E. Prodan
A practical solution for the mathematical problem of functional calculus with Laplace-Beltrami operator on surfaces with axial symmetry is found. A quantitative analysis of the spectrum is presented.
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In quantum mechanics the kinetic energy term for a single particle is usually written in the form of the Laplace-Beltrami operator. This operator is a factor ordering of the classical kinetic energy. We investigate other relatively simple factor orderings and show that the {\it only} other solution for a conformally flat metric is the conformally invariant Laplace-Beltrami operator. For non-conformally-flat metrics this type of factor ordering fails, by just one term, to give the conformally invariant Laplace-Beltrami operator.
Frame Representation of the First-Order Part of the Laplace-Beltrami Operator
研究定向黎曼流形上拉普拉斯 - 贝尔特拉米算子一阶部分几何内容，通过正交标架相关构造及变换，消除其一阶部分并代以标量势，揭示其几何作用。
An explicit formula for the Laplace-Beltrami operator on the Stiefel manifold
本文基于约束流形的一般框架，推导了正交Stiefel流形上Laplace-Beltrami算子用环境欧氏坐标表示的显式公式，并将球面与特殊正交群作为特例。
We give a formula that represents magnetic Berezin transforms associated with generalized Bergman spaces as functions of the Laplace-Beltrami operator on the Bergman ball. In particular, we recover the result obtained by J. Peeter [J. Oper. Theory, 24, 1990].
We construct in this article a class of closed semi-bounded quadratic forms on the space of square integrable functions over a smooth Riemannian manifold with smooth boundary. Each of these quadratic forms specifies a semi-bounded self-adjoint extension of the Laplace-Beltrami operator. These quadratic forms are based on the Lagrange boundary form on the manifold and a family of domains parametrized by a suitable class of unitary operators on the boundary that will be called admissible. The corresponding quadratic forms are semi-bounded below and closable. Finally, the representing operators c…
We study combinatorial Laplacians on rectangular subgraphs of $ ε\mathbb{Z}^2 $ that approximate Laplace-Beltrami operators of Riemannian metrics as $ ε\rightarrow 0 $. These laplacians arise as follows: we define the notion of a Riemmanian metric structure on a graph. We then define combinatorial free field theories and describe how these can be regarded as finite dimensional approximations of scalar field theory. We focus on the Gaussian field theory on rectangular subgraphs of $ \mathbb{Z}^2 $ and study its partition function by computing the asymptotic determinant of the discrete laplacian…
In this paper we deal with the problems of the weak localization of the eigenfunction expansions related to Laplace-Beltrami operator on unit sphere. The conditions for weak localization of Fourier-Laplace series are investigated by comparing the Riesz and Cesaro methods of summation for eigenfunction expansions of the Laplace-Beltrami operator. It is shown that the weak localization principle for the integrable functions f(x) at the point x depends not only on behavior of the function around x but on the behavior of the function around diametrically opposite point \overline{x}.
For bicovariant differential calculi on quantum matrix groups a generalisation of classical notions such as metric tensor, Hodge operator, codifferential and Laplace-Beltrami operator for arbitrary k-forms is given. Under some technical assumptions it is proved that Woronowicz' external algebra of left-invariant differential forms either contains a unique form of maximal degree or it is infinite dimensional. Using Jucys-Murphy elements of the Hecke algebra the eigenvalues of the Laplace-Beltrami operator for the Hopf algebra O(SLq(N)) are computed.
We generalize the result of [Matveev-Topalov 2001] to all signatures: we show that in all signatures the Killing tensors constructed by projectively equivalent metrics correspond to commuting differential operators
Eigendecomposition of the Laplace-Beltrami operator is instrumental for a variety of applications from physics to data science. We develop a numerical method of computation of the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on a smooth bounded domain based on the relaxation to the Schrödinger operator with finite potential on a Riemannian manifold and projection in a special basis. We prove spectral exactness of the method and provide examples of calculated results and applications, particularly, in quantum billiards on manifolds.
Proposed is a new approach to finding exact solutions of nonlinear $p$-brane equations in $D$-dimensional Minkowski space based on the use of various initial value constraints. It is shown that the constraints $Δ^{(p)}\vec{x}=0$ and $Δ^{(p)}\vec{x}=-Λ(t,σ^r)\vec{x}$ give two sets of exact solutions.
