Érik Amorim, Eric A. Carlen
We specify the structure of completely positive operators and quantum Markov semigroup generators that are symmetric with respect to a family of inner products, also providing new information on the order strucure an extreme points in some previously studied cases.
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$L^p$ positivity preservation and self-adjointness on incomplete Riemannian manifolds
本文证明了不完备黎曼流形上薛定谔型算子的 $L^p$ 正性保持性质，并在控制势函数于 Cauchy 边界附近行为的假设下，确立了算子的本质自伴性及其在 $p\ eq 2$ 情形下的推广。
We show that a combination of well-known operators, namely $\Iτ\circ{H}\circ\Ps$ is self-adjoint and {\em ad-hoc} related to the $ζ$ function. Here $τ$ is an involution appearing in Weil's positivity criteria needed for hermitrization, $H$ a regularization operator introduced by Connes \cite{Co2} and $\Ps$ essentially Poisson summation. We elaborate on the Hilbert-Pólya conjecture, discuss why the Hermite-Biehler theorem, uncertainty relations and cohomologies are interesting in our scenario.
We consider the problem of quantum and stochastic confinement for drift-diffusion equations on domains $ Ω\subset \mathbb R^d$. We obtain various sufficient conditions on the behavior of the coefficients near the boundary of $Ω$ which ensure the essential self-adjointness or stochastic completeness of the symmetric form of the drift-diffusion operator, $-\frac{1}{ρ_\infty}\,\nabla\cdot ρ_\infty\mathbb D\nabla$. The proofs are based on the method developed in [29] for quantum confinement on bounded domains in $\mathbb R^d$. In particular for stochastic confinement we combine the Liouville prope…
We consider perturbed quadharmonic operators, $Δ^4 + V$, acting on sections of a Hermitian vector bundle over a complete Riemannian manifold, with the potential $V$ satisfying a bound from below by a non-positive function depending on the distance from a point. Under a bounded geometry assumption on the Hermitian vector bundle and the underlying Riemannian manifold, we give a sufficient condition for the essential self-adjointness of such operators. We then apply this to prove the separation property in $L^2$ when the perturbed operator acts on functions.
We consider general symmetric systems of first order linear partial differential operators on domains $Ω\subset \mathbb{R}^d$, and we seek sufficient conditions on the coefficients which ensure essential self-adjointness. The coefficients of the first order terms are only required to belong to $C^1(Ω)$ and there is no ellipticity condition. Our criterion writes as the completeness of an associated Riemannian structure which encodes the propagation velocities of the system. As an application we obtain sufficient conditions for confinement of energy for some wave propagation problems of classica…
In the present work we consider in $L^2(\mathbb{R}_+)$ the Schrödinger operator $\mathrm{H_{X,α}}=-\mathrm{\frac{d^2}{dx^2}}+\sum_{n=1}^{\infty}α_nδ(x-x_n)$. We investigate and complete the conditions of self-adjointness and nontriviality of deficiency indices for $\mathrm{H_{X,α}}$ obtained in \cite{karpiiKost}. We generalize the conditions found earlier in the special case $d_n:=x_{n}-x_{n-1}=1/n$, $n\in \mathbb{N}$, to a wider class of sequences $\{x_n\}_{n=1}^\infty$. Namely, for $x_n=\frac{1}{n^γ\ln^ηn}$ with $<γ,η>\in(1/2, 1)\times(-\infty,+\infty)\:\cup\:\{1\}\times(-\infty,1]$, the des…
Let $M$ be a complete Riemannian manifold and let $Ω^*(M)$ denote the space of differential forms on $M$. Let $d:Ω^*(M) \to Ω^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V(x):Ω^*(M)\to Ω^*(M)$ is a zero order differential operator) to be self-adjoint. Our result generalizes a theorem by Igor Oleinik about self-adjointness of a Schroedinger operator which acts on the space of scalar valued functions.
We consider first-order differential operators with locally bounded measurable coefficients on vector bundles with measurable coefficient metrics. Under a mild set of assumptions, we demonstrate the equivalence between the essential self-adjointness of such operators to a negligible boundary property. When the operator possesses higher regularity coefficients, we show that higher powers are essentially self-adjoint if and only if this condition is satisfied. In the case that the low-regularity Riemannian metric induces a complete length space, we demonstrate essential self-adjointness of the o…
We consider Schrödinger operators on possibly noncompact Riemannian manifolds, acting on sections in vector bundles, with locally square integrable potentials whose negative part is in the underlying Kato class. Using path integral methods, we prove that under geodesic completeness these differential operators are essentially self-adjoint on the space of smooth functions with compact support, and that the corresponding operator closures are semibounded from below. These results apply to nonrelativistic Pauli-Dirac operators that describe the energy of Hydrogen type atoms on Riemannian 3-manifo…
Sharp Asymptotics for $k$-Positivity Shifts under Singular Perturbations
本文提出了一种衡量量子映射偏离完全正性的层次结构方法，通过谱公式确定最优纠缠见证的构造条件。
By the Choi matrix criteria it is easy to determine if a specific linear matrix map is completely positive, but to establish whether a linear matrix map is positive is much less straightforward. In this paper we consider classes of linear matrix maps, determined by structural conditions on an associated matrix, for which positivity and complete positivity coincide. The basis of our proofs lies in a representation of $*$-linear matrix maps going back to work of R.D. Hill which enables us to formulate a sufficient condition in terms of surjectivity of certain bilinear maps.
