Abstract
We study propagation of phase space singularities for the initial value Cauchy problem for a class of Schrödinger equations. The Hamiltonian is the Weyl quantization of a quadratic form whose real part is non-negative. The equations are studied in the framework of projective Gelfand–Shilov spaces and their distribution duals. The corresponding notion of singularities is called the Gelfand–Shilov wave front set and means the lack of exponential decay in open cones in phase space. Our main result shows that the propagation is determined by the singular space of the quadratic form, just as in the framework of the Schwartz space, where the notion of singularity is the Gabor wave front set.
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1 Introduction
The goal of this paper is to study propagation of singularities for the initial value Cauchy problem for a Schrödinger type equation
where \(u_0\) is a Gelfand–Shilov distribution on \(\mathbf R^{d}\), \(q=q(x,\xi )\) is a quadratic form on the phase space \((x,\xi ) \in T^* \mathbf R^{d}\) with \(\mathrm{Re} \, q \geqslant 0\), and \(q^w(x,D)\) is a pseudodifferential operator in the Weyl quantization. This family of equations comprises the free Schrödinger equation where \(q(x,\xi )= i |\xi |^2\), the harmonic oscillator where \(q(x,\xi )= i (|x|^2 + |\xi |^2)\) and the heat equation where \(q(x,\xi )= |\xi |^2\).
The problem has been studied in the space of tempered distributions [22, 34] where the natural notion of singularity is the Gabor wave front set. This concept of singularity is defined as the conical subset of the phase space \(T^* \mathbf R^{d} {\setminus } \{0\}\) in which the short-time Fourier transform does not have rapid (superpolynomial) decay. The Gabor wave front set of a tempered distribution is empty exactly when the distribution is a Schwartz function so it measures deviation from regularity in the sense of both smoothness and decay at infinity comprehensively.
In this work we study propagation of singularities for the Eq. (1.1) in the functional framework of the Gelfand–Shilov test function spaces and their dual distribution spaces. More precisely we use the projective limit (Beurling type) Gelfand–Shilov space \(\Sigma _s(\mathbf R^{d})\) for \(s>1/2\) that consists of smooth functions satisfying
This means that \(\Sigma _s(\mathbf R^{d})\) is smaller than the Schwartz space, and hence its dual \(\Sigma _s'(\mathbf R^{d})\) is a space of distributions that contains the tempered distributions.
The natural concept of phase space singularities in the realm of Gelfand–Shilov spaces is the s-Gelfand–Shilov wave front set. The idea was introduced by Hörmander [16] under the name analytic wave front set for tempered distributions, and further developed by Cappiello and Schulz [2] and Cordero et al. [5] for Gelfand–Shilov distributions. These authors had an approach based on the inductive limit Gelfand–Shilov spaces as opposed to our concept that is based on the projective limit spaces.
A concept similar to the s-Gelfand–Shilov wave front set has been studied by Mizuhara [18]. This is the homogeneous wave front set of Gevrey order \(s>1\). It is included in the s-Gelfand–Shilov wave front set. Propagation results for the homogeneous Gevrey wave front set are proved in [18] for Schrödinger equations, albeit of a different type than ours.
In this paper the s-Gelfand–Shilov wave front set \(WF^s(u) \subseteq T^* \mathbf R^{d} {\setminus } \{0\}\) of \(u \in \Sigma _s'(\mathbf R^{d})\) for \(s>1/2\) is defined as follows: \(WF^s(u)\) is the complement in \(T^* \mathbf R^{d} {\setminus } \{0\}\) of the set of \(z_0 \in T^* \mathbf R^{d} {\setminus } \{0\}\) such that there exists an open conic set \(\Gamma \subseteq T^* \mathbf R^{d} {\setminus } \{ 0 \}\) containing \(z_0\), and
Here \(V_\varphi u\) denotes the short-time Fourier transform defined by \(\varphi \in \Sigma _s (\mathbf R^{d}) {\setminus } \{0\}\). Thus the short-time Fourier transform decays like \(e^{-A |z|^{1/s}}\) for any \(A>0\) in an open cone around \(z_0\). Note that this means that the decay can be close to but not quite like a Gaussian \(e^{-A |z|^2}\), due to our assumption \(s>1/2\).
For a tempered distribution, the s-Gelfand–Shilov wave front set contains the Gabor wave front set, and thus gives an enlarged notion of singularity.
Our main result on propagation of singularities for Schrödinger type equations goes as follows, where \(e^{-t q^w(x,D)}\) denotes the solution operator (propagator) of the Eq. (1.1). Let q be the quadratic form on \(T^*\mathbf R^{d}\) defined by \(q(x,\xi ) = \langle (x,\xi ), Q(x,\xi ) \rangle \) and a symmetric matrix \(Q \in \mathbf C^{2d \times 2d}\), \(\mathrm{Re} \, Q \geqslant 0\), \(F=\mathcal {J}Q\) where
and \(s > 1/2\). Then for \(u_0 \in \Sigma _s'(\mathbf R^{d})\)
where S is the singular space
of the quadratic form q. This result is verbatim the same as [22, Theorem 5.2] when \(u_0\) is restricted to be a tempered distribution and when the s-Gelfand–Shilov wave front set is replaced by the Gabor wave front set (cf. [34, Corollary 4.6]). Thus it gives a new manifestation of the importance of the singular space for propagation of phase space singularities for the considered class of equations of Schrödinger type.
The singular space has attracted much attention recently and occurs in several works on spectral and hypoelliptic properties of non-elliptic quadratic operators [12–14, 23, 24, 32, 33].
The paper is organized as follows. Section 2 contains notations and background material on Gelfand–Shilov spaces, pseudodifferential operators, the short-time Fourier transform and the Gabor wave front set. Section 3 gives a comprehensive discussion on three alternative families of seminorms for the projective Gelfand–Shilov spaces.
In Sect. 4 we define the s-Gelfand–Shilov wave front set and deduce some properties: independence of the window function, symplectic invariance, behavior under tensor product and composition with surjective matrices, and microlocality with respect to pseudodifferential operators with certain symbols. In this process we show the continuity of metaplectic operators on \(\Sigma _s(\mathbf R^{d})\) and on \(\Sigma _s'(\mathbf R^{d})\).
Section 5 gives a brief discussion on the solution operator to the Eq. (1.1), that is formulated for \(u_0 \in L^2(\mathbf R^{d})\) by means of semigroup theory. Exact propagation results are given for the case \(\mathrm{Re} \, Q = 0\). Section 6 treats propagation of the s-Gelfand–Shilov wave front set for a class of linear operators, continuous on \(\Sigma _s(\mathbf R^{d})\) and uniquely extendible to continuous operators on \(\Sigma _s'(\mathbf R^{d})\).
In Sect. 7 we discuss shortly Hörmander’s oscillatory integrals with quadratic phase function, and we prove the inclusion of the s-Gelfand–Shilov wave front set of such an oscillatory integral in the intersection of its corresponding positive Lagrangian in \(T^* \mathbf C^{d}\) with \(T^* \mathbf R^{d}\). Section 8 gives an account of Hörmander’s description of the Schwartz kernel of the propagator as an oscillatory integral and we show the continuity of the propagator on \(\Sigma _s(\mathbf R^{d})\) and on \(\Sigma _s'(\mathbf R^{d})\).
Finally in Sect. 9 we assemble the results of Sects. 6, 7 and 8 to prove our results on propagation of the s-Gelfand–Shilov wave front set for equations of the form (1.1).
2 Preliminaries
2.1 Notations and Basic Definitions
The gradient of a function f with respect to the variable \(x \in \mathbf R^{d}\) is denoted by \(f'_x\) and the mixed Hessian matrix \((\partial _{x_i} \partial _{y_j} f)_{i,j}\) with respect to \(x \in \mathbf R^{d}\) and \(y \in \mathbf R^{n}\) is denoted \(f_{x y}''\). The Fourier transform of \(f \in \mathscr {S}(\mathbf R^{d})\) (the Schwartz space) is normalized as
where \(\langle x, \xi \rangle \) denotes the inner product on \(\mathbf R^{d}\). The topological dual of \(\mathscr {S}(\mathbf R^{d})\) is the space of tempered distributions \(\mathscr {S}'(\mathbf R^{d})\). As conventional \(D_j = -i \partial _j\) for \(1 \leqslant j \leqslant d\).
We will make frequent use of the inequality
where
Since this inequality will be used only for \(s>1/2\) we may use the cruder estimate \(C_s=2\), which leads to the inequalities
The Japanese bracket is \(\langle x\rangle = (1+|x|^2)^{1/2}\). For a positive measurable weight function \(\omega \) defined on \(\mathbf R^{d}\), the Banach space \(L_{\omega }^1(\mathbf R^{d})\) is endowed with the norm \(\Vert f \Vert _{L_{\omega }^1}= \Vert f \omega \Vert _{L^1}\). The unit sphere in \(\mathbf R^{d}\) is denoted \(S_{d-1} = \{ x \in \mathbf R^{d}: \ |x|=1 \}\). For a matrix \(A \in \mathbf R^{d \times d}\), \(A \geqslant 0\) means that A is positive semidefinite, and \(A^t\) is the transpose. If A is invertible then \(A^{-t}\) denotes the inverse transpose. In estimates the notation \(f (x) \lesssim g (x)\) understands that \(f(x) \leqslant C g(x)\) holds for some constant \(C>0\) that is uniform for all x in the domain of f and g. If \(f(x) \lesssim g(x) \lesssim f(x)\) then we write \(f(x) \asymp g(x)\).
We denote the translation operator by \(T_x f(y)=f(y-x)\), the modulation operator by \(M_\xi f(y)=e^{i \langle y, \xi \rangle } f(y)\), \(x,y,\xi \in \mathbf R^{d}\), and the phase space translation operator by \(\Pi (z) = M_\xi T_x\), \(z=(x,\xi ) \in \mathbf R^{2d}\).
