Narrow Results

We consider the spectral problem associated with the Klein–Gordon equation for unbounded electric potentials such that the spectrum is contained in two disjoint real intervals related to positive and negative energies, respectively. If the two inner boundary points are eigenvalues, we show that these extremal eigenvalues are simple and possess strictly positive eigenfunctions. Examples of electric potentials satisfying these assumptions are given.

In this article, we extend Pólya's legendary inequality for the Dirichlet Laplacian to the fractional Laplacian. Pólya's argument is revealed to be a powerful tool for proving such extensions on tiling domains. As in the Dirichlet Laplacian case, Pólya's inequality for the fractional Laplacian on any bounded domain is still an open problem. Moreover, we also investigate the equivalence of several related inequalites for bounded domains by using the convexity, the Lieb–Aizenman procedure (the Riesz iteration), and some transforms such as the Laplace transform, the Legendre transform, and the Weyl fractional transform.

This paper deals with the massive three-dimensional Dirac operator coupled with a Lorentz scalar shell interaction supported on a compact smooth surface. The rigorous definition of the operator involves suitable transmission conditions along the surface. After showing the self-adjointness of the resulting operator, we switch to the investigation of its spectral properties, in particular, to the existence and non-existence of eigenvalues. In the case of an attractive coupling, we study the eigenvalue asymptotics as the mass becomes large and show that the behavior of the individual eigenvalues and their total number are governed by an effective Schrödinger operator on the boundary with an external Yang–Mills potential and a curvature-induced potential.

We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is the construction of generalized eigenfunctions of the time evolution operator. Roughly speaking, the generalized eigenfunctions are not square summable but belong to ${\ell }^{\infty }$-space on $Z$. Moreover, we derive a characterization of the set of generalized eigenfunctions in view of the time-harmonic scattering theory. Thus we show that the S-matrix associated with the quantum walk appears in the singularity expansion of generalized eigenfunctions.

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.

We give a lower bound for the first Dirichlet eigenvalue for a compact manifold with positive Ricci curvature in terms of the lower bound of the Ricci curvature and the interior radius. The result sharpens earlier estimates.

UHF flows are the flows obtained as inductive limits of flows on full matrix algebras. We will revisit universal UHF flows and give an explicit construction of such flows on a UHF algebra M_k^∞ for any k and also present a characterization of such flows. Those flows are UHF flows whose cocycle perturbations are almost conjugate to themselves.

For a bounded domain Ω with a piecewise smooth boundary in a complete Riemannian manifold M, we study eigenvalues of the Dirichlet eigenvalue problem of the Laplacian. By making use of a fact that eigenfunctions form an orthonormal basis of L²(Ω) in place of the Rayleigh–Ritz formula, we obtain inequalities for eigenvalues of the Laplacian. In particular, for lower order eigenvalues, our results extend the results of Chen and Cheng [D. Chen and Q.-M. Cheng, Extrinsic estimates for eigenvalues of the Laplace operator, J. Math. Soc. Japan 60 (2008) 325–339].

In this paper, we consider the boundary value problem

We show that for any two $n\times n$ matrices $X$ and $Y$ we have the inequality $s_j^2(I+XY)\le {\lambda }_j((I+X^{\ast }X)(I+Y^{\ast }Y)),$ where $s_j(T)$ and ${\lambda }_j(T)$ denote the decreasingly ordered singular values and eigenvalues of $T$. As an application, we show that for $2n\times 2n$ real positive definite matrices the symplectic eigenvalues $d_j,$ under some special conditions, satisfy the inequality $d_j(A+B)\ge d_j(A)+d_1(B)$.

In this article, we have determined the remainder term for Hardy–Sobolev inequality in H¹(Ω) for Ω a bounded smooth domain and studied the existence, non existence and blow up of first eigen value and eigen function for the corresponding Hardy–Sobolev operator with Neumann boundary condition.

In this paper, we prove the long-time existence of the CR Yamabe flow on the compact strictly pseudoconvex CR manifold with positive CR invariant. We also prove the convergence of the CR Yamabe flow on the sphere by proving that: the contact form which is pointwise conformal to the standard contact form on the sphere converges exponentially to a contact form of constant pseudo-Hermitian sectional curvature. We also show that the eigenvalues of some geometric operators are non-decreasing under the unnormalized CR Yamabe flow provided that the pseudo-Hermitian scalar curvature satisfies certain conditions.

