Positive Differential Operators Research Articles

On the problem of oscillation properties of positive differential operators with singular coefficients

A criterion for a highly singular positive fourth-order operator with separable boundary conditions to have oscillation properties, as well as sufficient conditions for similar higher-order operators to have oscillation properties, are obtained.

On the Problem of Oscillation Properties of Positive Differential Operators with Singular Coefficients

Устанавливается критерий осцилляционности для высокосингулярного положительного оператора четвертого порядка с распадающимися граничными условиями, а также достаточные условия осцилляционности для аналогичных операторов более высоких порядков. Библиография: 7 названий.

The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients

We investigate the problem of similarity to a self-adjoint operator for J -positive Sturm– Liouville operators L = 1 ω ( − d2 dx2 +q ) with 2π -periodic coefficients q and ω . It is shown that if 0 is a critical point of the operator L , then it is a singular critical point. This gives us a new class of J -positive differential operators with the singular critical point 0 . Also, we extend the Beals and Parfenov regularity conditions for the critical point ∞ to the case of operators with periodic coefficients. Mathematics subject classification (2010): 47E05, 34B24, 34B09, 34L10, 47B50.

On the Determination of Eigenpairs of 2D Positive Differential Operators with Periodic Boundary Conditions

An easy-to-implement and versatile method for computing the behaviour of an electron propagating in 2D periodic quantum structures has been introduced. Numerical results for various parameters of the problem has been presented. The resulting eigenvalues directly lead to the Brillouine dispersion diagrams of the periodic structures studied.

A Restriction Theorem for the Heisenberg Group

Since the discovery by E. M. Stein (see Fefferman [5]) that the Fourier transform of an LP-function on R' has a well-defined restriction to the unit sphere S`' which is square integrable on Sn-1, if p is close enough to 1, various new restriction theorems have been proved (see e.g. [10], [22], [27], [29]). The importance of such theorems in harmonic analysis (see e.g. [5], [6], [24], [28]) as well as in the theory of partial differential equations ([18], [19], [25]) has become evident. It has turned out [29] that the crucial step in proving the Stein-Tomas restriction theorem is the following: Let a denote the surface measure of S n 1, and let K denote the Fourier transform of a. Then the operator Af = f * K is bounded from LP to LP', if p is sufficiently close to 1. 9 can be interpreted in terms of the Laplacian A on Rn: Let -A = foX dEx be the spectral decomposition of A. Then, formally, 9 = const. f106(1 X) dEx, where 6 denotes Dirac's point measure. Therefore, it is natural to study analogues for 9, where A is replaced by a more general positive differential operator L. In the case of second order elliptic operators on compact manifolds, this has been done by Sogge [25].

On semigroups generated by left-invariant positive differential operators on nilpotent Lie groups

On semigroups generated by left-invariant positive differential operators on nilpotent Lie groups

On positive differential operators (deficiency indices, factorization, perturbations)

SynopsisA short survey is given of some recent results. The perturbations and stability of discrete spectrum and the problems of resolvent convergence of densely or non-densely defined elliptic differential operators are considered. The Courant theorem on variations of the domain is generalized. In connection with Berezanskiy's theorem on essential self-adjointness, the test for the finite velocity of propagation is extended. The Frobenius and the Krein-Heinz-Rellich factorization theorems and the Etgen Pawlowski oscillation criterion are generalized for equations of any order with operator-valued coefficients. Brusentsev's recent example of a two term fourth order differential operator with deficiency index 4 is discussed.

Einschlie�ungsaussagen bei Differentialoperatoren zweiter Ordnung durch punktweise Ungleichungen

We consider non-inverse positive differential operators of the second order and show that error estimates can be obtained through methods of inverse positivity alone. This means a significant simplification if the operator has only one negative eigenvalue. In addition we show how to obtain Range-Domain implications for operators with mixed boundary conditions.

Boundary Value Problems for Symmetric Positive Differential Operators of Odd Order

Boundary Value Problems for Symmetric Positive Differential Operators of Odd Order

Eigenfunction Expansions for Non-Symmetric Partial Differential Operators, I

In two previous papers under the same title ([7], [8]), the writer has constructed a theory of eigenfunction expansions of a generalized PlancherelWeyl type for general classes of realizations of a partial differential operator L on an open subset G of El, the subnormal realizations in [7] and the decomposable realizations in [8]. These results were established without any restrictions upon G, the type of L, or on the smoothness or behaviour of the coefficients of L at the boundary of G or at infinity. Results of the same type were obtained for expansions in solutions of the equations (L CB) u = 0 for positive differential operators B. It is the purpose of the present paper to extend and generalize these results in two separate directions. In the first place, it is very difficult to determine in any concrete case whether a particular realization of a partial differential operator under specific boundary conditions falls within either of the two classes which we have studied, except in the relatively simple case in which L is symmetric. For this reason, we have begun the study elsewhere ([5], [61, [9] ) of the simplest of the basically non-symmetric cases, elliptic operators on unbounded domains under Dirichlet boundary conditions, but the results obtained at this level of generality do not as yet permit us to apply the theory as previously constructed. On the other hand, for the regular case on a bounded domain, the completeness of the eigenfunctions of the Dirichlet problem was proved by the writer in [4] by a method which extends to the whole class of regular variational boundary-value problems. It is of interest, therefore, to obtain some results on the completeness of the eigenfunctions of L for L singular but elliptic, without any assumption on the boundedness or smoothness in the large of its coefficients. In section 1, we show that if f is a distribution with compact support in G and if L satisfies certain simple local conditions at each point of G, then the orthogonality of f to each eigenfunction