2.2 Gelfand–Shilov Spaces
Let \(h,s,t >0\). The space \(\mathcal S_{t,h}^s(\mathbf R^{d})\) is defined as all \(f\in C^\infty (\mathbf R^{d})\) such that
is finite. The supremum refers to all \(\alpha ,\beta \in \mathbf N^d\) and \(x\in \mathbf R^{d}\). We set \(\mathcal S_{s,h}=\mathcal S_{s,h}^s\).
The Banach space \(\mathcal S_{t,h}^s\) increases with h, s and t, and the embedding \(\mathcal S_{t,h}^s\subseteq \mathscr {S}\) holds for all \(h,s,t >0\). If \(s,t>1/2\), or \(s=t =1/2\) and h is sufficiently large, then \(\mathcal S_{t,h}^s\) contains all finite linear combinations of Hermite functions.
The Gelfand–Shilov spaces \(\mathcal S_{t}^s(\mathbf R^{d})\) and \(\Sigma _{t}^s(\mathbf R^{d})\) are the inductive and projective limits respectively of \(\mathcal S_{t,h}^s(\mathbf R^{d})\) with respect to \(h>0\). This means on the one hand
On the other hand it means that the topology for \(\mathcal S_{t}^s(\mathbf R^{d})\) is the strongest topology such that the inclusion \(\mathcal S_{t,h}^s(\mathbf R^{d}) \subseteq \mathcal S_{t}^s(\mathbf R^{d})\) is continuous for each \(h>0\), and the topology for \(\Sigma _{t}^s(\mathbf R^{d})\) is the weakest topology such that the inclusion \(\Sigma _{t}^s(\mathbf R^{d}) \subseteq \mathcal S_{t,h}^s(\mathbf R^{d})\) is continuous for each \(h>0\).
The space \(\Sigma _t^s(\mathbf R^{d})\) is a Fréchet space with seminorms \(\Vert \cdot \Vert _{\mathcal S_{s,h}^t}\), \(h>0\). The Gelfand–Shilov spaces are invariant under translation, modulation, dilation, linear coordinate transformations and tensor products. It holds \(\Sigma _t^s(\mathbf R^{d})\ne \{ 0\}\) if and only if \(s,t>0\), \(s+t \geqslant 1\) and \((s,t) \ne (1/2,1/2)\). We set \(\mathcal S_{s}=\mathcal S_{s}^s\) and \(\Sigma _{s}=\Sigma _{s}^s\). Then \(\mathcal S_s(\mathbf R^{d})\) is zero when \(s<1/2\), and \(\Sigma _s(\mathbf R^{d})\) is zero when \(s \le 1/2\). From now on we assume that \(s>1/2\) when considering \(\Sigma _s(\mathbf R^{d})\).
The Gelfand–Shilov distribution spaces \((\mathcal S_{t}^s)'(\mathbf R^{d})\) and \((\Sigma _{t}^s)'(\mathbf R^{d})\) are the projective and inductive limits respectively of \((\mathcal S_{t,h}^s)'(\mathbf R^{d})\). This implies that
The space \((\mathcal S_{t}^s)'(\mathbf R^{d})\) is the topological dual of \(\mathcal S_{t}^s(\mathbf R^{d})\), and if \(s>1/2\) then \((\Sigma _{t}^s)'(\mathbf R^{d})\) is the topological dual of \(\Sigma _{t}^s(\mathbf R^{d})\) [8].
In this paper we work with the spaces \(\Sigma _s(\mathbf R^{d})\) and \(\Sigma _s'(\mathbf R^{d}) = (\Sigma _s^s)'(\mathbf R^{d})\) for \(s>1/2\). These spaces are embedded with respect to the Schwartz space and the tempered distributions as
For \(s>1/2\) the (partial) Fourier transform extends uniquely to homeomorphisms on \(\mathscr {S}'(\mathbf R^{d})\), \(\mathcal S_s'(\mathbf R^{d})\) and \(\Sigma _s'(\mathbf R^{d})\), and restricts to homeomorphisms on \(\mathscr {S}(\mathbf R^{d})\), \(\mathcal S_s(\mathbf R^{d})\) and \(\Sigma _s(\mathbf R^{d})\).
2.3 Pseudodifferential Operators and the Gabor Wave Front Set
Let \(s>1/2\). Given a window function \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } \{ 0 \}\), the short-time Fourier transform (STFT) [9] of \(u \in \Sigma _s'(\mathbf R^{d})\) is defined by
where \((\cdot ,\cdot )\) denotes the conjugate linear action of \(\Sigma _s'\) on \(\Sigma _s\), consistent with the inner product \((\cdot ,\cdot )_{L^2}\) which is conjugate linear in the second argument. The function \(\mathbf R^{2d} \ni z \rightarrow V_\varphi u(z)\) is smooth.
Let \(\varphi ,\psi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\). By [29, Theorem 2.5] we have
and by [9, Lemma 11.3.3] we have
where \(V_\psi \varphi \in \Sigma _s(\mathbf R^{2d})\).
If \(\varphi \in \Sigma _s (\mathbf R^{d})\) and \(\Vert \varphi \Vert _{L^2}=1\), the STFT inversion formula [9, Corollary 11.2.7] reads
The Weyl quantization of pseudodifferential operators (cf. [7, 15, 27]) is the map from symbols \(a \in \mathscr {S}(\mathbf R^{2d})\) to operators acting on \(f \in \mathscr {S}(\mathbf R^{d})\) defined by
The conditions on a and f can be modified and relaxed in various ways. The Weyl quantization can be formulated in the framework of Gelfand–Shilov spaces [1]. For certain symbols the operator \(a^w(x,D)\) acts continuously on \(\Sigma _s(\mathbf R^{d})\) when \(s>1/2\).
If \(a \in \Sigma _s'(\mathbf R^{2d})\) the Weyl quantization extends a continuous operator \(\Sigma _s(\mathbf R^{d}) \rightarrow \Sigma _s'(\mathbf R^{d})\) that satisfies
where the cross-Wigner distribution is defined as
We have \(W(g,f) \in \Sigma _s(\mathbf R^{2d})\) when \(f,g \in \Sigma _s(\mathbf R^{d})\).
We need the following symbol classes for pseudodifferential operators that act on \(\mathscr {S}(\mathbf R^{d})\) in order to define the Gabor wave front set and explain its properties.
Definition 2.1
[27] For \(m\in \mathbf R\) the Shubin symbol class \(G^m\) is the subspace of all \(a \in C^\infty (\mathbf R^{2d})\) such that for every \(\alpha ,\beta \in \mathbf N^{d}\)
Definition 2.2
[15] For \(m\in \mathbf R\), \(0 \leqslant \rho \leqslant 1\), \(0 \leqslant \delta < 1\), the Hörmander symbol class \(S_{\rho ,\delta }^m\) is the subspace of all \(a \in C^\infty (\mathbf R^{2d})\) such that for every \(\alpha ,\beta \in \mathbf N^{d}\)
Both \(G^m\) and \(S_{\rho ,\delta }^m\) are Fréchet spaces with respect to their naturally defined seminorms.
The following definition involves conic sets in the phase space \(T^* \mathbf R^{d} {\setminus } 0 \simeq \mathbf R^{2d} {\setminus } 0\). A set is conic if it is invariant under multiplication with positive reals. Note the difference to the frequency-conic sets that are used in the definition of the (classical) \(C^\infty \) wave front set [15].
Definition 2.3
Given \(a \in G^m\), a point \(z_0 \in T^* \mathbf R^{d} {\setminus } 0\) is called non-characteristic for a provided there exist \(A,\varepsilon >0\) and an open conic set \(\Gamma \subseteq T^* \mathbf R^{d} {\setminus } 0\) such that \(z_0 \in \Gamma \) and
The Gabor wave front set is defined as follows where \({\text {char}}(a)\) is the complement in \(T^* \mathbf R^{d} {\setminus } 0\) of the set of non-characteristic points for a.
Definition 2.4
[16] If \(u \in \mathscr {S}'(\mathbf R^{d})\) then the Gabor wave front set WF(u) is the set of all \(z \in T^*\mathbf R^{d} {\setminus } 0\) such that \(a \in G^m\) for some \(m \in \mathbf R\) and \(a^w(x,D) u \in \mathscr {S}\) implies \(z \in {\text {char}}(a)\).
According to [16, Proposition 6.8] and [25, Corollary 4.3], the Gabor wave front set can be characterized microlocally by means of the STFT as follows. If \(u \in \mathscr {S}'(\mathbf R^{d})\) and \(\varphi \in \mathscr {S}(\mathbf R^{d}) {\setminus } 0\) then \(z_0 \in T^*\mathbf R^{d} {\setminus } 0\) satisfies \(z_0 \notin WF(u)\) if and only if there exists an open conic set \(\Gamma _{z_0} \subseteq T^*\mathbf R^{d} {\setminus } 0\) containing \(z_0\) such that
The most important properties of the Gabor wave front set include the following facts. Here the microsupport \(\mu {\text {supp}}(a)\) of \(a \in G^m\) is defined as follows (cf. [26]). For \(z_0 \in T^* \mathbf R^{d} {\setminus } 0\) we have \(z_0 \notin \mu {\text {supp}}(a)\) if there exists an open cone \(\Gamma \subseteq T^* \mathbf R^{d} {\setminus } 0\) containing \(z_0\) such that
-
(1)
If \(u \in \mathscr {S}'(\mathbf R^{d})\) then \(WF(u) = \emptyset \) if and only if \(u \in \mathscr {S}(\mathbf R^{d})\) [16, Proposition 2.4].
-
(2)
If \(u \in \mathscr {S}'(\mathbf R^{d})\) and \(a \in G^m\) then
$$\begin{aligned} WF( a^w(x,D) u)&\subseteq WF(u) \cap \mu {\text {supp}}(a) \\&\subseteq WF( a^w(x,D) u) \ \bigcup \ {\text {char}}(a). \end{aligned}$$ -
(3)
If \(a \in S_{0,0}^0\) and \(u \in \mathscr {S}'(\mathbf R^{d})\) then by [25, Theorem 5.1]
$$\begin{aligned} WF(a^w(x,D) u) \subseteq WF(u). \end{aligned}$$(2.9)In particular \(WF(\Pi (z) u) = WF(u)\) for any \(z \in \mathbf R^{2d}\).
