of an underlying Lie group. The coherent states discussed so far are related to the Heisenberg group. In his article, containing the proof of Wehrl’s conjecture, Lieb conjectured that the analog of Wehrl’s conjecture also holds for Bloch coherent states, that is, for a family of coherent states related to SU(2). After some partial results in [9,38], this conjecture was finally solved by Lieb and Solovej in [24]; see also [25] for a partially alternate proof. Again, they prove a generalized version of Lieb’s conjecture involving general convex functions
Let us be more specific. As is well known (see, e.g., [19, Chapter II] and [41, Section VIII.4]), the nontrivial irreducible representations of SU(2) are labeled by
Theorem 4
Let
and the supremum is attained for
Note that the value of the integral with
Corollary 5
Let
and the supremum is attained for
Our third main result concerns coherent states for certain representations of SU(1, 1). After initial results in [3,6,26], the analog of Wehrl’s conjecture was recently settled by Kulikov in [20], again for general convex functions
All nontrivial representations of SU(1, 1) are infinite-dimensional. Its nontrivial irreducible unitary representations consist of discrete, principal, and complementary series, as well as limits of the discrete series; see, e.g., [19, Chapters II and XVI; also (2.20)]. Here, we are only interested in one of the two discrete series. The results for the other one can be deduced from the results below by applying complex conjugation at the appropriate places.
Following the notation in [5], the discrete series representation under consideration is labeled by
Moreover, one has
where
For any
In this way, we obtain a family of vectors
In what follows, we denote by
Theorem 6
Let
and the supremum is attained for
Note that the value of the integral with
Theorem 6 proves the uniqueness part of a conjecture of Lieb and Solovej [26, Conjecture 5.2]. As we mentioned before, the inequality part is due to Kulikov [20].
Corollary 7
Let
and the supremum is attained for
For every real
The affine group (in one space dimension), also known as the
We fix a parameter
(Here, we follow the convention in [12]. What we call
Theorem 8
Let
and the supremum is attained for
Note that the value of the double integral with
Theorem 8 settles the equality part of a conjecture of Lieb and Solovej [26, Conjecture 3.1]. For strictly convex
We note that Theorem 8 has a version for
Corollary 9
Let
and the supremum is attained for
This concludes the description of our main results, but we would like to draw the reader’s attention also to Sections 3 and 5 where we prove, respectively, sharp reverse Hölder inequalities for analytic functions, thereby settling a conjecture of Bodmann [9], and optimal Faber-Krahn-type inequalities for coherent state transforms.
The method that we will be using is that from a recent, beautiful article by Kulikov [20]. He developed this method to solve the Lieb-Solovej conjectures for SU(1, 1) and the affine group. Here, we show that it can be adapted to deal with the Heisenberg and the SU(2) cases. We also push the characterization of optimizers a bit further than in [20], thus leading to the optimal results in Theorems 1, 4, 6, and 8.
Kulikov’s article in turn seems to be inspired by an equally beautiful recent article by Nicola and Tilli [29]. They were the first, as far as we know, to use the isoperimetric inequality in connection with the coherent state transform to obtain optimal functional inequalities. (Talenti [43] used a closely related method for comparison theorems for solutions of PDEs.) Kulikov proved his results by instead using the isoperimetric inequality in hyperbolic space, and we will prove Theorem 4 by using that on the sphere. While it is tempting to try to use the same method for more general groups, an obstacle will have to be overcome (see Section 4.6).
Nicola and Tilli proved Faber-Krahn-type inequalities for the Heisenberg coherent states. We will show that their main result (at least without the characterization of the cases of equality) follows from Theorem 1, and we will use this idea to prove analogs of their results for coherent states based on SU(2), SU(1, 1), and the affine group (see Section 5). For further developments started by [29], see, for instance, [18,30,35].
After this article was submitted for publication, we learned that Kulikov et al. had independently obtained similar results with similar techniques. These results have appeared in preprint form [21].
