We consider the perturbations of the harmonic oscillator operator by an odd pair of point interactions, z(δx−b−δx+b). The spectrum of such operators is analyzed as a set of roots of the properly constructed entire (meromorphic) function. If z = ir, r real, as r → ∞, the number of non-real eigenvalues tends to infinity.
For fixed positive alpha and beta, and a fixed integer n, n >= 2, we consider the family of matrices diag(a(1), a(2), . . . , a(n)) - diag(b(1), b(2), . . ., b(n)) Sigma(*), where all the a(k)'s and b(k)'s are positive, the geometric mean of the a(k)'s and b(k)'s must be a and beta, respectively, and Sigma(*) denotes the permutation matrix corresponding to the cycle (1, 2, . . . , n). C. Johnson, Z. Price, and I. Spitkovsky conjectured that in this family, the number of eigenvalues in the left half plane is maximized by alpha I - beta Sigma(*); we prove this conjecture. Moreover, the complete range of possibilities for the number of eigenvalues in the left half-plane is demonstrated: if alpha < beta, then any odd number between 1 and the maximum, inclusive, is attainable. (C) 2017 Elsevier Inc. All rights reserved.
We study the positive-definiteness of a family of \(L^2(\mathbb {R})\) integral operators with kernel \(K_{t, a} (x, y) = \pi ^{-1} (1 + (x - y)^2+ a(x^2 + y^2)^t)^{-1}\), for \(t > 0\) and \(a > 0\). For \(0 < t \le 1\) and \(a > 0\), the known theory of positive-definite kernels and conditionally negative-definite kernels confirms positive-definiteness. For \(t > 1\) and a sufficiently large, the integral operator is not positive-definite. For t not an integer, but with integer odd part, the integral operator is not positive-definite.
We study the positive-definiteness of a family of L^2(𝐑) integral operators with kernel K_t, a(x, y) = (1 + (x - y)^2 + a(x^2 + y^2)^t)^-1, with t > 0 and a > 0. When 0 < t ≤ 1, the known theory of positive-definite kernels ensures that the operator is positive-definite; when t > 1, constructions disprove positive-definiteness for many (t, a)-pairs.