As three basic examples of the Gabor wave front set we mention (cf. [25, Examples 6.4–6.6])
and
The canonical symplectic form on \(T^* \mathbf R^{d}\) is
With the matrix (1.2) this can be expressed with the inner product on \(\mathbf R^{2d}\) as
To each symplectic matrix \(\chi \in {\text {Sp}}(d,\mathbf R)\) is associated an operator \(\mu (\chi )\) that is unitary on \(L^2(\mathbf R^{d})\), and determined up to a complex factor of modulus one, such that
(cf. [7, 15]). The operator \(\mu (\chi )\) is a homeomorphism on \(\mathscr {S}\) and on \(\mathscr {S}'\).
The mapping \({\text {Sp}}(d,\mathbf R) \ni \chi \rightarrow \mu (\chi )\) is called the metaplectic representation [7, 28]. It is in fact a representation of the so called 2-fold covering group of \({\text {Sp}}(d,\mathbf R)\), which is called the metaplectic group and denoted \({\text {Mp}}(d,\mathbf R)\). The metaplectic representation satisfies the homomorphism relation modulo a change of sign:
According to [16, Proposition 2.2] the Gabor wave front set is symplectically invariant as
The work [4] contains a study of the propagation of the Gabor wave front set for linear Schrödinger equations, and [19, 21] contain studies of the same question for semilinear Schrödinger-type equations.
3 Seminorms on Gelfand–Shilov Spaces
We need to work with several families of seminorms on \(\Sigma _s(\mathbf R^{d})\) for \(s>1/2\) apart from the seminorms defined by (2.3). The next result shows that there are three families of seminorms for \(\Sigma _s(\mathbf R^{d})\) that are each equivalent to the family of seminorms \(\{\Vert f \Vert _{\mathcal S_{s,h}}, \ h > 0 \}\) defined by (2.3). The three families of seminorms are firstly \(\{\Vert f \Vert _A', \ \Vert \widehat{f} \Vert _B', \ A,B>0 \}\) where
secondly \(\{| f |_A, \ A>0 \}\) where
and thirdly \(\{\Vert f \Vert _A'', \ A>0 \}\) where
for \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) fixed. It will turn out that the choice of \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) in the definition of the seminorms \(\{\Vert f \Vert _A'', \ A>0 \}\) is arbitrary (see the proof of Proposition 3.1).
The essential arguments in the proof of the following proposition can be found in several places, e.g. [3, 8, 11, 20, 30]. Nevertheless we prefer to give a detailed account since it is a cornerstone for our results, and in order to give a self-contained narrative.
Proposition 3.1
Let \(s>1/2\). Then
and
Likewise
and
Finally
and
Proof
We start with (3.4). Let \(f \in \Sigma _s(\mathbf R^{d})\). From (2.3) we have for any \(h>0\)
This gives for any \(n \in \mathbf N\) and any \(\beta \in \mathbf N^{d}\)
which in turn gives with \(\beta =0\) and \(A=2^{-1}s (d^{1/2} h)^{-1/s}\)
For any \(A>0\) we thus have
if \(h_1=(s/(2A))^{s} d^{-1/2}\).
Since the Fourier transform is continuous on \(\Sigma _s(\mathbf R^{d})\) we get from (3.11) for any \(B>0\)
for some \(h_0, h_2>0\). Addition of (3.11) and (3.12) proves (3.4) for \(h=\min (h_1,h_2)\).
The second and longer argument of this proof serves to prove (3.5). The argument follows closely that of the proof of [20, Theorem 6.1.6]. For completeness’ sake we give the full details.
First we deduce two estimates that are needed. From (3.4) it follows that \(\Vert f \Vert _A' < \infty \) and \(\Vert \widehat{f} \Vert _B' < \infty \) for any \(A,B>0\) when \(f \in \Sigma _s(\mathbf R^{d})\). Thus for any \(A>0\) we have
which gives the estimate
Using \(|\alpha |! \leqslant d^{|\alpha |} \alpha !\) (cf. [20, Eq. (0.3.3)]) this gives in turn
Finally we take the \(L^2\) norm and estimate for an integer \(k>d/4\) with \(\varepsilon =4k-d>0\):
using \((\alpha +\gamma )! \leqslant 2^{|\alpha +\gamma |} \alpha ! \gamma !\) (cf. [20]) and considering k a fixed parameter.
From (3.13), \(\Vert \widehat{f} \Vert _B' < \infty \) for any \(B>0\), and Parseval’s theorem we obtain
Since \(\Vert f \Vert _{A}' \leqslant \Vert f \Vert _{A+A_0}'\) when \(A_0\geqslant 0\) and \(A>0\) we may use \(B=A\) when we now set out to prove (3.5). It suffices to assume \(h \leqslant 1\). We have for \(\alpha ,\beta \in \mathbf N^{d}\) arbitrary and \(f \in \Sigma _s(\mathbf R^{d})\), using the Cauchy–Schwarz inequality, Parseval’s theorem and the Leibniz rule
In an intermediate step we rewrite the expression for the \(L^2\) norm squared using integration by parts and estimate it as
Setting \(h = 2^{2s+5/2} (2d s/A)^s\) and using (3.13), (3.14) and \(\kappa ! = \kappa !^{2s -\delta }\) where \(\delta =2s-1>0\), we get
since we have assumed \(h \leqslant 1\), since \(|\mu | \leqslant 2 k\) which is a fixed constant, and since
We insert this into (3.15) which gives, using \(\mu ! \leqslant \mu !^{2s}\) and
This finally proves (3.5), since for any \(h>0\) we may take \(A = 2^{3+5/2s} h^{-1/s} d s\).
Next we show (3.6) and (3.7). We start with (3.6).
From (3.10) it follows that we have for \(A>0\)
provided \(A \leqslant 2^{-1}s (d^{1/2} h)^{-1/s}\). Thus
which gives
provided \(h \leqslant \min (A^{-1}, (s/(2A))^s d^{-1/2})\). Hence we have proved (3.6).
We continue with the proof of (3.7). From (3.6) we know that \(|f|_A < \infty \) for any \(A>0\) when \(f \in \Sigma _s(\mathbf R^{d})\). Hence for any \(A>0\), \(\beta \in \mathbf N^{d}\) and \(x \in \mathbf R^{d}\)
which gives
and thus
From this it follows that
for any \(h>0\) provided \(A \geqslant \max (h^{-1}, s d h^{-1/s})\). This proves (3.7).
It remains to show (3.8) and (3.9). We start with (3.8).
Let \(A>0\) and \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\). We have for \(f \in \Sigma _s(\mathbf R^{d})\)
using (2.2) and (3.4). From this estimate and \(|V_\varphi f (x,\xi )| = (2 \pi )^{-d} |V_{\widehat{\varphi }} \widehat{f} (\xi ,-x)|\) we also obtain
With the aid of (2.1) we may conclude
which gives
Combining with (3.4) we have proved (3.8).
We now show (3.9). For that purpose we use the strong version of the STFT inversion formula (2.6) and its Fourier transform, that is
where \(f \in \Sigma _s(\mathbf R^{d})\) and \(\varphi \in \Sigma _s(\mathbf R^{d})\) satisfies \(\Vert \varphi \Vert _{L^2} = 1\). From (3.16) we obtain for any \(A>0\)
which gives \(\Vert f \Vert _A' \lesssim \Vert f \Vert _{3A}''\).
From (3.17) we obtain for any \(A>0\)
which gives \(\Vert \widehat{f} \Vert _A' \lesssim \Vert f \Vert _{3A}''\). Thus \(\Vert f \Vert _A' + \Vert \widehat{f} \Vert _A' \lesssim \Vert f \Vert _{3A}''\) so combining with (3.5) we have proved (3.9).
Finally we show that the seminorms \(\{\Vert f \Vert _{A}'', \ A>0\}\) are equivalent to the same family of seminorms when the window function \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) is replaced by another function \(\psi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\). From (2.5) we obtain for \(A>0\)
using (3.8) applied to \(\varphi \in \Sigma _s(\mathbf R^{d})\), i.e. \(\Vert \varphi \Vert _{C}'' < \infty \) for all \(C>0\). This proves the claim that the window function \(\psi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) gives seminorms equivalent to those of \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\). \(\square \)
4 Definition and Properties of the s-Gelfand–Shilov Wave Front Set
For \(s>1/2\) and \(u \in \Sigma _s' (\mathbf R^{d})\) we define the s-Gelfand–Shilov wave front set \(WF^s (u)\) as follows, modifying slightly Cappiello’s and Schulz’s [2, Definition 3.1]. This concept is a coarsening of the Gabor wave front set WF(u) in the sense that \(WF(u) \subseteq WF^s(u)\) for all \(s > 1/2\) and all \(u \in \mathscr {S}'(\mathbf R^{d})\).
Definition 4.1
Let \(s > 1/2\), \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) and \(u \in \Sigma _s'(\mathbf R^{d})\). Then \(z_0 \in T^*\mathbf R^{d} {\setminus } 0\) satisfies \(z_0 \notin WF^s (u)\) if there exists an open conic set \(\Gamma _{z_0} \subseteq T^*\mathbf R^{d} {\setminus } 0\) containing \(z_0\) such that for every \(A>0\)
It follows that \(WF^s (u) = \emptyset \) if and only if \(u \in \Sigma _s'(\mathbf R^{d})\) satisfies
for any \(A>0\). By Proposition 3.1 (cf. [29, Proposition 2.4]) this is equivalent to \(u \in \Sigma _s(\mathbf R^{d})\).
The following lemma is needed in the proof of the independence of \(WF^s (u)\) of the window function \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\).