Thanks are due to Eric Carlen, Elliott Lieb, and Jan Philip Solovej for many discussions on the topics of this article.
It is my pleasure to dedicate this article to David Jerison on the occasion of his 70th birthday. His articles have been an inspiration for me, those on sharp inequalities [17] and others. I am particularly indebted to him for his remarks in the fall of 2008, which indirectly were a great motivation for work that eventually led to [14].
2 Inequalities for analytic functions
The main ingredient behind the results in the previous section are sharp inequalities for analytic functions and the characterization of their optimizers, which we discuss in this section.
2.1 Definitions and main result
There are three different types of inequalities, corresponding to the cases of the Heisenberg group, SU(2), and SU(1, 1). We refer to these different scenarios as Cases 1, 2, and 3. In Cases 2 and 3, there is a parameter
In Case 1, we consider functions from the Fock space
We recall that we write
This norm is finite for any
To treat all cases simultaneously, we set
and denote the norm in
Thus, the set on which the relevant functions are defined is
and the relevant measure is
To a function
The problem that we are interested in is to maximize, given a convex function
over all
and set
In Case 2, we consider the functions
and set
In Case 3, we consider the function
and set
In each case, it can be verified that
The following is the main result of this section.
Theorem 10
Let
and the supremum is attained in
We will prove this theorem in Subsection 2.2, after establishing some lemmas.
2.2 Proof of Theorem 10
We begin the proof of Theorem 10 by recalling a simple and well-known bound on the supremum of
Lemma 11
Let
with equality if and only if either
Proof
In Case 1, this is essentially [29, Proposition 2.1]. Indeed, there the inequality in the lemma is proved, and it is shown that
In Case 2, the inequality is mentioned in [9, Paragraph after Remark 3.2]. Since the function
In Case 3, the inequality and the fact that
We now come to the core of the proof of Theorem 10, which concerns a certain monotonicity of the measure of superlevel sets of
Lemma 12
Let
is nondecreasing on
Proof
Writing
Let
Taking
Thus,
For a curve
In particular, for the level set
As we argued before, the first term on the right side is
To compute the Laplacian of
Note that the right side is equal to a constant multiple of
To summarize, we have shown that
We now use the isoperimetric inequality to bound the left side from below; for references in the spherical and hyperbolic case, see, for instance, [31, (4.23)], as well as [8], [22, Third part, Chapter IV], [34,36,37]. We have
Using these inequalities with
These inequalities are equivalent to the monotonicity assertions in the lemma.
It remains to verify that this function is constant if
It follows that for
The last ingredient in the proof of Theorem 10 is an inequality due to Chebyshev [11] (see also [16, Theorems 43 and 236]). For a proof of the following lemma, with a slightly weaker assumption than monotonicity of one of the functions, see [20, Lemma 4.1].
Lemma 13
Let
We are finally in position to prove the main result of this section.
