Thomas T. Read

The linearization of boundary eigenvalue problems and reproducing kernel Hilbert spaces

The boundary eigenvalue problems for the adjoint of a symmetric relation S in a Hilbert space with finite, not necessarily equal, defect numbers, which are related to the selfadjoint Hilbert space extensions of S are characterized in terms of boundary coefficients and the reproducing kernel Hilbert spaces they induce.

Positivity and discrete spectra for differential operators

the property that the minimal operator generated by L on an open interval I is positive definite. We shall then apply this to give several different sets of conditions sufficient for self-adjoint operators generated by L on a weighted Hilbert space Lf&a, b) to be bounded below with discrete spectrum Here b may be either a finite or infinite singularity. We assume throughout that the coefficient pi in (1.1) is a real-valued function on I with with py-li locally absolutely continuous and that p, > 0.‘ The characterization of positivity appears as part (4) of Theorem 2.1. It takes the form of IZ inequalities involving decompositions of the coefficients and is a direct generalization of Theorem 2.1(c) of Read [ 131. In Section 3 the link with discreteness of the spectrum is outlined, and the characterization is applied to give a criterion for discreteness of the spectrum of some fourth-order operators with nonnegative coefficients in which further assumptions about each coefficient are made only on a sequence of disjoint intervals (Theorem 3.6). This criterion comes very close to including an example of Hinton and Lewis [8] where the spectrum is not discrete. Our principal result on discreteness of the spectrum is Theorem 5.1. It is somewhat in the spirit of the well-known result of Molchanov [lo] (see also Glazman [7, p. 901) for two-term expressions in that the bottom coefficient p. is assumed to approach infinity on the average as x approaches b, but differs in allowing intermediate terms and in allowing much more irregular behavior by all coefficients. In Section 6 we use an extension of Theorem 2 of Hinton and Lewis ]S] to modify Theorem 5.1 so that any coefficient pk, k = 0, I,..., ra 1, may play 1 ~22.0396/~2/010~1-27~02.OOi~

Balanced growth without constant returns to scale

Abstract The existence of a unique, relatively stable balanced growth path is established for the nonlinear input-output model x ( t +1)= H ( x ( t )) in R n under hypotheses which allow the growth rate on the balanced growth path to fluctuate. A class of examples in R 2 involving such fluctuations is constructed, and it is shown that the hypotheses of the theorems are satisfied in a situation which may be interpreted as eventually decreasing returns to scale.

The number of Dirichlet solutions of a fourth order differential equation

It is shown that the equation (p 2 y”)”–(p 1 y’)’+ p 0 y = 0 has exactly two linearly independent solutions on [0,∞) with finite Dirichlet integral when the coefficients are nonnegative and p 2 satisfies a condition which includes all nondecreasing functions. An inequality for the Dirichlet form is derived and used to extend characterizations of the domains of certain self-adjoint operations associated with the differential expression to arbitrary symmetric boundary conditions at 0.

Essential Self-Adjointness for Powers of Schrödinger Operators

Conditions are given which are sufficient for the minimal operator T 0 associated with a second order elliptic differential expression on n to be essentially self-adjoint, and for all powers of T 0 to be self-adjoint. The conditions for T 0 which are similar to the most general conditions from the one-dimensional case, allow some coefficients not covered by other criteria and avoid some of their technical assumptions. The conditions for powers of T 0 allow faster growth by the leading terms than previous results.

Dirichlet Solutions of Fourth Order Differential Equations

The equation y (4) - (p 1 y′)′ + p 0 y = 0 has exactly two linearly independent solutions on [0,∞) with finite Dirichlet integral. Some applications to the determination of the domains of self - adjoint operators associated with the differential expression and to the minimization of a quadratic functional are discussed.