Lemma 4.2
Let \(s>1/2\) and let f be a measurable function on \(\mathbf R^{d}\) that satisfies
for some \(M \geqslant 0\). Suppose there exists a non-empty open conic set \(\Gamma \subseteq \mathbf R^{d} {\setminus } 0\) such that
for all \(A>0\). If
then for any open conic set \(\Gamma ' \subseteq \mathbf R^{d} {\setminus } 0\) such that \(\overline{\Gamma ' \cap S_{d-1}} \subseteq \Gamma \), we have
for all \(A>0\).
Proof
By (2.1) and the assumptions (4.1) and (4.3) we have
so it suffices to assume \(|x| \geqslant L\) for some large number \(L>0\).
Let \(\varepsilon >0\). We estimate and split the convolution integral as
Consider \(I_1\). Since \(\langle y\rangle \leqslant \varepsilon \langle x\rangle \) we have \(x-y \in \Gamma \) if \(x \in \Gamma '\), \(|x| \geqslant 1\), and \(\varepsilon >0\) is chosen sufficiently small. The assumptions (4.2), (4.3), and (2.2) give
for any \(A>0\). Next we estimate \(I_2\) using (4.1) and \(\langle y\rangle > \varepsilon \langle x\rangle \). The latter inequality implies that \(|y|^{1/s} \geqslant |x|^{1/s} \varepsilon ^{1/s}/2\) when \(|x| \geqslant L\) if \(L>0\) is sufficiently large. This gives for any \(A>0\)
again using (4.3). A combination of (4.5) and (4.6) proves (4.4) for \(A>0\) arbitrary. \(\square \)
Using Lemma 4.2 we show next that Definition 4.1 does not depend on the choice of the window function \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\).
Proposition 4.3
Suppose \(s>1/2\) and \(u \in \Sigma _s'(\mathbf R^{d})\). The definition of the s-Gelfand–Shilov wave front set \(WF^s(u)\) does not depend on the window function \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\).
Proof
Let \(\varphi ,\psi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\). By (2.4) we have for some \(M \geqslant 0\)
and by (2.5) we have
By Proposition 3.1 (cf. [29, Theorem 2.4]) we have
for any \(A>0\), and hence
From Lemma 4.2 we may now draw the following conclusion. If \(|V_\varphi u(z)|\) decays like \(e^{-A |z|^{1/s} }\) for any \(A>0\) in a conic set \(\Gamma \subseteq T^* \mathbf R^{d} {\setminus } 0\) containing \(z_0 \ne 0\) then we get decay like \(e^{-A |z|^{1/s} }\) for any \(A>0\) in a smaller cone containing \(z_0\) for \(|V_\psi u(z)|\). Hence, by symmetry, decay of order \(e^{-A |z|^{1/s} }\) for any \(A>0\) in an open cone around a point in \(T^* \mathbf R^{d} {\setminus } 0\) happens simultaneously for \(V_\varphi u\) and \(V_\psi u\). \(\square \)
The s-Gelfand–Shilov wave front set \(WF^s (u)\) decreases when the index s increases:
From \(WF(u) \subseteq WF^s(u)\) for \(u \in \mathscr {S}'(\mathbf R^{d})\) and (2.10) we have for any \(s > 1/2\)
To see the opposite inclusion we note that if \(x_0 \in \mathbf R^{d} {\setminus } 0\) and \(\xi _0 \in \mathbf R^{d}\) then \((x_0,\xi _0) \in \Gamma = \{(x,\xi ) \in T^* \mathbf R^{d} {\setminus } 0: |\xi | < C |x| \}\) for some \(C>0\), which is an open conic subset of \(T^* \mathbf R^{d} {\setminus } 0\).
Let \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) and let \(\varepsilon >0\). Since \(|V_\varphi \delta _0 (x, \xi )| = |\varphi (-x)|\) it follows from Proposition 3.1 that for any \(A>0\)
This shows that \((x_0,\xi _0) \notin WF^s(\delta _0)\), and proves
Hence
Next we show continuity of the metaplectic operators when they act on \(\Sigma _{s}(\mathbf R^{d})\) for \(s>1/2\). We note that continuity of metaplectic operators acting on \(\mathcal S_s (\mathbf R^{d})\) for \(s \geqslant 1\) is contained in [6, Proposition 3.5], and G. Tranquilli [31, Theorem 32] has shown continuity on \(\mathcal S_s (\mathbf R^{d})\) for \(s \geqslant 1/2\). Our proof is inspired by hers.
Proposition 4.4
If \(s>1/2\) and \(\chi \in {\text {Sp}}(d,\mathbf R)\) then the metaplectic operator \(\mu (\chi )\) acts continuously on \(\Sigma _{s}(\mathbf R^{d})\), and extends uniquely to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\).
Proof
By [7, Proposition 4.10] each matrix \(\chi \in {\text {Sp}}(d,\mathbf R)\) is a finite product of matrices of the form
for \(A \in {\text {GL}}(d,\mathbf R)\) and \(B \in \mathbf R^{d \times d}\) symmetric. To show that \(\mu (\chi )\) is continuous on \(\Sigma _{s}(\mathbf R^{d})\) it thus suffices to show that \(\mu (\chi )\) is continuous on \(\Sigma _{s}(\mathbf R^{d})\) when \(\chi \) has each of these three forms.
We have \(\mu (\mathcal {J}) = (2 \pi )^{-d/2} \mathscr {F}\), and \(\mu (\chi ) f(x) = |A|^{1/2} f(Ax)\) when \(A \in {\text {GL}}(d,\mathbf R)\) and
The Fourier transform and linear coordinate transformations are continuous operators on \(\Sigma _{s}(\mathbf R^{d})\). Therefore it remains to prove that \(\mu (\chi )\) is continuous on \(\Sigma _{s}(\mathbf R^{d})\) when \(B \in \mathbf R^{d \times d}\) is symmetric and
We have \(\mu (\chi )f (x) = e^{i \langle B x,x \rangle /2} f(x)\) when (4.9) holds (cf. (2.11) and [7]).
Due to the continuity of coordinate transformations on \(\Sigma _{s}(\mathbf R^{d})\), it suffices to consider diagonal matrices B with non-negative entries. By an induction argument applied to the seminorms (2.3) it further suffices to work in dimension \(d=1\) and prove continuity on \(\Sigma _{s}(\mathbf R)\) of the multiplication operator \(f \rightarrow g f\) for \(g(x) = e^{i x^2/2}\).
It may be confirmed by induction that for any \(k \in \mathbf N\) we have \(D^k g = p_k g\) where \(p_k\) is the polynomial of order k
Using \(k! \leqslant 2^k (k-2m)! (2m)!\) we can estimate \(|p_k(x)|\) as
By Leibniz’ rule we have for any \(A>0\) and any \(B \geqslant A\)
As an intermediate step we compute and estimate, using \(m! = m!^{2s -\varepsilon }\) where \(\varepsilon =2s-1>0\),
If \(B \geqslant sA^{1/s}\) we thus obtain
Hence we have shown that for any \(A>0\) we have
for \(B \geqslant \max (2 s(5A)^{1/s}, 10A)\). In view of Proposition 3.1 this shows that the multiplication operator \(f \rightarrow g f\) with \(g(x) = e^{i x^2/2}\) is continuous on \(\Sigma _{s}(\mathbf R)\).
Thus \(\mu (\chi )\) is continuous on \(\Sigma _s(\mathbf R^{d})\) when \(\chi \) has the form (4.9) with B symmetric. We may now conclude that \(\mu (\chi )\) is continuous on \(\Sigma _s(\mathbf R^{d})\) for all \(\chi \in {\text {Sp}}(d, \mathbf R)\).
Finally the unique extension to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\) follows from the facts that \(\mu (\chi )\) is unitary and \(\Sigma _s(\mathbf R^{d})\) is dense in \(\Sigma _s'(\mathbf R^{d})\). \(\square \)
The combination of Propositions 4.3 and 4.4 gives the symplectic invariance of the s-Gelfand–Shilov wave front set, as follows.
Corollary 4.5
If \(s>1/2\) then
Proof
By the proof of [34, Lemma 3.7] we have
for \(\varphi \in \Sigma _s(\mathbf R^{d})\), \(u \in \Sigma _s'(\mathbf R^{d})\), \(\chi \in {\text {Sp}}(d,\mathbf R)\) and \(z \in \mathbf R^{2d}\). By Proposition 4.4 \(\mu (\chi ) \varphi \in \Sigma _s(\mathbf R^{d})\) so the result follows immediately from Proposition 4.3. \(\square \)
Example 4.6
A combination of (4.8) and \(\mu (\mathcal {J}) = (2\pi )^{-d/2} \mathscr {F}\) gives
If \(B \in \mathbf R^{d \times d}\) is symmetric then \(\chi \) defined by (4.9) defines the metaplectic multiplication operator \(\mu (\chi ) = e^{i \langle Bx,x \rangle /2}\). Corollary 4.5 combined with (4.10) yields
Next we show a result on the s-Gelfand–Shilov wave front set of a tensor product. The corresponding result for the Gabor wave front set is [16, Proposition 2.8]. With obvious modification of the proof given below you get an alternative proof of the latter result. Here we use the notation \(x=(x',x'') \in \mathbf R^{m+n}\), \(x' \in \mathbf R^{m}\), \(x'' \in \mathbf R^{n}\).
Proposition 4.7
If \(s>1/2\), \(u \in \Sigma _s'(\mathbf R^{m})\), and \(v \in \Sigma _s'(\mathbf R^{n})\) then
Proof
Let \(\varphi \in \Sigma _s(\mathbf R^{m}) {\setminus } 0\) and \(\psi \in \Sigma _s(\mathbf R^{n}) {\setminus } 0\). Suppose \((x_0,\xi _0) \in T^* \mathbf R^{m+n} {\setminus } 0\) does not belong to the set on the right hand side. Then either \((x_0',\xi _0') \notin WF^s(u) \cup \{ 0 \}\) or \((x_0'',\xi _0'') \notin WF^s(v) \cup \{ 0 \}\). For reasons of symmetry we may assume \((x_0',\xi _0') \notin WF^s(u) \cup \{ 0 \}\).
