发表机构
University of Ostrava; Charles University; VSB – Technical University of Ostrava（俄斯特拉发大学; 查理大学; 俄斯特拉发工业大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究一类梅利克 - 阿达米扬规范哈密顿量，通过规范变换简化表达式，利用双边解耦和截断估计给出本征值\(3/2\)阶矩界，适用于任意厄米矩阵系数，处理半直线边界条件，计算零质量最优规范能量，证明主相位扇区极小值存在。
AI 中文摘要
我们研究一类由梅利克 - 阿达米扬的规范微分表达式产生且出现在阿尔佩 - 戈伯格附录中的正定矩阵哈密顿量。设\(J\)和\(B\)是\(\mathbb{C}^{2n}\)上满足\(JB = -BJ\)的自伴对合，且\(\mathcal{H}>0\)满足\(\mathcal{H}J\mathcal{H}=J\)。对于\(m>0\)，我们在加权空间\(L^2_{\mathcal{H}}\)中考虑\(\mathcal{A}_{m,\mathcal{H}}=\mathcal{H}^{-1}\left(-iJ\frac{d}{dt}+mB\right)\)。一个表示\(\mathcal{H}\)的局部绝对连续\(J\)酉规范\(\Theta\)将此表达式简化为自由大质量狄拉克算子加上厄米系数\(P_{m,\Theta}=-i\Theta^*J\Theta'+m(\Theta^*B\Theta - B)\)。当此系数属于\(L^2\)时，相应的自伴实现，包括其算子域，与所选的表示规范无关。通过规范纤维上最小化\(\int{\rm Tr}|P_{m,\Theta}|^2\)定义一个本征能量。双边的比尔曼 - 施温格解耦与这里证明的截断伪相对论估计相结合，就该能量而言，给出了间隙\((-m,m)\)中所有本征值的\(3/2\)阶矩界。狄拉克估计适用于\(L^2\)中的任意厄米矩阵系数且不需要符号条件。在半直线上，我们处理每个自伴拉格朗日边界条件。两个反射兼容条件不需要端点修正，而任意条件最多贡献\(2nm^{3/2}\)。在零质量时，根据\(\mathcal{H}^{-1/2}\mathcal{H}'\mathcal{H}^{-1/2}\)明确计算最优规范能量。对于标量双曲旋转族，大质量规范最小化精确地简化为一维相位泛函。我们证明在主相位扇区中存在一个极小值，并给出一个明确的试验相位，每当相应的一阶变分为非零时，该试验相位严格且定量地改善正提升。
英文摘要
We study a class of positive matrix Hamiltonians arising from the canonical differential expressions of Melik--Adamyan and appearing in the appendix of Alpay--Gohberg. Let $J$ and $B$ be self-adjoint involutions on $\mathbb C^{2n}$ satisfying $JB=-BJ$, and let $H>0$ satisfy $HJH=J$. For $m>0$ we consider $$ \mathcal A_{m,H}=H^{-1}\left(-iJ\frac{d}{dt}+mB\right) $$ in the weighted space $L^2_H$. A locally absolutely continuous $J$-unitary gauge $Θ$ representing $H$ reduces this expression to the free massive Dirac operator plus the Hermitian coefficient $$ P_{m,Θ}=-iΘ^*JΘ'+m(Θ^*BΘ-B). $$ Whenever this coefficient belongs to $L^2$, the corresponding self-adjoint realization, including its operator domain, is independent of the chosen representing gauge. Minimizing $\int\mathrm{Tr}|P_{m,Θ}|^2$ over the gauge fibre defines an intrinsic energy. A two-sided Birman--Schwinger decoupling, combined with a truncated pseudo-relativistic estimate proved here, gives a $3/2$-moment bound for all eigenvalues in the gap $(-m,m)$ in terms of this energy. The Dirac estimate applies to arbitrary Hermitian matrix coefficients in $L^2$ and requires no sign condition. On the half-line we treat every self-adjoint Lagrangian boundary condition. Two reflection-compatible conditions require no endpoint correction, while an arbitrary condition contributes at most $2nm^{3/2}$. At zero mass, the optimal-gauge energy is computed explicitly in terms of $H^{-1/2}H'H^{-1/2}$. For a scalar hyperbolic-rotation family the massive gauge minimization reduces exactly to a one-dimensional phase functional. We prove existence of a minimizer in the principal phase sector and give an explicit trial phase that strictly and quantitatively improves the positive lift whenever the corresponding first variation is nonzero.
