Spectral projections and resolvent bounds for partially elliptic quadratic differential operators
We study resolvents and spectral projections for quadratic differential operators under an assumption of partial ellipticity. We establish exponential-type resolvent bounds for these operators, including Kramers-Fokker-Planck operators with quadratic potentials. For the norms of spectral projections for these operators, we obtain complete asymptotic expansions in dimension one, and for arbitrary dimension, we obtain exponential upper bounds and the rate of exponential growth in a generic situation. We furthermore obtain a complete characterization of those operators with orthogonal spectral projections onto the ground state.
Key words and phrases:Non-selfadjoint operator; resolvent estimate; spectral projections; quadratic differential operator; FBI-Bargmann transform
An extensive body of recent work has focused on the size of resolvent norms, semigroups, and spectral projections for non-normal operators, where these objects are not controlled by the spectrum of the operator; see . Rapid resolvent growth for quadratic operators such as
along rays inside the range of the symbol has been shown ,  and extended significantly , . Sharp upper bounds of exponential type were recently shown in . The spectral projections of these operators were explored in , where precise rates of exponential growth were found. We focus here on operators with purely quadratic symbols, which are useful as accurate approximations for many operators whose symbols have double characteristics.
A weaker hypothesis than ellipticity describes a broader class of operators which includes many operators important to kinetic theory , . Hypotheses on the so-called singular space of the symbol, particularly when that space is trivial, have been used successfully to describe semigroups generated by such operators , , , , .
The purpose of the present work is threefold: first, we extend the analysis of  to include these operators with trivial singular spaces, providing exponential-type upper bounds for resolvents. Second, we describe the spectral projections of elliptic and partially elliptic operators in a concrete way. Third, we exploit this description to obtain information related to spectral projections and their norms, including exponential upper bounds, the rate of exponential growth in a generic situation, a complete asymptotic expansion in dimension 1, and a characterization of those operators with orthogonal projection onto the ground state.
1.2. Background on quadratic operators
The structure of quadratic forms
and their associated differential operators is well-studied (see e.g. Chapter 21.5 of ), and here we recall much of the standard terminology which will be used throughout this work.
With the formulation
we can identify the semiclassical Weyl quantization of , viewed as an unbounded operator on , with the formula
The semiclassical parameter is generally considered to be small and positive. Homogeneity of the symbol and the unitary (on ) change of variables
give the relation
demonstrating that the semiclassical quantization of quadratic forms is unitarily equivalent to a scaling of the classical () quantization.
We have the standard symplectic form
Associated with is the “Hamilton map” or “fundamental matrix”
which is the unique linear operator on , antisymmetric with respect to in the sense that
We will write when the quadratic form is perhaps unclear.
We here consider which are partially elliptic both in that
and in that the so-called singular space of , defined in , is trivial:
We will say that is elliptic if there exists with
We note that any elliptic quadratic form has , and so the conditions (1.5), (1.6) generalize the elliptic case. We also recall that, aside from some degenerate cases only occuring when , the assumption suffices to establish that is elliptic for some with (see, e.g., Lemma 3.1 of ).
Under the assumption (1.6), define as the least nonnegative integer such that the intersection defining becomes trivial:
We write for the generalized eigenspace of corresponding to the eigenvalue . We then have the associated subspaces
which are Lagrangian, meaning that and . Furthermore, when is elliptic as in (1.7), we have that is positive in the sense that
In the corresponding sense, is negative. The extension of this fact to obeying (1.5) and (1.6) is essentially known in previous works; see the end of Section 2 of  and references therein. For completeness, we here include a proof in Proposition 2.1.
Then we have the formula
We also study the spectral projections of these operators. Following the notation of Theorem XV.2.1 of  (see also Chapter 6 of ), let us assume that is a closed densely defined operator on a Hilbert space, and , where is contained in a bounded Cauchy domain with . Let be the oriented boundary of . Then we call
the spectral (or Riesz) projection for and .
Because the spectra we will study, given by (1.13), are discrete, we will generally use the definition in the case that is finite. We emphasize that facts about the spectral projections are independent of the semiclassical parameter after scaling, as (1.3) provides that the projection for the classical operator and is unitarily equivalent to the projection for the semiclassical operator and :
Here is Lebesgue measure on , and we only need to consider the most elementary case where is quadratic when regarded as a function of , real-valued, and strictly convex. When functions do not need to be holomorphic, we refer to
and we will often omit where we hope it can be understood.
