I am doing research in Canton Bern, Switzerland and have Aebi and Widmer as names in my genealogy. The LDS church has copied most of the parish records on film that can be rented and viewed at a Family History Center near you. Film 2005878 has births covering the years you would need for Alexander. Film 2005880 has marriage(1635-1875) and death records (1728-1875) for the parish of Walterswil. The films should also help you to find the parentage of Elisabeth. I checked the films that contain the notes of Julius Billeter who did great amount of research in this area on many surmnames from the area, and was unable to find your Alexander. It appears that he may not have done the Abei Family from Walterswil.
I have also done reasearch in Telemark and have found the bygdeboks from there to be very helpful in tracing ancestry. Most of them can be found in Madison, WI at the Naeseth Norwegian American Library as well as at the Chester Fitz Library on the campus of the University of North Dakota at Grand Forks. Many of the church books from Norway are on line from Norwegian web sites. They are also available for rental at your local Family History Library. Any help I can give you, please ask. I have been a non- member volunteer at the local FHC for 25 years and have helped people do their research in most areas of the world.