Bobo Hua, Jun Masamune, Radosław K. Wojciechowski
We discuss connections between the essential self-adjointness of a symmetric operator and the constancy of functions which are in the kernel of the adjoint of the operator. We then illustrate this relationship in the case of Laplacians on both manifolds and graphs. Furthermore, we discuss the Green's function and when it gives a non-constant harmonic function which is square integrable.
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We derive Povzner--Wienholtz-type self-adjointness results for $m\times m$ matrix-valued Sturm--Liouville operators $T=R^{-1}\big[-\f{d}{dx}P\f{d}{dx}+Q\big]$ in $L^2((a,b);Rdx)^m$, $m\in\bbN$, for $(a,b)$ a half-line or $\bbR$.
We define the magnetic Schrödinger on an infinite graph by the data of a magnetic field, some weights on vertices and some weights on edges . We discuss essential self-adjointness of this operator for graphs of bounded degree. The main result is a discrete version of a result of two authors of the present paper.
We study the essential self-adjointness for real principal type differential operators. Unlike the elliptic case, we need geometric conditions even for operators on the Euclidean space with asymptotically constant coefficients, and we prove the essential self-adjointness under the null non-trapping condition.
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph. Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the weighted graph Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us …
Let $Ω$ be a bounded domain in $R^n$ with $C^2$-smooth boundary of co-dimension 1, and let $H=-Δ+V(x)$ be a Schrödinger operator on $Ω$ with potential V locally bounded. We seek the weakest conditions we can find on the rate of growth of the potential V close to the boundary which guarantee essential self-adjointness of H on $C_0^\infty(Ω)$. As a special case of an abstract condition, we add optimal logarithmic type corrections to the known condition $V(x)\geq \frac{3}{4d(x)^2}$, where $d(x)=dist(x,\partialΩ)$. The constant 1 in front of each logarithmic term in Theorem 2 is optimal. The proof…
We prove a general essential self-adjointness criterion for sub-Laplacians on complete sub-Riemannian manifolds, defined with respect to singular measures. As a consequence, we show that the intrinsic sub-Laplacian (i.e. defined w.r.t. Popp's measure) is essentially self-adjoint on the equiregular connected components of a sub-Riemannian manifold. This result holds under mild regularity assumptions on the singular region, and when the latter does not contain characteristic points.
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…
We obtain several essential self-adjointness conditions for a Schroedinger type operator D*D+V acting in sections of a vector bundle over a manifold M. Here V is a locally square-integrable bundle map. Our conditions are expressed in terms of completeness of certain metrics on M; these metrics are naturally associated to the operator. We do not assume a priori that M is endowed with a complete Riemannian metric. This allows us to treat e.g. operators acting on bounded domains in the euclidean space. For the case when the principal symbol of the operator is scalar, we establish more precise res…
The essential self-adjointness of the wave operator on radiative spacetimes
本文证明辐射时空中达朗贝尔算子的本质自伴性，涵盖引力辐射扰动的闵可夫斯基时空，利用微局部方法和Taira的分布正则性结果。
We give sufficient conditions for essential self-adjointness of magnetic Schrödinger operators on locally finite graphs. Two of the main theorems of the present paper generalize recent results of Torki-Hamza.
We prove essential self-adjointness for semi-bounded below magnetic Schrödinger operators on complete Riemannian manifolds with a given positive smooth measure which is fixed independently of the metric. Some singularities of the scalar potential are allowed. This is an extension of the Povzner--Wienholtz--Simader theorem. The proof uses the scheme of Wienholtz but requires a refined invariant integration by parts technique, as well as a use of a family of cut-off functions which are constructed by a non-trivial smoothing procedure due to Karcher.
We consider weighted graphs, we equip them with a metric structure given by a weighted distance, and we discuss essential self-adjointness for weighted graph Laplacians and Schrödinger operators in the metrically non complete case.
We study the question of magnetic confinement of quantum particles on the unit disk $\ID$ in $\IR^2$, i.e. we wish to achieve confinement solely by means of the growth of the magnetic field $B(\vec x)$ near the boundary of the disk. In the spinless case we show that $B(\vec x)\ge \frac{\sqrt 3}{2}\cdot\frac{1}{(1-r)^2}-\frac{1}{\sqrt 3}\frac{1}{(1-r)^2\ln \frac{1}{1-r}}$, for $|\vec x|$ close to 1, insures the confinement provided we assume that the non-radially symmetric part of the magnetic field is not very singular near the boundary. Both constants $\frac{\sqrt 3}{2}$ and $-\frac{1}{\sqrt …
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 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 study $H=D^*D+V$, where $D$ is a first order elliptic differential operator acting on sections of a Hermitian vector bundle over a Riemannian manifold $M$, and $V$ is a Hermitian bundle endomorphism. In the case when $M$ is geodesically complete, we establish the essential self-adjointness of positive integer powers of $H$. In the case when $M$ is not necessarily geodesically complete, we give a sufficient condition for the essential self-adjointness of $H$, expressed in terms of the behavior of $V$ relative to the Cauchy boundary of $M$.
