The caller had tipped off the authorities in Maine on Friday night: He told them that he had broken into the home of Shenna Bellows, the state’s top election official, a Democrat who one night earlier had disqualified former President Donald J. Trump from the primary ballot because of his actions during the Jan. 6 Capitol riot.

No one was home when officers arrived, according to Maine State Police, who labeled the false report as a “swatting” attempt, one intended to draw a heavily armed law enforcement response.

In the days since, more bogus calls and threats have rolled in across the country. On Wednesday, state capitol buildings in Connecticut, Georgia, Hawaii, Kentucky, Michigan, Minnesota, Mississippi and Montana were evacuated or placed on lockdown after the authorities said they had received bomb threats that they described as false and nonspecific. The F.B.I. said it had no information to suggest any threats were credible.

The incidents intensified a climate of intimidation and the harassment of public officials, including those responsible for overseeing ballot access and voting. Since 2020, election officials have confronted rising threats and difficult working conditions, aggravated by rampant conspiracy theories about fraud. The episodes suggested 2024 would be another heated election year.

Gabriel Sterling, a top election official in Georgia who debunked Mr. Trump’s election fraud claims in 2020 in the battleground state, urged caution on Wednesday.

“Do not jump to conclusions as to who is responsible,” Mr. Sterling wrote on social media. “There will be chaos agents sowing discord for 2024. They want to increase tensions. Don’t let them.”

Mr. Sterling later said he was among the public officials who have been swatted by callers falsely reporting crimes. Someone called 911 to report a shooting after a drug deal gone bad, he wrote on X, “Everyone is ok. But this is wrong.”

Other prominent Republicans have also been swatted in recent weeks, including Representative Majorie Taylor Greene of Georgia and Senator Rick Scott of Florida.

Ms. Greene, a right-wing provocateur and Trump ally, was home on Christmas morning when a man in New York called a Georgia suicide hotline and claimed that he had just shot his girlfriend at Ms. Greene’s residence and was about to kill himself, The Associated Press reported.

Two days later, on Dec. 27, it was Mr. Scott’s turn, he said. He was having dinner with his wife when it happened.

The week before, several justices on the Colorado Supreme Court, which had blocked Mr. Trump from the ballot in that state in December, had also faced threats, but details have not been disclosed.

Mr. Trump’s place on the ballot in several other states remains unsettled, adding another layer of uncertainty to his electoral and legal status — and the potential for backlash.

Ms. Bellows, Maine’s secretary of state, condemned the instigator of the armed law enforcement response to her home. Maine State Police described the caller as an unknown male.

“It’s designed to scare not only me but also others into silence, to send a message,” she wrote in a Facebook post on Saturday.

According to Maine State Police, officers conducted a sweep inside Ms. Bellows’s home at her request after checking outside. The agency said that the incident was still under investigation and that it was working with law enforcement partners to provide special attention to appropriate locations.

Ms. Bellows drew the scorn of Mr. Trump and his supporters after she ruled on Dec. 28 that he was ineligible to be on the primary ballot under the third section of the 14th Amendment, which prohibits people who have engaged in insurrection from holding office.

Mr. Trump is appealing the 34-page ruling, which made Maine the second state to block him from the ballot.

In Colorado, the first state to do so, the Denver Police Department said in a statement on Dec. 23 that it was providing extra patrols around the residences of Colorado Supreme Court justices in that city in response to reports of threats and harassment.

Vikki Migoya, a spokeswoman for the F.B.I.’s Denver field office, said the bureau was aware of the situation and working with local law enforcement authorities.

“We will vigorously pursue investigations of any threat or use of violence committed by someone who uses extremist views to justify their actions regardless of motivation,” she said in an email on Wednesday.

A spokesman for the Trump campaign did not respond on Wednesday to a request for comment about the threats targeting the justices and Ms. Bellows.

Mr. Trump and the Colorado Republican Party have asked the United States Supreme Court to overturn Mr. Trump’s disqualification.

At the Georgia Capitol in Atlanta, an employee on Wednesday received a “hoax” email about a bomb threat at the building, Courtney Floyd, a spokeswoman for the Georgia Department of Public Safety, said in an email.

“A search was conducted, and an all-clear was given,” said Ms. Floyd, who did not release further details.

Mr. Sterling, who testified in 2022 before the Jan. 6 committee, chronicled the developments on social media.

Georgia has played an outsize role in both Mr. Trump’s election grievances and his deepening legal entanglements.

In August, the former president was indicted by an Atlanta-area grand jury on charges of trying to subvert the outcome of the 2020 election in Georgia, which Joseph R. Biden Jr. won. Republicans control the governor’s office and the Legislature in the state, but Georgia helped Democrats win the White House in 2020 and control of the U.S. Senate in 2021. The criminal case is one of four proceeding against Mr. Trump.

At the Michigan Capitol in Lansing, a bomb threat was sent to a general email account for the Michigan State Capitol Commission around 7:45 a.m. on Wednesday, the Michigan State Police said on X.

Law enforcement officers used dogs to sweep the building, which State Police said remained closed for the rest of the day out of an abundance of caution.

“The FBI will be taking over the investigation as multiple government agencies received the same threat,” Michigan State Police said.

Michigan has also been a flashpoint for Mr. Trump, who was defeated by Mr. Biden in the swing state in 2020 after winning there in 2016.

Last August, Matthew DePerno, a key orchestrator of efforts to help Mr. Trump try to overturn the 2020 election in Michigan and an unsuccessful candidate for state attorney general in 2022, was charged in an election breach.