Then \((x_0',\xi _0') \in \Gamma ' \subseteq T^* \mathbf R^{m} {\setminus } 0\) where \(\Gamma '\) is an open conic subset, and
for all \(A>0\). Define for \(C>0\) the open conic set
Then \((x_0,\xi _0) \in \Gamma \) for \(C>0\) sufficiently large since \((x_0',\xi _0') \ne 0\). By (2.4) we have for some \(M \geqslant 0\)
This gives for any \(A>0\)
It follows that \((x_0,\xi _0) \notin WF^s (u \otimes v)\). \(\square \)
We need in Sect. 7 the following result which is an adaptation of [16, Proposition 2.3] from the Gabor wave front set to the s-Gelfand–Shilov wave front set. Modified naturally the proof can be considered an alternative proof of the latter result.
Proposition 4.8
If \(s>1/2\), \(u \in \Sigma _s'(\mathbf R^{d}) {\setminus } 0\) and \(A \in \mathbf R^{d \times n}\) is a surjective matrix, then
Proof
Due to Corollary 4.5, and \(\mu (\chi ) f(x) = |B|^{1/2} f(Bx)\) when \(B \in {\text {GL}}(d,\mathbf R)\) and
it suffices to assume \(k := n - d > 0\) and \(A = ( I_d \ \ 0)\) where \(0 \in \mathbf R^{d \times k}\). We split variables as \(x = (x',x'') \in \mathbf R^{n}\) with \(x' \in \mathbf R^{d}\) and \(x'' \in \mathbf R^{k}\). We need to prove
where we use the notation \(0_n = 0\in \mathbf R^{n}\).
The inclusion
is a particular case of Proposition 4.7, combined with (4.10).
To prove the opposite inclusion, we first show
Let \(\varphi \in \Sigma _s(\mathbf R^{d}) {\setminus } 0\) satisfy \((u,\varphi ) \ne 0\) and let \(\psi \in \Sigma _s(\mathbf R^{k}) {\setminus } 0\) satisfy \(\widehat{\psi }(0) \ne 0\). If \(x'' \in \mathbf R^{k} {\setminus } 0\) then due to
we have for any \(t>0\)
Thus \(V_{\varphi \otimes \psi } u \otimes 1\) does not decay in any conical neighborhood of \((0,x'';0) \in T^* \mathbf R^{n}\), which proves (4.13).
To prove (4.12) it thus suffices to show the inclusion
Suppose \(0 \ne (x_0; \xi _0',0) \notin WF^s(u \otimes 1)\). If \((x_0',\xi _0') = 0\) then \((x_0',\xi _0') \notin WF^s(u)\), so we may assume \((x_0',\xi _0') \ne 0\). We have \((x_0; \xi _0',0) \in \Gamma \subseteq T^* \mathbf R^{n} {\setminus } 0\) where \(\Gamma \) is an open conic set such that
for all \(A>0\), cf. (4.14).
Define the open conic set
Then \((x_0',\xi _0') \in \Gamma '\) since \((x_0',\xi _0') \ne 0\). Let \(A>0\) be arbitrary. Define the functions
For some sequence \((x_n',\xi '_n)_{n \in \mathbf N} \subseteq \Gamma '\), where for each \(n \in \mathbf N\) there exists \(x_n'' \in \mathbf R^{k}\) such that \((x_n',x_n'',\xi _n',0) \in \Gamma \), we have
This means that \((x_0',\xi _0') \notin WF^s(u)\) which proves (4.15). \(\square \)
Let \(s>1/2\) and suppose \(a \in C^\infty (\mathbf R^{2d})\) satisfies the estimates
for all \(h>0\). According to [1, Theorem 3.10] \(a^w(x,D)\) is then a continuous operator on \(\Sigma _s (\mathbf R^{d})\) that extends uniquely to a continuous operator on \(\Sigma _s' (\mathbf R^{d})\). In particular \(WF^s (a^w(x,D) u)\) is well defined for \(u \in \Sigma _s' (\mathbf R^{d})\). The next result shows that these pseudodifferential operators are microlocal with respect to the s-Gelfand–Shilov wave front set. First we need a lemma.
Lemma 4.9
If \(\varphi \in \Sigma _s(\mathbf R^{2d}) {\setminus } 0\) and \(a \in C^\infty (\mathbf R^{2d})\) satisfies the estimates (4.16) for all \(h>0\), then for any \(A>0\)
Proof
We start by estimating a seminorm (2.3) of \(\overline{\varphi } \, T_{-x} a\). From (4.16) we obtain for any \(h>0\)
It follows that for any \(h>0\) we have the estimate
where \(C_h>0\). Note that the estimate is uniform over \(x \in \mathbf R^{2d}\).
By Proposition 3.1, or more precisely (3.4), we have for any \(A>0\)
for some \(C_A>0\). This gives finally for any \(A>0\)
\(\square \)
Proposition 4.10
If \(s>1/2\) and \(a \in C^\infty (\mathbf R^{2d})\) satisfies the estimates (4.16) for all \(h>0\) then
Proof
Pick \(\varphi \in \Sigma _s(\mathbf R^{d})\) such that \(\Vert \varphi \Vert _{L^2}=1\). Denoting the formal adjoint of \(a^w(x,D)\) by \(a^w(x,D)^*\), (2.6) gives for \(u \in \Sigma _s'(\mathbf R^{d})\) and \(z \in \mathbf R^{2d}\)
By e.g. [10, Lemma 3.1] we have
where \(\Phi \) is the Wigner distribution \(\Phi = W(\varphi ,\varphi ) \in \Sigma _s(\mathbf R^{2d})\). Defining
we thus obtain from Lemma 4.9
and
If \(0 \ne z_0 \in T^*\mathbf R^{d} {\setminus } WF^s(u)\) then there exists an open conic set \(\Gamma \subseteq T^* \mathbf R^{d} {\setminus } 0\) containing \(z_0\) such that for all \(A>0\)
By (2.4) we have for some \(M > 0\)
It now follows from (4.17) and Lemma 4.2 that for any open conic set \(\Gamma '\) containing \(z_0\) such that \(\overline{\Gamma ' \cap S_{2d-1}} \subseteq \Gamma \) we have for all \(A > 0\)
which proves that \(z_0 \notin WF^s( a^w(x,D) u)\). We have thus shown
\(\square \)
Corollary 4.11
Let \(s>1/2\) and \(u \in \Sigma _s'(\mathbf R^{d})\). For any \(z \in \mathbf R^{2d}\) we have \(WF^s( \Pi (z) u) = WF^s(u)\).
Proof
Since \(\Pi (-z) \Pi (z) = e^{i \langle x, \xi \rangle }\) for \(z=(x,\xi ) \in \mathbf R^{2d}\), it suffices to show \(WF^s( \Pi (z) u) \subseteq WF^s(u)\). The latter inclusion follows from Proposition 4.10 if we succeed in showing that the Weyl symbol for \(\Pi (z)\) is smooth and satisfies (4.16) for any \(h>0\).
We have \(\Pi (z) = a_z^w(x,D)\) where
(cf. the proof of [34, Lemma 3.7]). Thus
for any \(h>0\). The estimates (4.16) are thus satisfied. \(\square \)
5 Schrödinger Equations and Solution Operators
As stated in Sect. 1 the ultimate purpose of this paper is to prove results on propagation of the s-Gelfand–Shilov wave front set for the initial value Cauchy problem for a class of Schrödinger equations. More precisely we study the equation
where \(s>1/2\), \(u_0 \in \Sigma _s'(\mathbf R^{d})\), \(t \geqslant 0\) and \(x \in \mathbf R^{d}\). The Hamiltonian \(q^w(x,D)\) has the quadratic form Weyl symbol
where \(Q \in \mathbf C^{2d \times 2d}\) is a symmetric matrix with \(\mathrm{Re} \, Q \geqslant 0\). The special case \(\mathrm{Re} \, Q = 0\) will admit to study the equation for \(t \in \mathbf R\) instead of \(t \geqslant 0\).
According to [1, Theorem 3.10] \(q^w(x,D)\) extends to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\), and we will later prove that also the solution operator is continuous on \(\Sigma _s'(\mathbf R^{d})\) for each \(t \geqslant 0\) (see Corollary 8.2).
The Hamilton map F corresponding to q is defined by
where q(Y, X) is the bilinear polarized version of the form q, i.e. \(q(X,Y)=q(Y,X)\) and \(q(X,X)=q(X)\). The Hamilton map F is the matrix
where \(\mathcal {J}\) is the matrix (1.2).
For \(u_0 \in L^2(\mathbf R^{d})\) the equation (5.1) is solved for \(t \geqslant 0\) by
where the solution operator (propagator) \(e^{-t q^w(x,D)}\) is the contraction semigroup that is generated by the operator \(-q^w(x,D)\). Contraction semigroup means a strongly continuous semigroup with \(L^2\) operator norm \(\leqslant 1\) for all \(t \geqslant 0\) (cf. [35]). The reason why \(- q^w(x,D)\), or more precisely the closure \(M_{-q}\) as an unbounded linear operator in \(L^2(\mathbf R^{d})\) of the operator \(- q^w(x,D)\) defined on \(\mathscr {S}(\mathbf R^{d})\), generates such a semigroup is explained in [17, pp.425–26]. The contraction semigroup property is a consequence of \(M_{- q}\) and its adjoint \(M_{-\overline{q}}\) being dissipative operators [35]. For \(M_{-q}\) this means
\(D(M_{-q}) \subseteq L^2(\mathbf R^{d})\) denoting the domain of \(M_{-q}\), which follows from the assumption \(\mathrm{Re} \, Q \geqslant 0\). Note the feature \(M_{-\overline{q}} = M_{-q}^*\) that holds for the Weyl quantization.
Our objective is the propagation of the s-Gelfand–Shilov wave front set with \(s>1/2\) for the Schrödinger propagator \(e^{-t q^w(x,D)}\). This means that we seek inclusions for
in terms of \(WF^s(u_0)\), F and \(t \geqslant 0\) for \(u_0 \in \Sigma _s'(\mathbf R^{d})\).
