How to use index on function overlaps of postgres

I using postgresSql and using overlaps function find range time on table. My table look like

id room_id start_date start_time end_date    end_time delete_at
1  123    2021-01-09 10:00:00  2021-01-10   10:30:00 null
2  456    2021-08-10 15:00:00  2021-01-10   15:30:00 null
///some record
3600 789 2021-09-10  00:00:00 2021-09-10   15:00:00 null 

My table very large and have 38000 record. I write sql find logic overlap time look like :

    select
        *
from
        booking as b
where
        b.room_id in ('123', '456', '789')
        and b.delete_flag = 0
        and (cast('2021-01-09 11:30:00' as TIMESTAMP),
        cast('2021-09-10 11:30:00' as TIMESTAMP)) overlaps ((b.start_date + bk.start_time),
        (b.end_date + b.end_time));

I have create index start_date, start_time, end_date, end_time. But postgres cannot using my index . It only scan seq. I create new mutiple index look like :

CREATE INDEX start_date_time_idx ON booking ((start_date + start_time))  
   CREATE INDEX end_date_time_idx ON booking ((end_date + end_time))

But postgres cannot using start_date_time_idx and end_date_time_idx index. I can't understand why postgres cannot using index. I try change my code remove overlaps but no change

       select
        *
from
        booking as b
where
        b.room_id in ('123', '456', '789')
        and b.delete_flag = 0
        and (cast('2021-01-09 11:30:00' as TIMESTAMP) > (b.start_date + bk.start_time),
        cast('2021-09-10 11:30:00' as TIMESTAMP))<= (b.end_date + b.end_time));

How to create index for function overlap or how to using multiple index in mycase. Why index of start_date, start_time, end_date , end_time , start_date_time_idx, end_date_time_idx cannot using mycase. Please help. Thank you.