If he is engaging in this type behavior simply to annoy and disturb your girlfriend she could file criminal and civil charges for harassment.
The Modern Penal Code section 250.4(MPC) defines harassment as a petty misdemeanor if with purpose is to harass another, he: 1) makes a phone call without a legitimate purpose; or 2) insults, taunts or challenges another in a manner likely to provoke violent or disorderly response; or 3) calls at inconvenient hours or in offensive language; or 4) subjects another to offensive touching; or 5) engages in any other course of alarming conduct serving no legitimate purpose of the actor.
The texts and emails would likely fall under the telephone calls or the "alarming conduct" provisions.
Under the United States Code Title 18 Subsection 1514(c)1. Harassment is defined as "a course of conduct directed at a specific person that causes substantial emotional distress in such a person and serves no legitimate purpose".
Put simply, anything that people do for no good reason or lawful reason or purpose and they know it distresses you is harassment.
She could file a civil harassment suit against him and be awarded monetary damages and injunctive action if she can show proof he is harassing her with his actions. If he was threatening with his communications a criminal complaint would be appropriate. A judge would likely issue a restraining order against him for no communications to be directed towards her.
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