If \(\mathrm{Re} \, Q=0\) then the propagator is given by means of the metaplectic representation. To wit, if \(\mathrm{Re} \, Q=0\) then \(e^{-t q^w(x,D)}\) is a group of unitary operators, and we have by [7, Theorem 4.45]
In this case F is purely imaginary and \(i F \in {\text {sp}}(d,\mathbf R)\), the symplectic Lie algebra, which implies that \(e^{-2 i t F} \in {\text {Sp}}(d,\mathbf R)\) for any \(t \in \mathbf R\) [7]. According to Corollary 4.5 we thus have if \(s > 1/2\)
The propagation of the s-Gelfand–Shilov wave front set is thus exact when \(\mathrm{Re} \, Q=0\). In the rest of the paper we study the more general assumption \(\mathrm{Re} \, Q \geqslant 0\). Under this assumption we will show in Sect. 8 that the propagator \(e^{-t q^w(x,D)}\) is a continuous operator on \(\Sigma _s(\mathbf R^{d})\) and extends uniquely to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\) when \(s>1/2\).
6 Propagation of the s-Gelfand–Shilov Wave Front Set for Certain Linear Operators
In this section we prepare for the results on propagation of the s-Gelfand–Shilov wave front set for \(e^{-t q^w(x,D)}\) in Sect. 9. We show propagation of singularities for linear operators in terms of their Schwartz kernels.
For \(s>1/2\) a kernel \(K \in \Sigma _s'(\mathbf R^{2d})\) defines a continuous linear map \(\mathscr {K}: \Sigma _s (\mathbf R^{d}) \rightarrow \Sigma _s'(\mathbf R^{d})\) by
Let \(\varphi \in \Sigma _s(\mathbf R^{d})\) satisfy \(\Vert \varphi \Vert _{L^2} = 1\) and set \(\Phi = \varphi \otimes \varphi \in \Sigma _s(\mathbf R^{2d})\). By [34, Lemma 4.1] we have for \(u,\psi \in \Sigma _s(\mathbf R^{d})\)
In the following results we need a definition from [16], adapted from the Gabor to the s-Gelfand–Shilov wave front set. For \(K \in \Sigma _s'(\mathbf R^{2d})\) we define
Lemma 6.1
If \(s>1/2\), \(K \in \Sigma _s'(\mathbf R^{2d})\) and \(WF_1^s(K) = WF_2^s(K) = \emptyset \), then there exists \(C > 1\) such that
Proof
Suppose
does not hold for any \(C>0\). Then for each \(n \in \mathbf N\) there exists \((x_n,y_n,\xi _n,\eta _n) \in WF^s(K)\) such that \(|(y_n,\eta _n)| \geqslant n |(x_n,\xi _n)|\). We may assume that \(|(x_n,y_n,\xi _n,\eta _n)| = 1\) for each \(n \in \mathbf N\) since \(WF^s(K)\) is conic. Thus \((x_n,\xi _n) \rightarrow 0\) as \(n \rightarrow \infty \). Passing to a subsequence (without change of notation) and using the closedness of \(WF^s(K)\) gives
for some \((y,\eta ) \in S_{2d-1}\). This implies \((y,-\eta ) \in WF_2^s(K)\) which is a contradiction.
Similarly one shows
for some \(C>0\) using \(WF_1^s(K) = \emptyset \). \(\square \)
In the next result we use the conventional notation (cf. [16, 17]) for the reflection operator in the fourth \(\mathbf R^{d}\) coordinate on \(\mathbf R^{4d}\)
Lemma 6.2
Suppose \(s>1/2\), \(K \in \Sigma _s'(\mathbf R^{2d})\) and \(WF_1^s(K) = WF_2^s(K) = \emptyset \). Suppose that the linear map \(\mathscr {K}: \Sigma _s (\mathbf R^{d}) \rightarrow \Sigma _s'(\mathbf R^{d})\) defined by (6.1) is continuous \(\mathscr {K}: \Sigma _s (\mathbf R^{d}) \rightarrow \Sigma _s(\mathbf R^{d})\) and extends uniquely to a continuous linear operator \(\mathscr {K}: \Sigma _s' (\mathbf R^{d}) \rightarrow \Sigma _s'(\mathbf R^{d})\). If \(\varphi \in \Sigma _s(\mathbf R^{d})\) satisfies \(\Vert \varphi \Vert _{L^2} = 1\) and \(\Phi = \varphi \otimes \varphi \) then (6.2) extends to \(u \in \Sigma _s'(\mathbf R^{d})\) and \(\psi \in \Sigma _s(\mathbf R^{d})\).
Proof
Let \(\varphi \in \Sigma _s(\mathbf R^{d})\) satisfy \(\Vert \varphi \Vert _{L^2} = 1\) and let \(u \in \Sigma _s'(\mathbf R^{d})\). By (2.4) we have for some \(M \geqslant 0\)
Define for \(n \in \mathbf N\)
In order to verify \(u_n \in \Sigma _s(\mathbf R^{d})\) we use the seminorms (3.3) for \(A>0\). We have for any \(A>0\)
It follows that \(u_n \in \Sigma _s(\mathbf R^{d})\) for \(n \in \mathbf N\).
To prove that \(u_n \rightarrow u\) in \(\Sigma _s'(\mathbf R^{d})\) as \(n \rightarrow \infty \) we pick \(\psi \in \Sigma _s(\mathbf R^{d})\). From (6.4), the estimate (cf. Proposition 3.1 and [29, Theorem 2.4])
we obtain by means of dominated convergence and (2.6)
This proves the claim that \(u_n \rightarrow u\) in \(\Sigma _s'(\mathbf R^{d})\) as \(n \rightarrow \infty \).
We also need the estimate (cf. [9, Eq. (11.29)])
which in view of (6.4) and (6.5) with \(\psi \) replaced by \(\varphi \) and conjugation of the window function gives the bound
that holds uniformly over \(n \in \mathbf N\).
We are now in a position to assemble the arguments into a proof of formula (6.2) for \(u \in \Sigma _s'(\mathbf R^{d})\) and \(\psi \in \Sigma _s(\mathbf R^{d})\). Using (6.2) for \(u_n\) gives
Since \(V_{\overline{\varphi }} u_n(y,\eta ) \rightarrow V_{\overline{\varphi }} u(y,\eta )\) as \(n \rightarrow \infty \) for all \((y,\eta ) \in \mathbf R^{2d}\), the formula (6.2) follows from dominated convergence if we can show that the modulus of the integrand in (6.7) is bounded by an integrable function that does not depend on \(n \in \mathbf N\).
For \(C > 1\) define the open conic set
If C is chosen properly then we have \(WK^s (K) \subseteq \Gamma \) by Lemma 6.1.
Let \(\varepsilon >0\). For the integral in (6.7) over \(\Gamma '\) we may estimate, using the estimate
for some \(B \geqslant 0\) (cf. (2.4)), and (6.5), (6.6), for any \(A>0\)
in the final inequality assuming that \(A>0\) is sufficiently large.
Since \(\Gamma \subseteq T^* \mathbf R^{2d} {\setminus } 0\) is open and \(WK^s (K) \subseteq \Gamma \) we have for any \(A>0\)
This gives for the integral in (6.7) over \(\mathbf R^{4d} {\setminus } \Gamma '\), again using (6.6),
provided \(A>0\) is sufficiently large.
Combined, (6.9) and (6.10) show our claim that the modulus of the integrand in (6.7) is bounded by an integrable function that does not depend on \(n \in \mathbf N\). \(\square \)
Since
we obtain from Lemma 6.2 with \(\psi = \Pi (t,\theta ) \varphi \) for \((t,\theta ) \in \mathbf R^{2d}\), \(u \in \Sigma _s'(\mathbf R^{d})\), \(\varphi \in \Sigma _s(\mathbf R^{d})\) and \(\Vert \varphi \Vert _{L^2} = 1\)
This formula will be useful in the proof of Theorem 6.3.
The following result concerns propagation of singularities for linear operators and is a version of Hörmander’s [16, Proposition 2.11] adapted to the s-Gelfand–Shilov wave front set. We use the relation mapping notation
Theorem 6.3
Let \(s>1/2\) and let \(\mathscr {K}\) be the continuous linear operator (6.1) defined by the Schwartz kernel \(K \in \Sigma _s'(\mathbf R^{2d})\). Suppose \(\mathscr {K}: \Sigma _s(\mathbf R^{d}) \rightarrow \Sigma _s(\mathbf R^{d})\) is continuous and extends uniquely to a continuous linear operator \(\mathscr {K}: \Sigma _s' (\mathbf R^{d}) \rightarrow \Sigma _s' (\mathbf R^{d})\), and suppose
Then for \(u \in \Sigma _s'(\mathbf R^{d})\) we have
Proof
It follows from (2.4) that (6.8) is satisfied for some \(B \geqslant 0\) if \(\Phi \in \Sigma _s(\mathbf R^{2d}) {\setminus } 0\).
Denote by
the projections \(\mathbf R^{4d} \rightarrow \mathbf R^{2d}\) onto the first and the third \(\mathbf R^{d}\) coordinate, and onto the second and the fourth \(\mathbf R^{d}\) coordinate with a change sign in the latter, respectively.
By Lemma 6.1 there exists \(c>1\) such that
and, defining
we thus have
We show the inclusion (6.13) by showing that
implies \((t_0,\theta _0) \notin WF^s (\mathscr {K}u)\). Thus we suppose (6.15). By [34, Lemma 4.2] we may assume that \((t_0,\theta _0) \in \Omega _0\) and \(\overline{\Omega }_0 \cap WF^s(K)' \circ \overline{\Omega }_2 = \emptyset \) where \(\Omega _0, \Omega _2 \subseteq T^* \mathbf R^{d} {\setminus } 0\) are conic, open and \(WF^s (u) \subseteq \Omega _2\). (The assumption of [34, Lemma 4.2] corresponds to the assumption (6.12).) Here we use the notation \(\overline{\Omega }\subseteq T^* \mathbf R^{d} {\setminus } 0\) for the closure in the usual topology in \(T^* \mathbf R^{d} {\setminus } 0\) of a conical subset \(\Omega \subseteq T^* \mathbf R^{d} {\setminus } 0\).