Essential self-adjointness of non-semibounded Schrödinger operators on infinite graphs
本文研究无穷加权图上未必下半有界的离散薛定谔算子的本质自伴性，给出了相关充分条件，并推导得到势函数满足特定二次下界时算子本质自伴的推论。
Self-adjoint extensions of singular Sturm-Liouville operators on graphs and Weyl's law
本研究针对图上二阶Sturm-Liouville型微分算子的自伴延拓问题，建立了自伴性与哈密顿向量场非完整轨迹的关联，定义了顶点处的Kirchhoff条件以保证自伴延拓，并推导了谱渐近的Weyl定律。
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.
Let $M$ be a strongly pseudoconvex complex manifold which is also the total space of a principal $G$-bundle with $G$ a Lie group and compact orbit space $\bar M/G$. Here we investigate the $\bar\partial$-Neumann Laplacian on $M$. We show that it is essentially self-adjoint on its restriction to compactly supported smooth forms. Moreover we relate its spectrum to the existence of generalized eigenforms: an energy belongs to $σ(\square)$ if there is a subexponentially bounded generalized eigenform for this energy. Vice versa, there is an expansion in terms of these well-behaved eigenforms so tha…
The purpose of this note is to present several criteria for essential self-adjointness. The method is based on ideas due to Shubin. This note is divided into two parts. The first part deals with symmetric first order systems on the line in the most general setting. Such a symmetric first order system of differential equations gives rise naturally to a symmetric linear relation in a Hilbert space. In this case even regularity is nontrivial. We will announce a regularity result and discuss criteria for essential self-adjointness of such systems. This part is based on joint work with Mark Malamud…
Essential Self-Adjointness of the Geometric Deformation Operator on a Compact Interval
本文在紧区间上定义了由光滑函数构造的二阶微分算子，通过验证对称性和计算为零的亏指数证明其本质自伴性，为后续谱分析奠定基础。
Let $G$ be a Métivier group and let $N$ be any homogeneous norm on $G$. For $α>0$ denote by $w_α$ the function $e^{-N^α}$ and consider the weighted sub-Laplacian $\mathcal{L}^{w_α}$ associated with the Dirichlet form $ϕ\mapsto \int_{G} |\nabla_\mathcal{H}ϕ(y)|^2 w_α(y)\, dy$, where $\nabla_\mathcal{H}$ is the horizontal gradient on $G$. Consider $\mathcal{L}^{w_α}$ with domain $C_c^\infty$. We prove that $\mathcal{L}^{w_α}$ is essentially self-adjoint when $α\geq 1$. For a particular $N$, which is the norm appearing in $\mathcal{L}$'s fundamental solution when $G$ is an H-type group, we prove …
The notions of magnetic difference operator defined on weighted graphs or magnetic exterior derivative are discrete analogues of the notionof covariant derivative on sections of a fibre bundle and its extension on differential forms. In this paper, we extend this notion to certain 2-simplicial complexes called triangulations, in a manner compatible with changes of gauge. Then we study the magnetic Gauss-Bonnet operator naturally defined in this context and introduce the geometric hypothesis of $χ-$completeness which ensures the essential self-adjointness of this operator. This gives also the e…
The Landau-Dirac operator with shell interactions: self-adjointness and clustering
本文研究带壳相互作用的二维朗道-狄拉克算子，分析其自伴性，证明非临界情形下本征值随曲线对数容量在朗道-狄拉克能级处积累，还得到外边界值问题的本征值聚类结果。
We give a sufficient condition for the essential self-adjointness of a perturbation of the square of the magnetic Laplacian on an infinite weighted graph. The main result is applicable to graphs whose degree function is not necessarily bounded. The result allows perturbations that are not necessarily bounded from below by a constant.
We develop relative oscillation theory for general Sturm-Liouville differential expressions of the form \[ \frac{1}{r}\left(-\frac{\mathrm d}{\mathrm dx} p \frac{\mathrm d}{\mathrm dx} + q\right) \] and prove perturbation results and invariance of essential spectra in terms of the real coefficients $p$, $q$, $r$. The novelty here is that we also allow perturbations of the weight function $r$ in which case the unperturbed and the perturbed operator act in different Hilbert spaces.
