SQL 2012 makes pretty short work of this type of problem:

WITH PartitionedSchedules AS

(

SELECT ScheduleID, PersonID, startDate, durationDays

,CalculatedEndDate=DATEADD(day, durationDays, startDate)

,row2startDate=LEAD(startDate, 1) OVER (PARTITION BY PersonID ORDER BY startDate)

FROM Schedules

)

SELECT ScheduleID, PersonID, startDate, durationDays, CalculatedEndDate

,row2startDate

,datedifference

,analysis=CASE SIGN(datedifference)

WHEN 0 THEN 'contiguous'

WHEN 1 THEN CAST(ABS(datedifference) AS VARCHAR) + ' days overlap'

ELSE CAST(ABS(datedifference) AS VARCHAR) + ' days gap'

END

FROM PartitionedSchedules a

CROSS APPLY

(

SELECT DATEDIFF(day, row2startDate, CalculatedEndDate)

) b (datedifference)

WHERE datedifference IS NOT NULL;

No more need for a self-join.