Hence
or, equivalently,
Due to assumption (6.12) we may strengthen this into
Since \(p_{1,3}^{-1} \, (\overline{\Omega }_0 \cup \{ 0 \} ) {\setminus } 0\) and \(p_{2,-4}^{-1} \, (\overline{\Omega }_2 \cup \{ 0 \} ) {\setminus } 0\) are closed conic subsets of \(\mathbf R^{4d} {\setminus } 0\), decreasing \(\Gamma _1 \subseteq \mathbf R^{4d} {\setminus } 0\) if necessary, there exist open conic subsets \(\Gamma _0, \Gamma _1, \Gamma _2 \subseteq \mathbf R^{4d} {\setminus } 0\) such that
and
Let \(\Sigma _0 \subseteq T^* \mathbf R^{d} {\setminus } 0\) be an open conic set such that \((t_0,\theta _0) \in \Sigma _0\) and \(\overline{\Sigma _0 \cap S_{2d-1}} \subseteq \Omega _0\). Let \(\varphi \in \Sigma _s(\mathbf R^{d})\), \(\Vert \varphi \Vert _{L^2} = 1\) and \(\Phi = \varphi \otimes \varphi \). From Lemma 6.2 we know that formula (6.11) holds. Therefore we have for any \(A>0\)
We will show that this integral is bounded when \((t,\theta ) \in \Sigma _0\) for any \(A>0\) which proves that \((t_0,\theta _0) \notin WF^s (\mathscr {K}u)\).
Consider first the right hand side integral over \((x,y,\xi ,-\eta ) \in \mathbf R^{4d} {\setminus } \Gamma _1\). We have for any \(b>0\)
due to \(WF^s(K) \subseteq \Gamma _1\) and \(\Gamma _1 \subseteq \mathbf R^{4d} {\setminus } 0\) being open. By (2.4) we have
for some \(M \geqslant 0\), and by Proposition 3.1 we have
for any \(c>0\). Thus
if \(b > 0\) is chosen sufficiently large. The estimate holds for all \((t,\theta ) \in \mathbf R^{2d}\).
It remains to estimate the right hand side integral (6.17) over \((x,y,\xi ,-\eta ) \in \Gamma _1\). By (6.14) and (6.16) we have \(\Gamma _1 \subseteq G_1 \cup G_2\) where
When \((x,y,\xi ,-\eta ) \in \Gamma _1\) we have \(|(x,\xi )|\asymp |(y,\eta )|\). First we study \((x,y,\xi ,-\eta ) \in G_1\). Then \((x,y,\xi ,-\eta ) \notin \Gamma _0\) which implies \((x,\xi ) \notin \Omega _0\). There exists \(\delta >0\) such that
Let \(\varepsilon >0\). For \((t,\theta ) \in \Sigma _0\) we obtain with the aid of (6.8) using \(|(x,\xi )|\asymp |(y,\eta )|\), (6.19) and (6.20), where \(B,M \geqslant 0\) are fixed and \(c>0\) is arbitrary, with \(B_1>0\) a new constant that depends on \(B, M, \varepsilon , s\),
provided \(c \geqslant B_1 \delta ^{-1/s}\).
Finally we study \((x,y,\xi ,-\eta ) \in G_2\). Then \((x,y,\xi ,-\eta ) \notin \Gamma _2\) so we have \((y,\eta ) \notin \Omega _2\). Hence \((y,\eta ) \in G\) where \(G \subseteq T^* \mathbf R^{d}\) is closed, conic and does not intersect \(WF^s(u)\). We obtain with the aid of (6.8) for any \((t,\theta ) \in T^* \mathbf R^{d}\), using \(|(x,\xi )|\asymp |(y,\eta )|\) and (6.20), for \(B \geqslant 0\) fixed and \(B_1>0\) a new constant that depends on \(A, B, \varepsilon , s\),
We can now combine (6.17), (6.21), \(\Gamma _1 \subseteq G_1 \cup G_2\), (6.22) and (6.23) to conclude
for any \(A>0\). Thus \((t_0,\theta _0) \notin WF^s (\mathscr {K}u)\). \(\square \)
7 The s-Gelfand–Shilov Wave Front Set of Oscillatory Integrals
7.1 Oscillatory Integrals
We need to describe a class of oscillatory integrals with quadratic phase functions introduced by Hörmander [17]. This is useful due to the fact that the Schwartz kernel of the Schrödinger propagator \(e^{-t q^w(x,D)}\) is an integral of this form, as we will describe in Sect. 8. Our discussion on oscillatory integrals is brief. For a richer account we refer to [17, 22].
Let p be a complex-valued quadratic form on \(\mathbf R^{d + N}\),
where \(P \in \mathbf C^{(d+N) \times (d+N)}\) is the symmetric matrix
where \(P_{xx} \in \mathbf C^{d \times d}\), \(P_{x \theta } \in \mathbf C^{d \times N}\) and \(P_{\theta \theta } \in \mathbf C^{N \times N}\).
Suppose P satisfies the following two conditions.
-
(1)
\(\mathrm{Im} \, P \geqslant 0\);
-
(2)
the row vectors of the submatrix
$$\begin{aligned} \left( \begin{array}{ll} P_{\theta x}&P_{\theta \theta } \end{array} \right) \in \mathbf C^{N \times (d+N)} \end{aligned}$$are linearly independent over \(\mathbf C\).
Under these circumstances the oscillatory integral
can be given a unique meaning as an element in \(\mathscr {S}'(\mathbf R^{d})\), by means of a regularization procedure [17, 22]. Due to the embedding \(\mathscr {S}'(\mathbf R^{d}) \subseteq \Sigma _s'(\mathbf R^{d})\) (\(s>1/2\)), the oscillatory integral defines a unique element \(u \in \Sigma _s'(\mathbf R^{d})\).
An oscillatory integral of the form (7.3) is, up to multiplication with an element in \(\mathbf C{\setminus } 0\), bijectively associated with a Lagrangian subspace \(\lambda \subseteq T^* \mathbf C^{d}\), that is positive in the sense of
The positive Lagrangian associated with the oscillatory integral (7.3) is
The integer N in the integral (7.3) is not uniquely determined by u. In fact, it may be possible to decrease N and obtain the same u times a nonzero complex constant. This procedure may be iterated until the term that is quadratic in \(\theta \) disappears. The form p is then (cf. [17, Propositions 5.6 and 5.7])
where \(\rho \) is a quadratic form, \(\mathrm{Im} \, \rho \geqslant 0\) and \(L \in \mathbf R^{d \times N}\) is an injective matrix. The matrix L is uniquely determined by the Lagrangian \(\lambda \) modulo invertible right factors, and similarly the values of \(\rho \) on \({\text {Ker}}L^t\) are uniquely determined.
The oscillatory integral is (cf. [17, Proposition 5.7])
where \(\delta _0 = \delta _0(\mathbf R^{N})\).
7.2 The s-Gelfand–Shilov Wave Front Set of an Oscillatory Integral
Theorem 7.1
Let \(u \in \mathscr {S}'(\mathbf R^{d})\) be the oscillatory integral (7.3) with the associated positive Lagrangian \(\lambda \) given by (7.4). Then for \(s > 1/2\)
Proof
First we assume \(N \geqslant 1\) in (7.3) and in the end we will take care of the case \(N=0\).
As discussed above we may assume that p is of the form (7.5) where \(\rho \) is a quadratic form, \(\mathrm{Im} \, \rho \geqslant 0\) and \(L \in \mathbf R^{d \times N}\) is injective, and u is given by (7.6). The positive Lagrangian (7.4) associated to u is hence
By [22, Proposition 3.4] and the uniqueness part of [17, Proposition 5.7] we may assume
If \(x \in \mathbf R^{d}\), \(\theta \in \mathbf C^{N}\) and \(0 = \mathrm{Im} \, (\rho '(x) + L \theta ) = \mathrm{Im} \, (\rho ')(x) + L \mathrm{Im} \, \theta \), we may thus conclude \(\mathrm{Im} \, (\rho ')(x)=0\) and \(L \mathrm{Im} \, \theta =0\), so the injectivity of L forces \(\theta \in \mathbf R^{N}\). Hence
According to Proposition 4.8 and (4.8)
We write \(\rho (x) = \rho _r(x) + i \rho _i(x)\) and
with \(R_r, R_i \in \mathbf R^{d \times d}\) symmetric and \(R_i \geqslant 0\). The function \(e^{ i \rho _r(x)}\), considered as a multiplication operator, is the metaplectic operator
where
The function \(g(x) = e^{- \rho _i(x) }\) satisfies the estimates (4.16) for all \(h>0\). In fact, since \(R_i \geqslant 0\) it suffices to verify the estimates for \(g(Ux) = \Pi _{j=1}^n e^{-\mu _j x_j^2}\) where \(U \in \mathbf R^{d \times d}\) is an orthogonal matrix and \(\mu _j>0\) for \(1 \leqslant j \leqslant n \leqslant d\). The function \(x \rightarrow g(Ux)\) clearly satisfies (4.16) for all \(h>0\) since it is a tensor product of a Gaussian on \(\mathbf R^{n}\), that belongs to \(\Sigma _s(\mathbf R^{n})\), and the function one on \(\mathbf R^{d-n}\).
If we consider \(e^{- \rho _i(x) }\) as a function of \((x,\xi ) \in T^* \mathbf R^{d}\), constant with respect to the \(\xi \) variable, then the corresponding Weyl pseudodifferential operator is multiplication with \(e^{- \rho _i(x) }\). Proposition 4.10 gives
Piecing these arguments together, using Corollary 4.5 and (7.9), gives
This ends the proof when \(N \geqslant 1\).
