Spectra and semigroup smoothing for non-elliptic quadratic operators
In this paper, the authors study spectral and semigroup properties for pseudodifferential operators defined by the Weyl quantization formula \[ q(x,\xi)^{w}u(x)=\frac{1}{(2 \pi)^{n}}\int_{\mathbb{R}^{2n}}e^{i(x-y)\cdot\xi} q \biggl(\frac{x + y}{2},\xi\biggl)u(y) \,dy \,d\xi \] for some symbols \(q(x,\xi)\), \((x,\xi)\in\mathbb{R}^{n}\times\mathbb{R}^{n}\), which are complex-valued quadratic forms. First, it is pointed out the existence of a linear subspace \(S\) in \(\mathbb{R}_{x}^{n}\times\mathbb{R}_{\xi}^{n}\), called a singular space, which is defined in terms of the Hamilton map of the Weyl symbol \(q\), such that when \(S\) has a symplectic structure, the associated heat equation \[ \frac{\partial u}{\partial t}-q(x,\xi)^w u=0 \] with \(u(t,\cdot)/_{t=0}=u_{0}\in L^{2}(\mathbb{R}^{n})\) is smoothing in every direction of the orthogonal complement \(S^{\sigma \bot}\) of \(S\) with respect to the canonical symplectic form \(\sigma\) in \(\mathbb{R}^{2 n}\). The second result concerns the structure of the spectra for the considered pseudodifferential operators. It is proved that, if the real part of \(q\) is a non-positive quadratic form and if \(q\) is elliptic on \(S\), then the spectrum of the quadratic pseudodifferential operator \(q (x,\xi)^{w}\) is only composed of eigenvalues of finite multiplicity, and \(\sigma (q (x,\xi)^{w})\) coincides with the set of all points represented in the form \(\sum_{\lambda} (r_{\lambda} + 2 k_{\lambda})(-i \lambda)\), where the sum is taken over all \(\lambda \in \sigma (F)\) such that \(-i \lambda \in \mathbb{C}_{-} \cup (\sum (q \mid s)\backslash \{ 0 \})\), \(F\) is the Hamilton map associated to the quadratic form \(q\), \(r_{\lambda}\) denotes the dimension of the space of generalized eigenvalues of \(F\) in \(\mathbb{C}^{2 n}\) belonging to the eigenvalue \(\lambda \in \mathbb{C}\), \(\sum(q\mid S)=\overline{q (S)}\), and \(\mathbb{C}_{-}=\{z\in \mathbb{C}:\text{Re} z < 0 \}\). The large time behavior of contraction semigroups generated by \(q (x,\xi)^{w}\) is also investigated. The corresponding results are the following. If \(f\) is a complex-valued quadratic form with a non-positive real part and if its singular space \(S\) has a symplectic structure, then the following assertions are equivalent. (i) The norm of the contraction semigroup generated by the operator \(q (x,\xi)^{w}\) decays exponentially in time, i.e., \[ \| e^{t q (x,\xi)^{w}} \|_{\mathcal{L} (L^{2})} \leq M e^{-a t} \quad (t \geq 0) \] for some \(M > 0\) and \(a > 0\). (ii) The real part of the symbol \(q\) is a nonzero quadratic form, i.e., there exists \((x,\xi)\in\mathbb{R}^{2n}\) such that \(\text{Re}q(x,\xi)\neq 0\). (iii) The singular space \(S\) is distinct from the whole phase space \(\mathbb{R}^{2 n}\).
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