First of all, we will fill repository by nodes with mixin type 'mix:title' and different values of 'jcr:description' property.
root
document1 (mix:title) jcr:description = "The quick brown fox jumped over the lazy dogs"
document2 (mix:title) jcr:description = "Brown fox live in forest."
document3 (mix:title) jcr:description = "Fox is a nice animal."
Let's see analyzers effect closer. In first case, we use base jcr settings, so, as mentioned above, string "The quick brown fox jumped over the lazy dogs" will be transformed to set {[the] [quick] [brown] [fox] [jumped] [over] [the] [lazy] [dogs] }
// make SQL query
QueryManager queryManager = workspace.getQueryManager();
String sqlStatement = "SELECT * FROM mix:title WHERE CONTAINS(jcr:description, 'the')";
// create query
Query query = queryManager.createQuery(sqlStatement, Query.SQL);
// execute query and fetch result
QueryResult result = query.execute();
NodeIterator will return "document1".
Now change the default analyzer to org.apache.lucene.analysis.StopAnalyzer. Fill repository again (new Analyzer must process nodes properties) and run the same query again. It will return nothing, because stop words like "the" will be excluded from parsed string set.