Summation formula inequalities for eigenvalues of Schrödinger operators

Abstract

Abstract. We derive inequalities for sums of eigenvalues of Schrödinger operators on finite intervals and tori. In the first of these cases, the inequalities converge to the classical trace formulae in the limit as the number of eigenvalues considered approaches infinity.

Estimates for sums of eigenvalues of the free plate via the fourier transform

“…Proof of Corollary 2. We return our attention to the estimate of (12). Combining with (13) we deduce (14) Λ m+1 (Ω) ≤ nω n |Ω| r n+4 n+4 + τ r n+2 n+2 ω n |Ω|r n − m(2π) n = F (r), r > 2π m ω n |Ω| In the case τ = 0, it is easy to check that the derivative F ′ (r) vanishes precisely when (ω n |Ω|r n − m(2π) n ) nω n |Ω|r n+3 − nω n |Ω| r n+4 n + 4 nω n |Ω|r n−1 = 0 and solving this equation for r gives r = 2π m(n + 4) 4ω n |Ω| 1 n .…”

“…Estimates for the eigenvalues {µ j (Ω)} and for their sums in terms of the geometry of Ω have been obtained by many authors (see [3,4,5,10,11,12,14,15,16,17,18], for instance; see also [13,16,19,20] and the references therein for analogous estimates for the fixed membrane and [1,2,8,9,21,23,24] for analogous estimates for the clamped plate). For the purposes of this note, we simply recall the following estimate of Kröger [14] for sums of eigenvalues:…”

“…Proof of Corollary 2. We return our attention to the estimate of (12). Combining with (13) we deduce (14) Λ…”

A Gelfand-Levitan trace formula for generic quantum graphs

“…Of interest to us here are the extensions to the case of quantum graphs, where this type of result may be traced back to a paper by Roth in 1983 [Rot83], with further developments in several directions such as those in [BER15,BK13,FK16,Nic87]. Quantum graphs have also received much attention in the literature within the past 30 years and, in particular, there have been several attempts at generalizing Gelfand and Levitan's result to this setting.…”