Let $H = H_0 + P$ denote the harmonic oscillator on $\mathbb{R}^d$ perturbed by an isotropic pseudodifferential operator $P$ of order $1$ and let $U(t) = \operatorname{exp}(- it H)$. We prove a Gutzwiller–Duistermaat–Guillemin type trace formula for $\operatorname{Tr} U(t).$ The singularities occur at times $t \in 2 \pi \mathbb{Z}$ and the coefficients involve the dynamics of the Hamilton flow of the symbol $\sigma(P)$ on the space $\mathbb{CP}^{d-1}$ of harmonic oscillator orbits of energy $1$. This is a novel kind of sub-principal symbol effect on the trace. We generalize the averaging technique of Weinstein and Guillemin to this order of perturbation, and then present two completely different calculations of $\operatorname{Tr} U(t)$. The first proof directly constructs a parametrix of $U(t)$ in the isotropic calculus, following earlier work of Doll–Gannot–Wunsch. The second proof conjugates the trace to the Bargmann–Fock setting, the order $1$ of the perturbation coincides with the 'central limit scaling' studied by Zelditch–Zhou for Toeplitz operators.

中文翻译：

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Schrödinger迹不变式，用于谐振子的齐次扰动

令$ H = H_0 + P $表示$ \ mathbb {R} ^ d $上的谐波振荡器，被阶数为$ 1 $的各向同性伪微分算子$ P $扰动，并且让$ U（t）= \ operatorname {exp}（-它H）$。我们证明了$ \ operatorname {Tr} U（t）的Gutzwiller–Duistermaat–Guillemin类型跟踪公式。$奇异点有时在$ t \ in 2 \ pi \ mathbb {Z} $时出现，并且系数涉及符号$ \ sigma（P）$在能量$ 1 $的谐振子轨道$ \ mathbb {CP} ^ {d-1} $上的汉密尔顿流。这是一种对痕迹的新的次主符号作用。我们将Weinstein和Guillemin的平均技术推广到此扰动阶次，然后给出$ \ operatorname {Tr} U（t）$的两种完全不同的计算。第一个证明直接在各向同性演算中构造$ U（t）$的参数，遵循Doll–Gannot–Wunsch的早期工作。第二个证明使轨迹与Bargmann-Fock设置共轭，扰动的阶次$ 1 $与Zelditch-Zhou为Toeplitz算符研究的“中心极限比例”相吻合。