The women are sleeping when it happens.

The man enters their homes through unlocked doors. He takes off some of his clothes. He crawls into bed with the women.

Then he touches them--in a way they describe as nonsexual, police say. The women awake and start screaming, and he runs out the door.

Yesterday, police said they are searching for a man who doesn't injure his victims or steal their belongings: He creeps into Capitol Hill apartments through the door and lies down in bed.

"People write volumes in books on psychiatric problems," said 1st District Sgt. Jeff Christy. "We don't know why he's doing it."

It has happened twice. The first incident took place about 3 a.m. June 14 in the 500 block of Second Street SE, Christy said. The second happened at 2:20 a.m. Oct. 22 in the 300 block of Sixth Street SE.

"I woke up with this guy in my bed, with his head over my mouth. It was frightening," said the woman who was attacked in June. "I was panicking. I thought I was a goner."

The woman, 32, said she had returned from a party that night. The back door of her apartment was closed, but not locked.

"It's really hard to open anyway," she said. "I never expected that someone would get in there."

She said she went to sleep and later opened her eyes to find a man hunched over her.

"He kept saying 'Don't say anything. Don't say anything,' in this so-called sexy kind of voice," she recalled.

She reached for her briefcase to offer him money. Then she started screaming and ran out of the house. She screamed as she ran down the block, waking neighbors. The man disappeared.

"I can't believe I got away," she said. "He was very slow. Something was kind of wrong with him. He didn't show any kind of weapon, but he scared the hell out of me."

For a while, the woman said she had a hard time getting police to pay attention to the case. She said she went to several neighborhood meetings and called police many times before getting help. "I've never done anything like this," she said. "But I don't want this to happen to anyone else."

Police said yesterday they are paying close attention to the case.

First District Detective Barry Fine said the man does not appear to know the victims, single women between 20 and 40, and the women appear to be strangers to each other, he said.

The neighborhood where the incidents took place is residential and has few crime problems, police said. "Overall, every time I've been in the neighborhood, it's been quiet," Fine said.

Police yesterday released a sketch of the man, who they described as white, about 30 years old, about 6 feet tall and 150 pounds. During the last incident, he was wearing dark pants and a T-shirt and smelled of alcohol.

Police are asking anyone with information to call the 1st District at 202-727-4595.

CAPTION: Police are looking for a man who broke into two women's homes in Southeast Washington.