The purpose of this paper is two-fold. Firstly, we state and prove a Berezin–Li–Yau-type estimate for the sums of eigenvalues of , the fractional Laplacian operators restricted to a bounded domain Ω ⊂ ℝ^d for d ≥ 2 and α ∈ (0, 2]. Secondly, we provide an improvement to this estimate by using a pure analytical approach.

In this paper we will use variational methods and limiting approaches to give a complete solution to the maximization problems of the Dirichlet/Neumann eigenvalues of the one-dimensional p-Laplacian with integrable potentials of fixed L¹-norm.

A key basis for seeking solutions of the Camassa–Holm equation is to understand the associated spectral problem

In [6], J. Dixmier posed six problems for the Weyl algebra A₁ over a field K of characteristic zero. Problems 3, 5,and 6 were solved respectively by Joseph and Stein [7]; the author [1]; and Joseph [7]. Problems 1, 2, and 4 are still open.

For an arbitrary algebra A, Dixmier's problem 6 is essentially aquestion: whether an inner derivation of the algebra A of the type ad f(a), a ∈ A, f(t) ∈ K[t], deg_t(f(t)) > 1, has a nonzero eigenvalue. We prove that the answer is negative for many classes of algebras (e.g., rings of differential operators on smooth irreducible algebraic varieties, all prime factor algebras of the universal enveloping algebra of a completely solvable algebraic Lie algebra ). This gives an affirmative answer (with a short proof) to an analogue of Dixmier's Problem 6 for certain algebras of small Gelfand–Kirillov dimension, e.g. the ring of differential operators on a smooth irreducible affine curve X, Usl(2), etc. (see [3] for details). In this paper an affirmative answer is given to an analogue of Dixmier's Problem 3 but for the ring .

For any positive integer $k$, let ${{𝒢}}_k$ denote the set of finite groups $G$ such that all Cayley graphs $\mathrm{\text{Cay}}(G,S)$ are integral whenever $\left|S\right|\le k$. Estélyi and Kovács [On groups all of whose undirected Cayley graphs of bounded valency are integral, Electron. J. Combin.21 (2014) #P4.45.] classified ${{𝒢}}_k$ for each $k\ge 4$. In this paper, we characterize the finite groups each of whose cubic Cayley graphs is integral. Moreover, the class ${{𝒢}}_3$ is characterized. As an application, the classification of ${{𝒢}}_k$ is obtained again, where $k\ge 4$.

The purpose of this paper is to establish a real Nullstellensatz, a Positivstellensatz and a Nichtnegativstellensatz for matrices over a commutative ring. The Stellensätze in this paper may be regarded as certain generalizations of the abstract Stellensätze for commutative rings.

We discuss an algebraic problem (Separation and Irreducibility Conjecture) which arises from the study of the nonlinear Schrödinger equation (NLS for short). This problem is about separation and irreducibility (over the ring of integers) of the characteristic polynomials of the graphs, describing blocks of a normal form for the NLS. For the cubic NLS the problem has been completely solved (see [C. Procesi, M. Procesi and B. Van Nguyen, The energy graph of the nonlinear Schrödinger equation, Rend. Lincei Mat. Appl.24(2) (2013) 229–301]), meanwhile for higher degree NLS it is still open, even in small dimensions (see [C. Procesi, The energy graph of the non-linear Schrödinger equation, open problems. Int. J. Algebra Comput.23(4) (2013) 943–962]). In this work, the author will give a partial answer for this problem, in particular, the author will prove Separation and Irreducibility Conjecture for NLS of arbitrary degree on one-dimensional and two-dimensional tori.

The zero-divisor graph $\mathrm{\Gamma }(R)$ of a commutative ring $R$ is the graph whose vertices are the nonzero zero divisors in $R$ and two vertices $x$ and $y$ are adjacent if and only if $xy=0$. We study the adjacency and Laplacian eigenvalues of the zero-divisor graph $\mathrm{\Gamma }(R)$ of a finite commutative von Neumann regular ring $R$. We prove that $\mathrm{\Gamma }(R)$ is a generalized join of its induced subgraphs. Among the $\left|Z{(R)}^{\ast }\right|$ eigenvalues (respectively, Laplacian eigenvalues) of $\mathrm{\Gamma }(R)$, exactly $|B(R)|-2$ are the eigenvalues of a matrix obtained from the adjacency (respectively, Laplacian) matrix of $\mathrm{\Gamma }(B(R))$-the zero-divisor graph of nontrivial idempotents in $R$. We also determine the degree of each vertex in $\mathrm{\Gamma }(R)$, hence the number of edges.