Proof of Theorem 10
We begin with some preliminary remarks concerning convex functions
Next, we argue that we may assume that
After these preliminaries, we begin with the main part of the argument. Let
Then, the quantity we are interested in can be written as follows:
Here,
We now write the quantity on the right side as follows:
where, according to Lemma 12,
Let us set
and for
Then, the normalization (4) can be equivalently written as follows:
while the quantity to be maximized is
where, for
Since
where, for
Thus, we have shown the upper bound as follows:
where, for
Our goal now is to show that
In order to prove the monotonicity of
From these expressions, one can easily deduce that
We now compute
Since
According to (7), the right side is equal to zero in all cases. This proves that
As mentioned before, the monotonicity of
Finally, assume that
As a consequence, if
To compute the latter, we may choose an arbitrary element in
Changing variables
2.3 Extension to density matrices
In this subsection, we generalize the inequality in Theorem 10, and under a slightly stronger assumption on
Given an operator
We then set
It is easily checked that this is well-defined. (Note, in particular, the nonuniqueness of the above decomposition of
Corollary 14
Let
and the supremum is attained for
Proof
We use the above expansion of
Thus, with
Since, by Theorem 10,
Now assume that
Assuming now that
2.4 Another inequality of Kulikov
For later purposes, in this subsection, we record another inequality from [20], which corresponds, in some sense, to the limiting case
The underlying Hilbert space is the Hardy space
To emphasize the analogy with Theorem 10, we denote this space by
Proposition 15
Let
and the supremum is attained in
By
Proof
The first part is a special case of [20, Theorem 1.1]. The second part can be obtained by an inspection of the proof of the first part, but since this is not explicitly stated in [20], we provide some details. As in the proof of Theorem 10, we may assume that
where
As shown in [20, Theorem 3.1], we have
where the first term on the right side corresponds to the value of the supremum. Thus, if this term is finite and
To compute the value of the supremum, we can proceed exactly as in Case 3 of Theorem 10, setting
3 Reverse Hölder inequalities for analytic functions
The material in this section is an extension of that in the previous section. It is not relevant for the proof of the results in Section 1.
In Theorem 10, we were working under a constraint on a Hilbertian norm. It turns out that this is an unnecessary restriction. We will now prove a generalization of Theorem 10 with a constraint on a more general norm or quasinorm. This will allow us to settle a conjecture by Bodmann [9, Conjecture 3.5].
We continue to use the notation of Section 2. For
This is a norm for
Theorem 16
Let
and the supremum is attained in
For
Taking
Corollary 17
Let
with equality if and only if
This corollary in Case 1 is due to Carlen [10, Theorem 2]. In fact, Carlen proves a more general inequality including an additional parameter. Carlen’s method of proof depends on the logarithmic Sobolev inequality and an identity for analytic functions. It is different from ours. Corollary 17 in Case 2 has been conjectured by Bodmann [9, Conjecture 3.5], who proved it in the special case where
We now turn to the proof of Theorem 16. The main new ingredient is the following generalization of Lemma 11. In Case 3, this is well known [20, (1.1)] and probably also in Case 1, but in Case 2 it might be new.
Lemma 18
Let
with equality if and only if either
Proof
We begin by showing that
with equality if and only if
We multiply by
where
The measure
where the second inequality comes from Jensen’s inequality. Since the exponential function is strictly convex, Jensen’s inequality is strict unless
We now claim that, for any
if and only if
Inequality (9) implies the inequality in the lemma. Now assume that
Thus, it suffices to prove the asymptotic vanishing of
Proof of Theorem 16
Given Lemma 18, which replaces Lemma 11, the proof is a minor variation of that of Theorem 10. We only sketch the major steps. The task is to maximize
under the constraint
with
4 Proof of the main results
In this section, we prove the main results stated in Section 1. In each case, we will work with a concrete representation of the group action that involves analytic functions. The inequalities will then be deduced from Theorem 10.
4.1 Proof of Theorem 1
By scaling, it suffices to prove the theorem for a single value of
with
It is well known and easy to see that
where the last identity is the completeness relation of the coherent states. In particular,
so Theorem 1 follows immediately from Theorem 10 in Case 1. Similarly, Corollary 2 follows from Corollary 14.
4.2 Proof of Theorem 4
Let
where
This representation restricted to
We may choose the space
Consequently, the phases of
with
Since
and so, by a change of variables,
Thus, Theorem 4 follows immediately from Theorem 10 in Case 2. Similarly, Corollary 5 follows from Corollary 14.
4.3 Proof of Theorem 6
Let
where
This representation restricted to
We may choose the space
with
Since
so Theorem 6 follows immediately from Theorem 10 in Case 3. Similarly, Corollary 7 follows from Corollary 14.
4.4 The limit of the discrete series
Two other irreducible unitary representations of SU(1, 1) are not in the discrete series, but are closely related to it, the so-called limits of discrete series [19, Chapter II]. They are typically not considered in the context of coherent states, since they are not square-integrable, but the questions discussed in this article make perfect sense for them and can be completely answered.
