There's a passenger list for the SS Lucania from Liverpool 23 Sep. 1905 to New York Sep. 29 with this:
Schwartzman, Margaret, age 44, female, married, nationality - England, last permanent residence - Sunderland, going to Newark, NJ, previously in the US in Newark in 1905, going to home at 98 South XI (?) St., Newark, NJ
Schwartzman, Lewis, age 9, male, single, son, US born, passage paid by mother
Also, the WWII Draft Registration Card for William Louis Schwartzman (note that the WWI & WWII Draft Registration Cards do not mean that he enlisted or had to serve in the military - all men of a certain age were required to register for the draft) says he was born in New York City, which seems to fit in with their marriage & them being in Manhattan on the 1900 Census.