When working in weighted spaces or , we assume unless otherwise stated that derivatives are holomorphic, meaning that
We finish this section with some brief remarks on notation. We use for a symmetric (bilinear) inner product, usually the dot product on , and for a Hermitian (sesquilinear) inner product on a Hilbert space . We frequently refer to adjoints of operators on a Hilbert space. When the space needs to be emphasized, we add it as a subscript, for example writing . We use to denote the set of bounded linear operators mapping to itself with the usual operator norm. We frequently use a superscript to indicate that an object is “dual” in a loose sense, but the formal meaning may change from instance to instance.
Finally, when we say that a unitary operator quantizes a canonical transformation , we mean that
for appropriate symbols and an appropriate definition of the semiclassical Weyl quantization. In this work, we only apply this notion to (complex) linear canonical transformations and to symbols which are homogeneous polynomials of degree no more than 2, in which case formulas like (1.1) may be used. We therefore use only the most rudimentary aspects of the theory of metaplectic operators; see for instance the Appendix to Chapter 7 of  or Chapter 3.4 of .
1.3. Statement of results
We are now in a position to formulate the four main results of this work.
First, we extend the central result of  to include partially elliptic operators, at the price of more rapid exponential growth. In fact, the result here is identical to the main result in  save that exponential growth in is replaced by exponential growth in . A remarkable recent estimate of Pravda-Starov  provides a subelliptic estimate sufficient to establish the following theorem, which gives exponential-type semiclassical resolvent bounds when the spectral parameter is bounded and avoids a rapidly shrinking neighborhood of the spectrum.
If is diagonalizable, then for any there exist sufficiently small and sufficiently large where, if , ,
and , we have the resolvent bound
If is not assumed to be diagonalizable, then for any there exist sufficiently small and sufficiently large where, if , ,
and , we have the resolvent bound
We also have in  a unitary equivalence between and a weighted space of entire functions, defined in (1.16), which reduces the symbol to a normal form ; we review this in Section 2.2. In that weighted space, we have a simple characterization of the spectral projections for as truncations of the Taylor series.
Let be quadratic and partially elliptic with trivial singular space as in (1.5) and (1.6). Let be as defined in (1.12). As described in Proposition 2.2, the operator acting on is unitarily equivalent to acting on for some real-valued, quadratic, and strictly convex. Using the notation (1.14), write
The motivation behind establishing Theorem 1.2 is to provide information about the spectral projections, particularly the operator norms thereof. The approach of using dual bases for eigenvectors was used in  in finding exact rates of exponential growth for the operators described in Examples 2.6 and 3.6; we follow a similar approach here. The most tractable projections seem to be for eigenvalues with multiplicity 1, meaning the expansion in (1.17) consists of a single term. Note that this is true for every simultaneously if and only if the eigenvalues of which lie in the upper half-plane are rationally independent, which is a generic condition.
As explained above in (1.15), there is no reason to describe the norms of spectral projections semiclassically; we therefore state the result with . We furthermore see in Proposition 4.1 that the set of which may be obtained from Proposition 2.2 is exactly the set of strictly convex real-valued quadratic forms . We therefore treat such a as the object of study in the following theorem.
Let be strictly convex, real-valued, and quadratic. Write
Then there exists another quadratic strictly convex weight and a constant for which, for all , we have the formula
This result has three simple corollaries, which we formally state in Section 1.4. First, we have exponential upper bounds for the spectral projections of any elliptic or partially elliptic operator. Second, we have a complete asymptotic expansion for spectral projections in dimension one, where eigenvalues are automatically simple. Finally, we have a formula for the rate of exponential growth, regardless of dimension, in the generic situation when eigenvalues are simple.
It is useful for analysis of to have some orthogonal decomposition of into -invariant subspaces. That collections of Hermite functions of fixed degree form such a decomposition for Kramers-Fokker-Planck operators with quadratic potential was known since , as described in Section 5.5 of . We explore one such operator in Example 2.7, and we have the same decomposition for an operator whose Hamilton map has Jordan blocks in Example 2.8.