Laplace-Beltrami operator on the orthogonal group in Cartesian coordinates
本研究借助嵌入梯度向量场方法推导约束流形上Laplace-Beltrami算子的环境坐标通用公式，进一步给出正交群上该算子的显式表达式，并完成相关函数的应用验证。
The self-adjointness of the reduced Hamiltonian operators arising from the Laplace-Beltrami operator of a complete Riemannian manifold through quantum Hamiltonian reduction based on a compact isometry group is studied. A simple sufficient condition is provided that guarantees the inheritance of essential self-adjointness onto a certain class of restricted operators and allows us to conclude the self-adjointness of the reduced Laplace-Beltrami operators in a concise way. As a consequence, the self-adjointness of spin Calogero-Sutherland type reductions of `free' Hamiltonians under polar actions…
The Laplacian on the Lie groups U(N) and SU(N) is given in a parametrized edition for practical purposes. The radial part is often seen in work on lattice gauge theory, but here is derived also the off-diagonal part which in SU(3) and U(3) is expressed via the well known Gell-Mann matrices but with a more easily memorized notation. Relations to I, U and V spin are also shown.
Upper block triangular form for the Laplace-Beltrami operator on the special orthogonal group acquired through a flag of trace polynomials spaces
该论文显式计算了特殊正交群上迹多项式的拉普拉斯算子，通过构造不变空间旗得到上块三角矩阵，从而迭代求解特征值，并给出SO(3)和SO(4)的完整特征值及特征标表示。
It is natural to investigate if the quantization of an integrable or superintegrable classical Hamiltonian systems is still integrable or superintegrable. We study here this problem in the case of natural Hamiltonians with constants of motion quadratic in the momenta. The procedure of quantization here considered, transforms the Hamiltonian into the Laplace-Beltrami operator plus a scalar potential. In order to transform the constants of motion into symmetry operators of the quantum Hamiltonian, additional scalar potentials, known as quantum corrections, must be introduced, depending on the Ri…
We investigate a mixed boundary value problem for the stationary heat transfer equation in a thin layer with a mid hypersurface $\mathcal{C}$ in $\mathbb{R}^3$ with the boundary. The main object is to trace what happens in $Γ$-limit when the thickness of the layer converges to zero. The limit Dirichlet BVP for the Laplace-Beltrami equation on the surface is described explicitly and we show how the Neumann boundary conditions in the initial BVP transform in the $Γ$-limit. For this we apply the variational formulation and the calculus of Günter's tangential differential operators on a hypersurfa…
Finite element method to solve the spectral problem for arbitrary self-adjoint extensions of the Laplace-Beltrami operator on manifolds with a boundary
本文提出一种计算带边界的 manifold 上大规模自共轭扩展拉普拉斯-贝尔特里算子谱的数值方案，基于其关联二次型特征，证明了算法收敛性，并通过二维实现与数值例子展示其统一处理多种边界条件的能力。
In this paper, we propose a new trace finite element method for the {Laplace-Beltrami} eigenvalue problem. The method is proposed directly on a smooth manifold which is implicitly given by a level-set function and require high order numerical quadrature on the surface. A comprehensive analysis for the method is provided. We show that the eigenvalues of the discrete Laplace-Beltrami operator coincide with only part of the eigenvalues of an embedded problem, which further corresponds to the finite eigenvalues for a singular generalized algebraic eigenvalue problem. The finite eigenvalues can be …
We review the theory of orthogonal separation of variables on pseudo-Riemannian manifolds of constant non-zero curvature via concircular tensors and warped products. We then apply this theory simultaneously to both the three-dimensional Hyperbolic and de Sitter spaces, obtaining an invariant classification of the thirty-four orthogonal separable webs on each space, modulo action of the respective isometry groups. The inequivalent coordinate charts adapted to each web are also determined and listed. The results obtained for Hyperbolic 3-space agree with those in the literature, while the result…
We study a discrete model of the Laplacian in $\mathbb{R}^2$ that preserves the geometric structure of the original continual object. This means that, speaking of a discrete model, we do not mean just the direct replacement of differential operators by difference ones but also a discrete analog of the Riemannian structure. We consider this structure on the appropriate combinatorial analog of differential forms. Self-adjointness and boundness for a discrete Laplacian are proved. We define the Green function for this operator and also derive an explicit formula of the one.
