Bounds for Schrödinger Operators on the Half-Line Perturbed by Dissipative Barriers

We consider Schrödinger operators of the form $$H_R = - \,\text {{d}}^2/\,\text {{d}}x^2 + q + i \gamma \chi _{[0,R]}$$ H R = - d 2 / d x 2 + q + i γ χ [ 0 , R ] for large $$R>0$$ R > 0 , where $$q \in L^1(0,\infty )$$ q ∈ L 1 ( 0 , ∞ ) and $$\gamma > 0$$ γ > 0 . Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.

The Drury–Arveson Space on the Siegel Upper Half-space and a von Neumann Type Inequality

In this work we study what we call Siegel–dissipative vector of commuting operators $$(A_1,\ldots , A_{d+1})$$ ( A 1 , … , A d + 1 ) on a Hilbert space $${{\mathcal {H}}}$$ H and we obtain a von Neumann type inequality which involves the Drury–Arveson space DA on the Siegel upper half-space $${{\mathcal {U}}}$$ U . The operator $$A_{d+1}$$ A d + 1 is allowed to be unbounded and it is the infinitesimal generator of a contraction semigroup $$\{e^{-i\tau A_{d+1}}\}_{\tau <0}$$ { e - i τ A d + 1 } τ < 0 . We then study the operator $$e^{-i\tau A_{d+1}}A^{\alpha }$$ e - i τ A d + 1 A α where $$A^{\alpha }=A_1^{\alpha _1}\cdots A^{\alpha _d}_d$$ A α = A 1 α 1 ⋯ A d α d for $$\alpha \in {\mathbb N}_0^d$$ α ∈ N 0 d and prove that can be studied by means of model operators on a weighted $$L^2$$ L 2 space. To prove our results we obtain a Paley–Wiener type theorem for DA and we investigate some multiplier operators on DA as well.