The question of orthogonal spectral projections for partially elliptic operators has been raised in the recent work , which focuses on semigroup bounds for such operators. Working under the assumptions that the ground state of matches that of and that the operator is totally real, the authors of  show strong similarity, on the level of semigroups, between the behavior of the spectral projection for and and the behavior of the orthogonal projection onto the span of the corresponding eigenfunction.
Inspired by this work, we observe that the analysis here beginning at Theorem 1.2 and leading towards Theorem 1.3 puts us in a position to describe necessary and sufficient conditions on for this projection to be orthogonal.
Let be quadratic and partially elliptic with trivial singular space as in (1.5) and (1.6). Recall the definitions of in (1.10) and in (1.12). Let be the spectral projection for and , as in (1.14). Then the following are equivalent:
the ground states of the operator and the adjoint match,
the stable manifolds associated with are conjugate, ; and
the projection is orthogonal on .
A further decomposition immediately follows if any of these conditions hold. Studying the unitarily equivalent operator acting on , we have that the spaces of polynomials homogeneous of fixed degree,
are orthogonal -invariant subspaces of which together have dense span. We also have that
Some illustrations using this decomposition may be found in Section 2.6.
1.4. Corollaries on the growth of spectral projections
First, we have an exponential upper bound for spectral projections for the quadratic operators we have been considering. We note that, following Remark 3.7, we do not expect this bound to be sharp in dimension in general.
In (spatial) dimension 1, we have a complete asymptotic expansion for spectral projections as the size of the eigenvalue becomes large.
Let be quadratic and partially elliptic with trivial singular space as in (1.5) and (1.6). By (1.9), there exists only one (algebraically simple) eigenvalue of with positive imaginary part; call this eigenvalue . Let
Using (1.14), write
as , for some a sequence of real numbers depending only on . We furthermore compute that
In the case of higher dimensions, the maximization problem leading to Corollary 1.7 is much more difficult. In the generic case of simple eigenvalues, we are nonetheless able to identify the rate of exponential growth for spectral projections along rays for fixed as .
While this provides significant information on the exponential growth of spectral projections for a broad class of non-normal quadratic operators, the author feels that this result in higher dimensions is rather preliminary and hopes to return to the subject in later work.
Let be strictly convex, real-valued, and quadratic. Write
Let be the dual weight as in Theorem 1.3.
Consider normalized so that . For those for which , we have the following exponential rate of growth in the limit :
As with multi-indices, we define .
Furthermore, consider quadratic and partially elliptic with trivial singular space as in (1.5) and (1.6), with defined in (1.12). Due to the unitary equivalence in Proposition 2.2 with provided therein, the same rate of growth holds for the norm of the classical () spectral projections
so long as we assume that the eigenvalue is simple.
1.5. Plan of the paper
Section 2 is devoted to proving Theorem 1.1 and recapitulating the necessary machinery used in . Also included are examples in Section 2.3 and illustrations of partial ellipticity in Section 2.6. Section 3 contains the proof of Theorem 1.2 as well as an elementary exponential upper bound for spectral projections which is related to the work . Section 4 focuses on the properties of dual bases for projection onto monomials in weighted spaces, and it contains proofs of Theorems 1.3 and 1.4. Finally, Section 5 contains computations based on these results which prove Corollaries 1.6, 1.7, and 1.8 and numerical computations based on Corollary 1.8.
2. Resolvent bounds in the partially elliptic case
One may extend the upper bounds obtained in  for resolvents of elliptic quadratic operators to upper bounds for partially elliptic quadratic operators after two steps: duplicating the reduction to normal form and finding some replacement for an elliptic estimate. The former can be done after demonstrating that the stable (linear Lagrangian) manifolds defined in (1.10) are positive and negative Lagrangian planes as defined in (1.11), which follows more or less directly from reasoning in . A subelliptic estimate, sufficient to establish Theorem 1.1, may be deduced from a remarkable recent result of Pravda-Starov .
In this section, we will assume that our quadratic symbol
We begin by proving sign definiteness of . Afterwards, we recall the reduction to normal form in  and remark on some additional information which may be derived from this reduction. Following this, we present three examples which will be used throughout the rest of the paper. Next, we prove the weak elliptic estimate for high-energy functions. Finally, recalling the low-energy finite dimensional analysis of , we are able to prove Theorem 1.1.