In the context of a weighted graph with vertex set $V$ and bounded vertex degree, we give a sufficient condition for the essential self-adjointness of the operator $Δ_σ+W$, where $Δ_σ$ is the magnetic Laplacian and $W\colon V\to\mathbb{R}$ is a function satisfying $W(x)\geq -q(x)$ for all $x\in V$, with $q\colon V\to [1,\infty)$. The condition is expressed in terms of completeness of a metric that depends on $q$ and the weights of the graph. The main result is a discrete analogue of the results of I. Oleinik and M. A. Shubin in the setting of non-compact Riemannian manifolds.
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 extend relative oscillation theory to the case of Sturm--Liouville operators $H u = r^{-1}(-(pu')'+q u)$ with different $p$'s. We show that the weighted number of zeros of Wronskians of certain solutions equals the value of Krein's spectral shift function inside essential spectral gaps.
The main purpose of this paper is to investigate the formal deficiency indices ${\cal N}_{\pm}(I)$ of a symmetric first order system $$ Jf'+Bf=λ{\cal H} f $$ on an interval $I$, where $I=\mathbb{R}$ or $I=\mathbb{R}_\pm.$ Here $J,B,{cal H}$ are $n\times n$ matrix valued functions and the Hamiltonian ${\cal H}\ge 0$ may be singular even everywhere. We obtain two results for such a system to have minimal numbers ${\cal N}_\pm(\mathbb{R})=0$ (resp. ${\cal N}_\pm(\mathbb{R}_\pm)=n$) and a criterion for their maximality ${\cal N}_{\pm}(\mathbb{R}_+)=2n.$ Some conditions for a canonical system to ha…
It is shown the stability of the essential self-adjointness, and an inclusion of the essential spectra of Laplacians under the change of Riemannian metric on a subset K of M. The set K may have infinite volume measured with the new metric and its completion may contain a singular set such as fractal, to which the metric is not extendable.
Essential self-adjointness of strongly singular homogeneous polyharmonic operators
研究强奇异齐次多项调和算子在 $ L^2 $ 空间中的本质自 adjoint 性，推导了不同维度下算子自 adjoint 的临界常数条件。
We study perturbations of the self-adjoint periodic Sturm--Liouville operator \[ A_0 = \frac{1}{r_0}\left(-\frac{\mathrm d}{\mathrm dx} p_0 \frac{\mathrm d}{\mathrm dx} + q_0\right) \] and conclude under $L^1$-assumptions on the differences of the coefficients that the essential spectrum and absolutely continuous spectrum remain the same. If a finite first moment condition holds for the differences of the coefficients, then at most finitely many eigenvalues appear in the spectral gaps. This observation extends a seminal result by Rofe-Beketov from the 1960s. Finally, imposing a second moment c…
We study the relationship between the adjacency matrix and the discrete Laplacian acting on 1-forms. We also prove that if the adjacency matrix is bounded from below it is not necessarily essentially self-adjoint. We discuss the question of essential self-adjointness and the notion of completeness.
We study positivity, localization of binding and essential self-adjointness properties of a class of Schroedinger operators with many anisotropic inverse square singularities, including the case of multiple dipole potentials.
Essential self-adjointness of even-order, strongly singular, homogeneous half-line differential operators
本文证明偶数阶强奇异齐次半线微分算子的本质自伴性存在阈值常数c_n，并给出其下界及前几项显式值，揭示其与多项式根及代数数性质的复杂联系。
Positivity, essential self-adjointness, and spectral properties of a class of Schroedinger operators with multipolar inverse-square potentials are discussed. In particular a necessary and sufficient condition on the masses of singularities for the existence of at least a configuration of poles ensuring the positivity of the associated quadratic form is established.
Eigenvalues in the essential spectrum of a weighted Sturm-Liouville operator are studied under the assumption that the weight function has one turning point. An abstract approach to the problem is given via a functional model for indefinite Sturm-Liouville operators. Algebraic multiplicities of eigenvalues are obtained. Also, operators with finite singular critical points are considered.