Finally, if \(N=0\) then the (degenerate) oscillatory integral (7.3) is
where \(\rho = \rho _r + i \rho _i\) is given by (7.10) with \(R_r, R_i \in \mathbf R^{d \times d}\) symmetric and \(R_i \geqslant 0\). By (7.4) the corresponding positive Lagrangian is
which gives
Since \(WF^s(1)= (\mathbf R^{d} {\setminus } 0) \times \{0\}\) (cf. (4.10)) we obtain, again using (7.12) and recycling the argument above,
\(\square \)
Remark 7.2
From \(WF(u) \subseteq WF^s(u)\) for \(u \in \mathscr {S}'(\mathbf R^{d})\) and \(s>1/2\) it follows that Theorem 7.1 is a sharpening of [22, Theorem 3.6].
8 The Schwartz Kernel of the Schrödinger Propagator
Let q be a quadratic form on \(T^*\mathbf R^{d}\) defined by a symmetric matrix \(Q \in \mathbf C^{2 d \times 2 d}\) with Hamilton map \(F=\mathcal {J}Q\) and \(\mathrm{Re} \, Q \geqslant 0\). According to [17, Theorem 5.12] the Schrödinger propagator is
where \(\mathscr {K}_{e^{-2 i t F}}: \mathscr {S}(\mathbf R^{d}) \rightarrow \mathscr {S}'(\mathbf R^{d}) \) is the linear continuous operator with Schwartz kernel
with \(T = e^{-2 i t F}\). This kernel is an oscillatory with respect to a quadratic form p on \(\mathbf R^{2 d + N}\) as discussed in Sect. 7.1. The positive Lagrangian associated to the Schwartz kernel \(K_{e^{-2 i t F}}\) is
By [22, Lemma 4.2] the Lagrangian \(\lambda = \lambda _{e^{-2 i t F}}\) given by (8.2) is a positive twisted graph Lagrangian defined by the matrix \(e^{-2 i t F} \in {\text {Sp}}(d,\mathbf C)\). When a twisted graph Lagrangian defined by a matrix \(T \in {\text {Sp}}(d,\mathbf C)\) is positive, also the matrix is called positive [17]. This means
Since \(K_{e^{-2 i t F}} \in \mathscr {S}'(\mathbf R^{2d})\) the propagator is a continuous operator \(e^{-t q^w(x,D)}: \mathscr {S}(\mathbf R^{d}) \rightarrow \mathscr {S}' (\mathbf R^{d})\). We have in fact continuity \(e^{-t q^w(x,D)}: \mathscr {S}(\mathbf R^{d}) \rightarrow \mathscr {S}(\mathbf R^{d})\). This follows from [17, Proposition 5.8 and Theorem 5.12] which says that \(\mathscr {K}_T: \mathscr {S}(\mathbf R^{d}) \rightarrow \mathscr {S}(\mathbf R^{d})\) is continuous for any positive matrix \(T \in {\text {Sp}}(d,\mathbf C)\). The next result shows that \(\mathscr {K}_T: \Sigma _s (\mathbf R^{d}) \rightarrow \Sigma _s (\mathbf R^{d})\) is continuous and \(\mathscr {K}_T\) extends uniquely to a continuous operator \(\mathscr {K}_T: \Sigma _s' (\mathbf R^{d}) \rightarrow \Sigma _s' (\mathbf R^{d})\).
Proposition 8.1
Suppose \(T \in {\text {Sp}}(d,\mathbf C)\) is positive and let \(\mathscr {K}_T: \mathscr {S}(\mathbf R^{d}) \rightarrow \mathscr {S}'(\mathbf R^{d})\) be the continuous linear operator having Schwartz kernel \(K_T \in \mathscr {S}'(\mathbf R^{2d})\) defined by (8.1). For \(s > 1/2\) the operator \(\mathscr {K}_T\) is continuous on \(\Sigma _s(\mathbf R^{d})\) and \(\mathscr {K}_T\) extends uniquely to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\).
Proof
Due to the above mentioned continuity of \(\mathscr {K}_T\) on \(\mathscr {S}(\mathbf R^{d})\), we have for some integer \(L \geqslant 0\)
Using the seminorms (2.3) it can be seen readily that the operators \(f \rightarrow D^\beta f\) and \(f \rightarrow x^\alpha f\) are continuous operators on \(\Sigma _s(\mathbf R^{d})\). By Proposition 3.1 we therefore have for any \(A>0\)
for some \(B>0\), where we use the seminorm (3.1). This gives
which gives a desired continuity estimate for one of the two families of seminorms \(\{\Vert f \Vert _A', \ \Vert \widehat{f} \Vert _B', \ A,B>0 \}\).
To prove that \(\mathscr {K}_T\) is continuous on \(\Sigma _s(\mathbf R^{d})\) it remains to estimate \(\Vert \mathscr {F}(\mathscr {K}_T f) \Vert _A'\) for any \(A>0\). The Fourier transform has Schwartz kernel \(K(x,y)=e^{-i \langle x, y \rangle }\) which is a degenerate oscillatory integral with \(N=0\), cf. (7.3). The corresponding Lagrangian is by (7.4)
Due to the uniqueness, modulo multiplication with a nonzero complex number, of the correspondence between oscillatory integrals and Lagrangians, it follows that \(\mathscr {F}= (2 \pi )^{d/2} \mu (\mathcal {J}) = c \mathscr {K}_{\mathcal {J}}\) for some \(c \in \mathbf C{\setminus } 0\).
We may hence write for some \(c \in \mathbf C{\setminus } 0\), using the semigroup property of \(T \rightarrow \mathscr {K}_T\) modulo sign, when \(T \in {\text {Sp}}(d,\mathbf C)\) is positive (cf. [17, Proposition 5.9])
Since \(\mathcal {J}T \mathcal {J}\in {\text {Sp}}(d,\mathbf C)\) is positive, and since the estimate (8.3) holds for any positive \(T \in {\text {Sp}}(d,\mathbf C)\) for some \(B>0\), we obtain for any \(A>0\)
for some \(B>0\). Combining (8.3) and (8.4) and referring to Proposition 3.1, we have proved that \(\mathscr {K}_T\) is continuous on \(\Sigma _s(\mathbf R^{d})\).
Finally we show that \(\mathscr {K}_T\) extends uniquely to a continuous operator on \(\Sigma _s'(\mathbf R^{d})\). The formal adjoint of \(\mathscr {K}_T\) is \(\mathscr {K}_{\overline{T}^{-1}}\) which is indexed by the inverse conjugate matrix \(\overline{T}^{-1} \in {\text {Sp}}(d,\mathbf C)\) [17]. The positivity of \(\overline{T}^{-1}\) is an immediate consequence of the assumed positivity of T. Thus \(\mathscr {K}_T\) may be defined on \(\Sigma _s' (\mathbf R^{d})\) by
which gives a uniquely defined extension of \(\mathscr {K}_T\) as a continuous operator on \(\Sigma _s' (\mathbf R^{d})\). \(\square \)
Corollary 8.2
The Schrödinger propagator \(e^{-t q^w(x,D)}\) has Schwartz kernel \(K_{e^{-2 i t F}} \in \mathscr {S}'(\mathbf R^{2d})\). For \(t \geqslant 0\) and \(s>1/2\) it is a continuous operator on \(\Sigma _s (\mathbf R^{d})\), and it extends uniquely to a continuous operator on \(\Sigma _s' (\mathbf R^{d})\).
9 Propagation of the s-Gelfand–Shilov Wave Front set for Schrödinger Equations
Since the Schwartz kernel of the Schrödinger propagator \(e^{-t q^w(x,D)}\) is an oscillatory integral corresponding to the positive Lagrangian (8.2), an appeal to Theorem 7.1 gives the following result. For \(s>1/2\) the s-Gelfand–Shilov wave front set of the Schwartz kernel \(K_{e^{-2 i t F}}\) of the propagator \(e^{-t q^w(x,D)}\) for \(t \geqslant 0\) obeys the inclusion
Combining this with Corollary 8.2 and Theorem 6.3 now gives a result on propagation of singularities. The assumptions of the latter theorem are satisfied for \(\mathscr {K}= \mathscr {K}_{e^{-2 i t F}} = e^{-t q^w(x,D)}\), since \(WF_1^s ( K_{e^{-2 i t F}} ) = WF_2^s ( K_{e^{-2 i t F}} )= \emptyset \) follows from (9.1) and the invertibility of \(e^{-2 i t F} \in \mathbf C^{2d \times 2d}\), cf. (6.3).
Corollary 9.1
Suppose q is a quadratic form on \(T^*\mathbf R^{d}\) defined by a symmetric matrix \(Q \in \mathbf C^{2d \times 2d}\), \(\mathrm{Re} \, Q \geqslant 0\), \(F=\mathcal {J}Q\), and \(s > 1/2\). Then for \(u \in \Sigma _s'(\mathbf R^{d})\)
As in the proof of [22, Theorem 5.2], the latter inclusion can be sharpened using the semigroup modulo sign property of the propagator [17]
In fact, using this property and some elementary arguments one obtains the inclusions
where the singular space
of the quadratic form q plays a crucial role.
Corollary 9.2
Suppose q is a quadratic form on \(T^*\mathbf R^{d}\) defined by a symmetric matrix \(Q \in \mathbf C^{2d \times 2d}\), \(\mathrm{Re} \, Q \geqslant 0\), \(F=\mathcal {J}Q\), and \(s > 1/2\). Then for \(u \in \Sigma _s'(\mathbf R^{d})\)
Change history
08 April 2021
A Correction to this paper has been published: https://doi.org/10.1007/s00041-021-09824-3
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Communicated by Michael Ruzhansky.
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Carypis, E., Wahlberg, P. Propagation of Exponential Phase Space Singularities for Schrödinger Equations with Quadratic Hamiltonians. J Fourier Anal Appl 23, 530–571 (2017). https://doi.org/10.1007/s00041-016-9478-6
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DOI: https://doi.org/10.1007/s00041-016-9478-6
Keywords
- Schrödinger equation
- Heat equation
- Propagation of singularities
- Phase space singularities
- Gelfand–Shilov wave front set