Afterwards, in Section 2.6, we see some evidence that the elliptic estimate in Proposition 2.10 may not give a sharp rate of growth in Theorem 1.1. However, the phenomenon of subellipticity formalized in  appears to be sharp, presenting a genuine obstacle in adapting the standard ellipticity argument found in Proposition 2.10.
2.1. Sign definiteness of
We also recall that, when obeys (1.5) and (1.6), there exists and a continuous family of complex linear canonical transformations acting on beginning with the identity, , and positive constants for which
enjoy the relation
It immediately follows that is a continuous family of Lagrangian planes. When , positivity of and negativity of follow from ellipticity of . To apply a deformation argument in , we wish to show that .
We know from Lemma 3.7 of  that implies that and are orthogonal with respect to , and therefore that are Lagrangian planes. (This may also be seen by applying to .)
Because generalized eigenspaces of an operator are invariant under that operator, we see that implies that . Since is Lagrangian, we see that
If we assume furthermore that , we have that and so as well. But then
By induction we therefore see that, whenever , we have that
We have already seen that contains , and so we conclude that, whenever , we have .
We may then appeal to the deformation argument following Lemma 3.8 in , which shows that if is a continuous family of Lagrangian planes for which , then all the are positive so long as one is. Since is positive for , we know that is positive. The same reasoning provides that is a negative Lagrangian plane, completing the proof. ∎
2.2. Review of reduction to normal form
Having established sign definiteness of , a reduction to normal form may then proceed exactly as in Section 2 of . We state the result as a proposition, following Proposition 2.1 in that work, and review the proof to record some minor details. We then make some minor remarks providing further information which will be used in the sequel. The relevant symbol classes are
However, as mentioned in Section 1.2, in this work we only require the use of symbols which are polynomials in .
for block-diagonal with each block being a Jordan one. Furthermore, the eigenvalues of are precisely those of in the upper half-plane. Associated with the transformation are a real-valued quadratic strictly convex weight function and a unitary operator
quantizing in that
We repeat the proof of Proposition 2.1 in  solely to make certain small details and minor changes of notation explicit. There are three pieces in the reduction to normal form: quantizing a real canonical transformation straightening , an FBI-Bargmann transform reducing to a polynomial simultaneously homogeneous of degree 1 in and of degree 1 in , and a change of variables reducing the matrix in the resulting symbol to Jordan normal form.
That is a negative Lagrangian plane is equivalent to having
with the last in the sense of positive definite matrices. A real linear canonical transformation such as
gives . We have that may be quantized by a standard unitary operator on , which reduces to accordingly.
Since is real canonical, remains positive, and so for some symmetric with positive definite imaginary part. Straightening
while simultaneously straightening
is accomplished by an FBI-Bargmann transform
This FBI-Bargmann transform quantizes the canonical transformation, in block matrix form,
We rearrange as follows:
We will use the expression
We may see that is strictly convex through the following useful computation, recalling that is symmetric:
We then see through a change of variables that positive definiteness of is equivalent to positive definiteness of . We therefore have that for all . Then, by the Cauchy-Schwarz inequality, when we have
establishing strict convexity of .
The canonical transformation (2.5) relates symbols with FBI-side symbols , with
where derivatives here are holomorphic. After conjugation with the FBI-Bargmann transform (2.4), we have reduced to , where is not necessarily in Jordan normal form.
Finally, for some invertible chosen so that is in Jordan normal form, we use a final linear change of variables
quantizing the canonical transformation
The resulting weight is , which is strictly convex since is.
We note that a real-valued quadratic form is uniquely determined by the two matrices and , since in this case
(See Section 4.1 for more details.) We therefore only need to record that
As in , we note that is an isomorphism
and , we see that acting on and acting on are similar linear operators and therefore isospectral. By the definition (1.10) of , we then have
We furthermore remark that the change of variables (2.10) is a degree-preserving isomorphism on polynomials. For this reason, it will sometimes be simpler to work on instead of . ∎
From Section 4 of  we record the specific formula
As usual, the are the eigenvalues of for which . We furthermore remark that it is clear from the fact that is in Jordan normal form that when .
In order to see how complex Gaussians