In these three lectures we will discuss some fundamental aspects of the theory of self-adjoint extensions of the covariant Laplace-Beltrami and Dirac operators on compact Riemannian manifolds with smooth boundary emphasizing the relation with the theory of global boundary conditions. Self-adjoint extensions of symmetric operators, specially of the Laplace-Beltrami and Dirac operators, are fundamental in Quantum Physics as they determine either the energy of quantum systems and/or their unitary evolution. The well-known von Neumann's theory of self-adjoint extensions of symmetric operators is n…
Axial Symmetric Navier Stokes Equations and the Beltrami /anti Beltrami spectrum in view of Physics Informed Neural Networks
本文研究了轴对称流的贝特拉米/反贝特拉米谱，并提出利用物理感知神经网络优化算法求解纳维-斯托克斯方程的理论基础。
On the Theory of Self-Adjoint Extensions of the Laplace-Beltrami Operator, Quadratic Forms and Symmetry
本文研究了紧黎曼流形上拉普拉斯-贝尔特里米算子的自共轭扩展构造及二次型的作用，提出了一种计算任意维度特征值和特征向量的数值方案，并探讨了对称算子下不变性的角色。
Pseudo-Riemannian Lie algebras with coisotropic ideals and integrating the Laplace-Beltrami equation on Lie groups
本文研究了一类左不变伪黎曼度量，其拉普拉斯-贝尔特里方程可简化为一阶偏微分方程并具有精确解。通过非交换积分方法，将方程转化为可显式积分的一阶线性偏微分方程，并揭示了非局部对称算子的存在。
The Heuristic Approach to General Relativity in the Laplace-Beltrami Formalism
本文在Laplace-Beltrami形式下，对爱因斯坦场方程进行全阶分析，通过变分方法和度量假设探索该形式在致密双星系统之外的适用性，并展示一阶分解中矢量场和标量场在弯曲时空中的力学机制。
It is known (E.L. Green (1997), O. Post (2003)) that for an arbitrary $m\in\mathbb{N}$ one can construct a periodic non-compact Riemannian manifold $M$ with at least $m$ gaps in the spectrum of the corresponding Laplace-Beltrami operator $-Δ_M$. In this work we want not only to produce a new type of periodic manifolds with spectral gaps but also to control the edges of these gaps. The main result of the paper is as follows: for arbitrary pairwise disjoint intervals $(\a_j,\b_j)\subset[0,\infty)$, $j=1,...,m$ ($m\in\mathbb{N}$), for an arbitrarily small $δ>0$ and for an arbitrarily large $L>0$ …
In this paper we study the almost everywhere convergence of the expansions related to the self-adjoint extension of the Laplace operator. The sufficient conditions for summability is obtained. For the orders of Riesz means, which greater than critical index $\frac{N-1}{2}$ we established the estimation for maximal operator of the Riesz means. Note that when order $α$ of Riesz means is less than critical index then for establish of the almost everywhere convergence requests to use other methods form proving negative results. We have constructed different method of summability of Laplace series,…
The classical and quantum particle on a flag manifold
针对旗流形上的测地线与Laplace–Beltrami算子谱问题，本文给出一族可显式求解的不变度量，并用自旋链型有限维空间逼近函数Hilbert空间以确定谱。
In a previous paper we established Cwikel-type estimates on noncommutative tori and used them to get analogues in this setting of the Cwikel-Lieb-Rozenblum (CLR) and Lieb-Thirring inequalities for negative eigenvalues of fractional Schrödinger operators. In this paper, we focus on "curved" NC tori, where the role of the usual Laplacian is played by Laplace-Beltrami operators associated with arbitrary Riemannian metrics. The Cwikel-type estimates of our previous paper are extended to pseudodifferential operators and powers of Laplace-Beltrami operators. There are several applications of these e…
On $\mathbb{Z}$-invariant self-adjoint extensions of the Laplacian on quantum circuits
本文重新审视了Riemann流形上Laplace-Beltrami算子存在群不变自伴扩张的准则，据此刻画了量子电路上Laplace-Beltrami算子在$\mathbb{Z}$作用下的不变自伴扩张，并通过研究不变性性质确定了具体例子的谱与广义特征函数。
Spectral Bounds on Hyperbolic 3-Manifolds: Associativity and the Trace Formula
该研究利用基于共形自举结合性条件与Selberg迹公式的两类谱恒等式，结合线性规划推导三维闭双曲流形与轨形上拉普拉斯算子低能谱的严格上界，得到通用谱隙界并验证界的紧性。
We compute the Wodzicki residue of the inverse of a conformally rescaled Laplace operator over a 4-dimensional noncommutative torus. We show that the straightforward generalization of the Laplace-Beltrami operator to the noncommutative case is not the minimal operator.
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperboli…
The critical case for the concentration of eigenfunctions on singular Riemannian manifolds
本文研究带边界的奇异黎曼流形上Laplace-Beltrami算子特征函数的集中现象，在临界情形下证明其平均密度分布于所有长度尺度，并给出精确描述。
We consider the Laplace-Beltrami operator in tubular neighbourhoods of curves on two-dimensional Riemannian manifolds, subject to non-Hermitian parity and time preserving boundary conditions. We are interested in the interplay between the geometry and spectrum. After introducing a suitable Hilbert space framework in the general situation, which enables us to realize the Laplace-Beltrami operator as an m-sectorial operator, we focus on solvable models defined on manifolds of constant curvature. In some situations, notably for non-Hermitian Robin-type boundary conditions, we are able to prove ei…
On the intertwining map between Coulomb and hyperbolic scattering
本文提出一个将库仑散射与双曲散射间本征函数空间联系起来的单位元算子，推广了福克变换到连续谱，并解释了两种设置中散射矩阵相同的原因。
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