Geometric Criteria for Essential Self-Adjointness of Discrete Hodge Laplacians on Weighted Simplicial Complexes
该研究针对带权图与单纯复形建立离散Hodge拉普拉斯算子的范数界，推导了本质自伴性，通过线复形归约得到正则图的尖锐界，并结合周期格的Floquet-Bloch计算确定了具体范数值。
We propose a general condition, to ensure essential self-adjointness for the Gauß-Bonnet operator, based on a notion of completeness as Chernoff. This gives essential self-adjointness of the Laplace operator both for functions or 1-forms on infinite graphs. This is used to extend Flanders result concerning solutions of Kirchhoff's laws.
The characteristic function for a contraction is a classical complete unitary invariant devised by Sz.-Nagy and Foias. Just as a contraction is related to the Szego kernel $k_S(z,w) = (1 - z\ow)^{-1}$ for $|z|, |w| < 1$, by means of $(1/k_S)(T,T^*) \ge 0$, we consider an arbitrary open connected domain $Ω$ in $\BC^n$, a complete Nevanilinna-Pick kernel $k$ on $Ω$ and a tuple $T = (T_1, ..., T_n)$ of commuting bounded operators on a complex separable Hilbert space $\clh$ such that $(1/k)(T,T^*) \ge 0$. For a complete Pick kernel the $1/k$ functional calculus makes sense in a beautiful way. It t…
Simple Hartree-type equations lead to dynamics of a subsystem that is not completely positive in the sense accepted in mathematical literature. In the linear case this would imply that negative probabilities have to appear for some system that contains the subsystem in question. In the nonlinear case this does not happen because the mathematical definition is physically unfitting as shown on a concrete example.
Complete positivity and distance-avoiding sets
本文引入完全正函数锥，将最大密度距离避免集刻画为凸优化问题的最优解，从而重新证明并改进了球面与欧几里得空间中距离避免集的相关结果。
The Biharmonic Heat Equation with General Dynamic Boundary Conditions
本文研究了在有界空间域中受动态边界条件影响的四阶热方程，结合共轭线性形式方法和半群理论，探讨了方程的定性性质，包括算子的自伴随性、解集的紧性以及谱性质，还分析了半群的生成和主要性质。
Entrywise powers of symmetric matrices preserving positivity, monotonicity or convexity with respect to the Loewner ordering arise in various applications, and have received much attention recently in the literature. Following FitzGerald and Horn [J. Math. Anal. Appl., 1977], it is well-known that there exists a critical exponent beyond which all entrywise powers preserve positive definiteness. Similar phase transition phenomena have also recently been shown by Hiai (2009) to occur for monotonicity and convexity. In this paper, we complete the characterization of all the entrywise powers below…
Complete monotonicity is a strong positivity property for real-valued functions on convex cones. It is certified by the kernel of the inverse Laplace transform. We study this for negative powers of hyperbolic polynomials. Here the certificate is the Riesz kernel in Garding's integral representation. The Riesz kernel is a hypergeometric function in the coefficients of the given polynomial. For monomials in linear forms, it is a Gel'fand-Aomoto hypergeometric function, related to volumes of polytopes. We establish complete monotonicity for sufficiently negative powers of elementary symmetric fun…
A noncommutative generalization of Hunter's positivity theorem
本文将Hunter关于偶数次完全齐次对称多项式正定性的定理推广到非交换情形，用厄米算子替代标量变量，并给出了精确下界与厄米平方和表示。
Fractional $k$-positivity: a continuous refinement of the $k$-positive scale
本文提出分数$k$-正性概念，通过连续细化$k$-正尺度，揭示了新的量子映射结构和计算方法。
With Charles Read we have introduced and studied a new notion of (real) positivity in operator algebras, with an eye to extending certain C*-algebraic results and theories to more general algebras. As motivation note that the `completely' real positive maps on C*-algebras or operator systems are precisely the completely positive maps in the usual sense; however with real positivity one may develop a useful order theory for more general spaces and algebras. This is intimately connected to new relationships between an operator algebra and the C*-algebra it generates. We have continued this work …
A new proof is presented of a theorem of L.~Gurvits, which states that the cone of positive block-Toeplitz matrices with matrix entries has no entangled elements. The proof of the Gurvits separation theorem is achieved by making use of the structure of the operator system dual of the operator system $C(S^1)^{(n)}$ of $n\times n$ Toeplitz matrices over the complex field, and by determining precisely the structure of the generators of the extremal rays of the positive cones of the operator systems $C(S^1)^{(n)}\omin\B(\H)$ and $C(S^1)_{(n)}\omin\B(\H)$, where $\H$ is an arbitrary Hilbert space a…
Positivity conditions on the annulus via the double-layer potential kernel
本文在环面上引入并研究了一组受Nagy和Foiaş的ρ-压缩类启发的算子类，通过双层位势积分算子的压缩性进行定义，并在完全压缩性假设下获得了完全刻画与K-谱估计。
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