The number of self-adjoint extensions of a symmetric operator acting on a complex Hilbert space is characterized by its deficiency indices. Given a locally finite unoriented simple tree, we prove that the deficiency indices of any discrete Schrödinger operator are either null or infinite. We also prove that almost surely, there is a tree such that all discrete Schrödinger operators are essentially self-adjoint. Furthermore, we provide several criteria of essential self-adjointness. We also adress some importance to the case of the adjacency matrix and conjecture that, given a locally finite un…
On two-dimensional Dirac operators with $δ$-shell interactions supported on unbounded curves with straight ends
本文研究支撑在具有直端点无界曲线上的二维Dirac算子，刻画了带静电、Lorentz标量和反常磁δ-壳相互作用的自伴实现与本质谱，并证明了几何诱导离散特征值的存在性。
We introduce the weighted graph Laplacian and the notion of Schrödinger operator on a locally finite weighted graph . Concerning essential self-adjointness, we extend Wojciechowski's and Dodziuk's results for graphs with vertex constant weight. The main result in this work states that on any metrically complete weighted graph with bounded degree, the Laplacian is essentially self-adjoint and the same holds for the Schrödinger operator provided the associated quadratic form is bounded from below. We construct for the proof a strictly positive and harmonic function which allows us to write any S…
In the first part of this paper, we show that the Cauchy problem for wave propagation in some static spacetimes presenting a singular time-like boundary is well posed, if we only demand the waves to have finite energy, although no boundary condition is required. This feature does not come from essential self-adjointness, which is false in these cases, but from a different phenomenon that we call the alternative well-posedness property, whose origin is due to the degeneracy of the metric components near the boundary. Beyond these examples, in the second part, we characterize the type of degener…
We consider a non self-adjoint Laplacian on a directed graph with non symmetric weights on edges. We give a criterion for the m-accretiveness and the m-sectoriality of this Laplacian. Our results are based on a comparison of this operator with its symmetric part for which we can apply dierent results concerning essential self-adjointness of a symmetric Laplace operator on an innite graph. This gives results on the heat operator related to our non-symmetric Laplacian.
The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the $J$-selfadjoint operator $JA$ is real and it follows that an interval $(a,b)\subset\mathbb R^+$ is a gap in the essential spectrum of $A$ if and only if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of the $J$-selfadjoint operator $JA$. As one of the main results it is …
Spectral Theory for Sturm-Liouville operators with measure potentials through Otelbaev's function
本文利用 Otelbaev 函数分析带测度位势的 Sturm-Liouville 算子，建立特征值谱分布与本质谱下界的双边估计，并给出谱离散性及预解式属于 Schatten 类的判据。
The indefinite Sturm-Liouville operator $A = (\sgn x)(-d^2/dx^2+q(x))$ is studied. It is proved that similarity of $A$ to a selfadjoint operator is equivalent to integral estimates of Cauchy integrals. Also similarity conditions in terms of Weyl functions are given. For operators with a finite-zone potential, the components $\Aess$ and $\Adisc$ of $A$ corresponding to essential and discrete spectrums, respectively, are considered. A criterion of similarity of $\Aess$ to a selfadjoint operator is given in terms of Weyl functions for the Sturm-Liouville operator $-d^2/dx^2+q(x)$ with a finite-zo…
Perturbation and spectral theory for singular indefinite Sturm-Liouville operators
本文研究了权函数在奇异端点附近变号的奇异不定 Sturm-Liouville 算子，分析了其本质谱及离散特征值的聚集性质，并探讨了不定权扰动与周期算子的 $L^1$ 扰动。
Spectral properties of Sturm-Liouville operators on infinite metric graphs
本文研究无限度量图上 Sturm-Liouville 算子的谱性质，将经典 Allegretto-Piepenbrink 定理推广到量子图，并证明 Persson 型定理以刻画本质谱下确界。
As is known, for each fixed $ν\in\{0,1\},$ the spectra of two operators generated by $-y''(x)+q(x)y(x-a)$ and the boundary conditions $y^{(ν)}(0)=y^{(j)}(π)=0,$ $j=0,1,$ uniquely determine the complex-valued square-integrable potential $q(x)$ vanishing on $(0,a)$ as soon as $a\in[2π/5,π).$ Meanwhile, it actually became the main question of the inverse spectral theory for Sturm-Liouville operators with constant delay whether the uniqueness holds also for smaller values of $a.$ Recently, a negative answer was given by the authors [Appl. Math. Lett. 113 (2021) 106862] for $a\in[π/3,2π/5)$ in the …
On two properties of positively perturbed discrete Schrödinger operators
本文研究离散加权图上非负势扰动对 Schrödinger 算子自伴性的影响，证明型和与 Friedrichs 扩张一致，并给出本质自伴性的稳定性条件。
Sturm-Liouville Problems And Global Bounds By Small Control Sets And applications to quantum graphs
本文发展了一维向量值自伴算子的Logvinenko-Sereda理论，通过小控制集上的范数给出特征函数的上界，并推广至带磁势的薛定谔算子，在量子图理论中有重要应用。
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