Showing papers by "William Desmond Evans published in 1990"
TL;DR: In this paper, a characterisation of all the regularly solvable operators and their adjoints generated by a general differential expression in the domain of symmetric M is given. But the results of these results are restricted to self-adjoint realisations of M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results in this case.
Abstract: A characterisation is obtained of all the regularly solvable operators and their adjoints generated by a general differential expression in . The domains of these operators are described in terms of boundary conditions involving the solutions of Mu = λwu and the adjoint equation . The results include those of Sun Jiong [15] concerning self-adjoint realisations of a symmetric M when the minimal operator has equal deficiency indices: if the deficiency indices are unequal the maximal symmetric operators are determined by the results herein. Another special case concerns the J -self-adjoint operators, where J denotes complex conjugation, and for this we recover the results of Zai-jiu Shang in [16].
29 citations
TL;DR: In this article, the authors investigated the decay rate of solutions of a second-order elliptic equation in an arbitrary domain Ω⊆R n which satisfy zero boundary conditions and applied the results to a study of the associated heat semigroup in various weighted L 2 spaces.
Abstract: We investigate the rate of decay of solutions of a second-order elliptic equation tf=λf in an arbitrary domain Ω⊆R n which satisfy zero boundary conditions, and then apply the results to a study of the behaviour of the associated heat semigroup in various weighted L 2 spaces. Our results concern the behaviour at infinity and at points on the boundary δΩ of Ω, especially irregular ones like corners and cusps. The decay rates we obtain are expressed in terms of weighted L 2 estimates which are governed by the sharpness of the cusps and the properties of the coefficients of τ. Also, the meaning of the zero boundary condition at a cusp is addressed
15 citations
TL;DR: In this paper, the problem of predicting the existence of only a finite number of bound states corresponding to eigenvalues below the essential spectrum was considered for a class of second-order elliptic operators, including atomic-type N-body operators for N>2.
Abstract: We consider a class of second-order elliptic operators which includes atomic-type N-body operators for N>2. Our concern is the problem of predicting the existence of only a finite number of bound states corresponding to eigenvalues below the essential spectrum. We obtain a criterion which is natural for the problem and easy to apply as is demonstrated with various examples. While the criterion applies to general second-order elliptic operators, sharp results are obtained when the Hamiltonian of an atom with an infinitely heavy nucleus of charge Z and N electrons of charge 1 and mass 1/2 is considered
12 citations