What you stated may very well be the case for some types of income; but the US tax code is not without exceptions to the rule.
As mentioned earned income may be excluded. Also, the tax treaty between the countries permits certain income to only be taxed by the state in which the person is resident (as in residing physically).
There is a technical explanation of that treaty at http://www.irs.gov/pub/irs-trty/swistech.pdf but it too states "
"For example, a U.S. citizen who pursuant to the "citizen/green card holder" rule is not considered to be a resident of the United States still is taxable on his worldwide income under the generally applicable rules of the Code."
Article 7 of the treaty does limit the other state from taxing business profit if there is not any permanent establishment in that other state. So a business operated completely outside the United States would not be subject to US income tax.
Likewise the explanation of Article 10 says " the Article prohibits a State from imposing a tax on dividends paid by a company resident in the other Contracting State that are neither paid to a resident of the first-mentioned State nor attributable to a permanent establishment in that State" so that dividends from a company not attributable to the other state will not be taxed.
Article 11 grants only the "the State of residence the exclusive right, subject to exceptions provided in paragraphs 3 and 6, to tax interest beneficially owned by its residents." So all interest earned while resident in Switzerland is not subject to US tax.
It may be well worth having an experienced practitioner assist with these exceptions and the exceptions to the exceptions when tax treaties are involved.
Hope that clarifies for you; but please ask if you need more help.
Please remember when you are asked to rate my service that you are rating me, personally, for the service and information that I have provided. You are not rating this website or the situation itself. A positive rating is the only way that I receive credit for my time and effort to assist you