Functional Analysis and Its Applications最新文献

We prove two-term spectral asymptotics for the Riesz means of the eigenvalues of the Laplacian on a Lipschitz domain with Robin boundary conditions. The second term is the same as in the case of Neumann boundary conditions. This is valid for Riesz means of arbitrary positive order. For orders at least one and under additional assumptions on the function determining the boundary conditions, we derive leading order asymptotics for the difference between Riesz means of Robin and Neumann eigenvalues.

For rescaled additive functionals of the sine process, upper bounds are obtained for their speed of convergence to the Gaussian distribution with respect to the Kolmogorov–Smirnov metric. Under scaling with coefficient (R>1), the Kolmogorov–Smirnov distance is bounded from above by (c/log R) for a smooth function, and by (c/R) for a function holomorphic in a horizontal strip.

In this paper, we consider the Bloch eigenvalues and spectrum of the non-self-adjoint differential operator (L) generated by the differential expression of odd order (n) with periodic (mathcal{PT})-symmetric coefficients, where (n>1). We study the localizations of the Bloch eigenvalues and the structure of the spectrum. Moreover, we find conditions on the norm of the coefficients under which the spectrum of (L) is purely real and coincides with the real line.

We show that for all (n,p>1), there exists a unitary operator (U) such that the tensor product (Uotimes U^potimesdotsotimes U^{p^{n-1}}) is a unitary operator with simple Lebesgue spectrum. Moreover, there exists an ergodic automorphism (T) such that the spectrum of (Todot T) is simple, while the spectrum of (Totimes Totimes T) is absolutely continuous.

We consider holomorphic realizations in (mathbb C^4) for a large family of 7-dimensional Lie algebras containing a 6-dimensional nilradical and one or two 4-dimensional abelian subalgebras. We show that for these Lie algebras, a natural condition of having tubular Levi-nondegenerate 7-dimensional orbits is rarely compatible with a straightened basis of one of the abelian subalgebras. In many cases, this incompatibility follows easily from the structure and properties of abelian ideals in 4-dimensional subalgebras of the algebras in question.

In the present work, we consider solvability of the generalized Dirichlet problem for the linear elliptic differential equation (Lu=f), where (L=Delta +langle B(x),nablarangle+c(x)) is a linear operator, ((B(x)) is a vector field of class (mathrm{C}(mathcal{M})), (c(x)leq0), (c(x)in mathrm{C}(mathcal{M}))), considered on a non-compact Riemannian manifold ((mathcal{M},g)). We develop the approach to this problem, based on equivalence classes, introduced by E. A. Mazepa, which allows to state the problem on non-compact manifolds in the absence of a natural geometric compactification. We introduce and study linear spaces (mathrm{CM}_b) and (mathrm{CM}) of such classes. We give a version of the well-known Perron’s method with boundary data in these classes, and establish signs of (L)-parabolicity and (L)-hyperbolicity of the ends of the manifold (mathcal{M}) depending on their geometric structure. The signs of hyperbolicity of a manifold play a key role in justifying solvability of the Dirichlet problem, while signs of parabolicity are important for establishing theorems of Liouville type for the manifold.

The aim of this note is to share the observation that the set of elementary operations of Turing on lattice knots can be reduced to just one type of simple local switches.

A Lax pair for the field analogue of the classical spin elliptic Calogero–Moser model is proposed. Namely, using the previously known Lax matrix, we suggest an ansatz for the accompanying matrix. The presented construction is valid when the matrix of spin variables ({mathcal S}inoperatorname{Mat}(N,mathbb C)) satisfies the condition ({mathcal S}^2=c_0{mathcal S}) with some constant (c_0inmathbb C). It is shown that the Lax pair satisfies the Zakharov–Shabat equation with unwanted term, thus providing equations of motion on the unreduced phase space. The unwanted term vanishes after additional reduction. In the special case (operatorname{rank}(mathcal S)=1), we show that the reduction provides the Lax pair of the spinless field Calogero–Moser model obtained earlier by Akhmetshin, Krichever, and Volvovski.

We consider two (S)-dual hyperspherical varieties of the group (G_2timesoperatorname{SL}(2)): an equivariant slice for (G_2) and the symplectic representation of (G_2 times operatorname{SL}_2) in the odd part of the basic classical Lie superalgebra (mathfrak{g}(3)). For these varieties, we check the equality of the numbers of irreducible components of their Lagrangian subvarieties (zero levels of the moment maps of Borel subgroups’ actions), conjectured by M. Finkelberg, V. Ginzburg, and R. Travkin.

In this article, we use the inverse function theorem for Banach spaces to interpolate a given real analytic spacelike curve (a) in the Lorentz–Minkowski space (mathbb{L}^3) to another real analytic spacelike curve (c), which is “close” enough to (a) in a certain sense, by constructing a maximal surface containing them. Throughout this study, the Björling problem and Schwarz’s solution to it play pivotal roles.