We restrict our attention to one of the limits of the discrete series, since the results for the other one can be deduced by appropriate complex conjugation. The construction of the coherent states is verbatim the same as for the discrete series, except that the value of
Proposition 19
Consider the irreducible limit of the discrete series representation of
and the supremum is attained for
Note that the value of the integral with
Proof of Proposition 19
We consider the same representation of SU(1, 1) on functions on
with
By the reproducing property of the kernel (seen, for instance, by expanding both functions in the integrand into a Fourier series), we see that
so Proposition 19 follows from Proposition 15.□
There is also an analog of Corollary 7 extending Proposition 19 (with convex
4.5 Proof of Theorem 8
Given
with
It is easy to see and known that
where the last identity is the completeness relation of the coherent states [12, (2.10)]. Consider the conformal map
Setting
we find that
Moreover, after a simple computation,
and therefore,
In addition, the coherent states
Thus, Theorem 8 follows immediately from Theorem 10 in Case 3 with
Remark 20
The functions
4.6 Limitations of the method
In this article, we have discussed the cases of the Heisenberg group, SU(2), SU(1, 1), and the affine group. It is a natural question, potentially of relevance for representation theory, to which extent the results can be generalized to arbitrary Lie groups.
While the method of the present article is able to treat various cases in a unified way, it will probably not be able to deal with the general case, as we argue now. One of the key ingredients in the argument is Lemma 12, whose proof uses the fact that the superlevel sets of the overlap of two coherent states are isoperimetric set. This may fail in general, as we are going to show.
Following Lieb and Solovej [25], we consider the case of symmetric representations of SU(N). We fix
where
Lieb and Solovej have solved the corresponding problem and shown that, for any convex
is attained for coherent states.
If we tried to reprove this through the method in the present article, we would consider the measure of the superlevel sets of the function
These are geodesic balls (see, e.g., [15, Example 2.110]), and if we want to use the method based on an isoperimetric inequality, they should be optimizers for the isoperimetric inequality. (More precisely, this isoperimetric property should hold for all
5 Faber-Krahn-type inequalities for the coherent state transform
The main result of the recent article [29] by Nicola and Tilli states that, for any measurable set
with equality if and only if
We claim that the inequality (10) follows by abstract arguments from Theorem 1. Of course, this is not too surprising, since Kulikov’s arguments, which we have adapted to yield a proof of Theorem 1, are inspired by those in [29]. Nevertheless, this observation will allow us to derive an analog of the Nicola-Tilli results in the SU(2), SU(1, 1) and
Proof of (10) given Theorem 1
Fixing
for any convex function
for any measurable set
This is symmetric decreasing around
By carefully going through the majorization argument, it should be possible to deduce the equality statement by Nicola and Tilli from the equality statement in Theorem 1, but we omit this here.
Obviously, the above argument can be generalized to the SU(2), SU(1, 1), and
Theorem 21
Let
Equality is attained if
Theorem 22
Let
where
Theorem 23
Let
where
In Theorems 22 and 23, by a “hyperbolic ball,” we mean a geodesic ball with respect to the hyperbolic metric on
Theorem 23, including a characterization of equality cases, has recently been proved in [35] by a direct adaptation of the method in [29]. Our proof, based on Theorem 8, is different.
Proof of Theorems 21, 22, and 23
As above, one can show that the left sides in the theorems are bounded by the supremum of the integral of
To complete the proof, we will need some explicit knowledge about the function
This, in fact, yields the shape for an arbitrary choice of
It remains to compute the supremum. It is convenient to do this in terms of the functions
Meanwhile, by the layer cake formula,
The first term on the right side is equal to
with
This gives the expression of the supremum in terms of
(In the last case, we used the fact that
-
Funding information: Partial support through the US National Science Foundation grant DMS-1954995, as well as through the Deutsche Forschungsgemeinschaft (German Research Foundation) through Germany’s Excellence Strategy EXC-2111-390814868, is acknowledged.
