Inverse problems for differential operators on trees with general matching conditions

Sturm–Liouville differential operators on compact trees with general matching conditions in internal vertices are studied. We establish properties of the spectral characteristics and investigate three inverse problems of recovering the operator either from the so-called Weyl functions, or from discrete spectral data or from a system of spectra. For these inverse problems, we prove the corresponding uniqueness theorems and obtain procedures for constructing their solutions by the method of spectral mappings.     

Inverse nodal problems for the Sturm-Liouville differential operators on a star-type graph

We study inverse nodal problems for the second order differential operators on a star-type graph satisfying the standard matching conditions at the interior vertex. We prove uniqueness theorems and obtain a constructive solution to the inverse problems of this class. Original Russian Text Copyright ? 2009 Yurko V. A. The author was supported by the Russian Foundation for Basic Research (Grant 07-01-00003) and the National Science Council of Taiwan (Grant 07-01-92000-NSC-a). __________ Saratov. Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 50, No. 2, pp. 469–475, March–April, 2009.     

Boundary problems for systems of partial differential operators of mixed order

On a compact n-dimensional C^∞ manifold $\text{Ω}$ with boundary Γ we consider a matrix A of linear partial differential operators that are not all of the same order. For such systems it is not evident what to regard as Cauchy data (on Γ). We introduce a definition of the so-called reduced Cauchy data, ranging in a vector bundle over Γ, which allows us (1) to set up a Cauchy problem (well posed in the noncharacteristic, analytic case) without the usual extra compatibility condition; (2) to construct boundary projectors in the space of reduced Cauchy data for Douglis-Nirenberg elliptic systems, with which the study of boundary problems is carried over to a (global) study of a pseudodifferential problems over Γ. Further applications (e.g. to spectral theory, outlined in [4]) will be given elsewhere.     

Singular perturbation problems for linear parabolic differential operators of arbitrary order

We consider the first initial-boundary value problem for $\text{(}\text{?u}\text{?t}\text{) + ?L}{}_1\text{u + L}{}_0\text{u = f(L}{}_0$ and L₁ are linear elliptic partial differential operators) and investigate the properties of u(x, t, ?) as ? ↓ 0 in the maximum norm. Special attention is paid to approximations obtained by the boundary layer method. We use a priori estimates.     

Inverse problems for first-order integro-differential operators

Mathematical Notes - Inverse spectral problems for first-order integro-differential operators on a finite interval are studied, the properties of spectral characteristics are established, and...     

Inverse spectral problems for Bessel operators

We study the inverse spectral problem for a class of Bessel operators given in L₂(0,1) by the differential expression

     

Singular perturbation problems for linear elliptic differential operators of arbitrary order. II. Degeneration to first order operators

We consider the singular perturbation problem (Dirichlet problem), $\text{?L}{}_1\text{u + L}{}_0\text{u  ?L}{}_1\text{u + (}\text{?}\text{?x}{}_1\text{+ g)u = f,}\text{?}{}^8\text{u}\text{?n}{}^8\text{= Ψ}{}_8$. (1¹) The asymptotic character of approximation, as given by the boundary layer method, is proven in the maximum norm. The asymptotic character of the derivatives is also investigated.     

Nonlinear boundary value problems for second order differential equations with causal operators

In this paper we deal with second order differential equations with causal operators. To obtain sufficient conditions for existence of solutions we use a monotone iterative method. We investigate both differential equations and differential inequalities. An example illustrates the results obtained.     

Inverse problems on star-type graphs: differential operators of different orders on different edges

We study inverse spectral problems for ordinary differential equations on compact star-type graphs when differential equations have different orders on different edges. As the main spectral characteristics we introduce and study the so-called Weyl-type matrices which are generalizations of the Weyl function (m-function) for the classical Sturm-Liouville operator. We provide a procedure for constructing the solution of the inverse problem and prove its uniqueness.     

Singular perturbation problems for linear elliptic differential operators of arbitrary order. I. Degeneration to elliptic operators

Dirichlet problems of singular perturbation type for linear elliptic differential operators of arbitrary order are studied. The asymptotic validity of approximations constructed by the boundary layer method is demonstrated in the maximum norm by means of a priori estimates.     

Inverse problems and index formulae for Dirac operators

We consider a Dirac-type operator DP on a vector bundle V over a compact Riemannian manifold (M,g) with a non-empty boundary. The operator DP is specified by a boundary condition P(u_|∂M)=0 where P is a projector which may be a non-local, i.e., a pseudodifferential operator. We assume the existence of a chirality operator which decomposes L²(M,V) into two orthogonal subspaces X₊⊕X₋. Under certain conditions, the operator DP restricted to X₊ and X₋ defines a pair of Fredholm operators which maps X₊→X₋ and X₋→X₊ correspondingly, giving rise to a superstructure on V. In this paper we consider the questions of determining the index of DP and the reconstruction of and DP from the boundary data on ∂M. The data used is either the Cauchy data, i.e., the restrictions to ∂M×R₊ of the solutions to the hyperbolic Dirac equation, or the boundary spectral data, i.e., the set of the eigenvalues and the boundary values of the eigenfunctions of DP. We obtain formulae for the index and prove uniqueness results for the inverse boundary value problems. We apply the obtained results to the classical Dirac-type operator in M×C⁴, M⊂R³.     

A note on differential operators of infinite order

A sufficient condition for entire functions f and g to be such that the series ∑_m=0^∞f^(m)(0)g^(m)/m! represents an entire function is established; and in that case, the growth of the resulting function is described.     

Equivariant first order differential operators for parabolic geometries

Let G be a real semisimple Lie group, P a parabolic subgroup, V and W irreducible representations of P, G × p V and G × p W the associated homogeneous vector bundles. The G-equivariant first order differential operators from the first to the second bundle are determined and described using methods of Lie theory.     

Inverse spectral problems for Sturm-Liouville operators in impedance form

The spectral properties of Sturm-Liouville operators with an impedance in , for some p∈[1,∞), are studied. In particular, a complete solution of the inverse spectral problem is provided.