Relative Oscillation Theory, Weighted Zeros of the Wronskian, and the Spectral Shift Function

We develop an analog of classical oscillation theory for Sturm-Liouville operators which, rather than measuring the spectrum of one single operator, measures the difference between the spectra of two different operators. This is done by replacing zeros of solutions of one operator by weighted zeros of Wronskians of solutions of two different operators. In particular, we show that a Sturm-type comparison theorem still holds in this situation and demonstrate how this can be used to investigate the finiteness of eigenvalues in essential spectral gaps. Furthermore, the connection with Krein's spectral shift function is established.Comment: 26 page

Novel 7-Aryliden-3,3a,4,5,6,7-(hexahydro-5-(2-ethoxyethyl)-2-phenyl- 3-aryl-2H-pyrazolo[4,3-c]pyridine Hydrochloride: Synthesis and Structure

A Claisen-Schmidt type reaction of 1-(2-ethoxyethyl)piperidin-4-one with differ­ent aromatic aldehydes led to corresponding dienones with a yield of 65‒71%. 7-Arylidene-3,3a,4,5,6,7-(hexahydro-5-(2-ethoxyethyl)-2-phenyl-3-aryl-2H-pyrazolo[4,3-c]pyridines were synthesized by heterocyclization of 3,5-dia­rylidene-piperidin-4-ones with phenylhydrazine hydrochloride in methanol at 70 °C over 4‒6 h. The X-ray crystal structure determination of 7-(p-methoxy­benzyliden)-3,3a,4,5,6,7-(hexahydro-5-(2-ethoxyethyl)-2-phenyl-3-(p-methoxy­phenyl)-2H-pyrazolo[4,3-c]pyridine hydrochloride (the deposition number is CCDC 862410) was completed. The piperidine and pyrazoline rings are close to a chair and envelope conformations, respectively.