-
Conflict of interest: The author states that there is no conflict of interest.
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- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
Articles in the same Issue
- Research Articles
- Asymptotic properties of critical points for subcritical Trudinger-Moser functional
- The existence of positive solution for an elliptic problem with critical growth and logarithmic perturbation
- On some dense sets in the space of dynamical systems
- Sharp profiles for diffusive logistic equation with spatial heterogeneity
- Generic properties of the Rabinowitz unbounded continuum
- Global bifurcation of coexistence states for a prey-predator model with prey-taxis/predator-taxis
- Multiple solutions of p-fractional Schrödinger-Choquard-Kirchhoff equations with Hardy-Littlewood-Sobolev critical exponents
- Improved fractional Trudinger-Moser inequalities on bounded intervals and the existence of their extremals
- The existence of infinitely many boundary blow-up solutions to the p-k-Hessian equation
- A priori bounds, existence, and uniqueness of smooth solutions to an anisotropic Lp Minkowski problem for log-concave measure
- Existence of nonminimal solutions to an inhomogeneous elliptic equation with supercritical nonlinearity
- Non-degeneracy of multi-peak solutions for the Schrödinger-Poisson problem
- Gagliardo-Nirenberg-type inequalities using fractional Sobolev spaces and Besov spaces
- Ground states of Schrödinger systems with the Chern-Simons gauge fields
- Quasilinear problems with nonlinear boundary conditions in higher-dimensional thin domains with corrugated boundaries
- A system of equations involving the fractional p-Laplacian and doubly critical nonlinearities
- A modified Picone-type identity and the uniqueness of positive symmetric solutions for a prescribed mean curvature problem
- On a version of hybrid existence result for a system of nonlinear equations
- Special Issue: Geometric PDEs and applications
- Preface for the special issue on “Geometric Partial Differential Equations and Applications”
- Convex hypersurfaces with prescribed Musielak-Orlicz-Gauss image measure
- Total mean curvatures of Riemannian hypersurfaces
- On degenerate case of prescribed curvature measure problems
- A curvature flow to the Lp Minkowski-type problem of q-capacity
- Aleksandrov reflection for extrinsic geometric flows of Euclidean hypersurfaces
- A note on second derivative estimates for Monge-Ampère-type equations
- The Lp chord Minkowski problem
- Widths of balls and free boundary minimal submanifolds
- Smooth approximation of twisted Kähler-Einstein metrics
- The exterior Dirichlet problem for the homogeneous complex k-Hessian equation
- A Carleman inequality on product manifolds and applications to rigidity problems
- Asymptotic behavior of solutions to the Monge-Ampère equations with slow convergence rate at infinity
- Pinched hypersurfaces are compact
- The spinorial energy for asymptotically Euclidean Ricci flow
- Geometry of CMC surfaces of finite index
- Capillary Schwarz symmetrization in the half-space
- Regularity of optimal mapping between hypercubes
- Special Issue: In honor of David Jerison
- Preface for the special issue in honor of David Jerison
- Homogenization of oblique boundary value problems
- A proof of a trace formula by Richard Melrose
- Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents
- Regularity properties of monotone measure-preserving maps
- Examples of non-Dini domains with large singular sets
- Sharp inequalities for coherent states and their optimizers
- Gradient estimates and the fundamental solution for higher-order elliptic systems with lower-order terms
- Propagation of symmetries for Ricci shrinkers
- Linear extension operators for Sobolev spaces on radially symmetric binary trees
- The Neumann problem on the domain in 𝕊3 bounded by the Clifford torus
- On an effective equation of the reduced Hartree-Fock theory
- Polynomial sequences in discrete nilpotent groups of step 2
- Integral inequalities with an extended Poisson kernel and the existence of the extremals
- On singular solutions of Lane-Emden equation on the Heisenberg group
