If I understand your question correctly, you are asking about a situation in which somebody confesses to a murder but the prosecutor has no evidence outside of the confession, not even any evidence that the murder was actually committted. Is that right? In such a situation, the prsoecutor would normally ask the police to conduct an investigation to find out whether the confession is true, whether the murder actually happened, and to obtain all possible evidence. The person who confessed might be kept under police surveillance but not actually arrested until there was more information. If a criminal charge were actually to go to trial with nothing to the prosecutor's case other than a totally uncorroborated confession, the prosecution might very well fail under the corpus delicti rule. But these are just general observations. Every case is unique and would be handled on its own facts.
Here is a link to an article that explains the corpus dilecti rule at length, as applicable to Florida. Obviously LA would have different rules, but for the general principle, this would be a good primer: http://www.floridabar.org/DIVCOM/JN/JNJournal01.nsf/c0d731e03de9828d852574580042ae7a/3769a908dd4beaa585256adb005d632a!OpenDocument&Highlight=0,*
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The "corpus delecti" rule holds that a confession is only admissible evidence if there is some other evidence to support it. The evidence need only be slight. For example, if a person was missing without a trace and someone confessed to murdering them, they might be charged with the murder if the prosecutor concluded that the confession was supported by the evidence of the disappearance.
In any criminal investigation, police can only arrest once their investigation has developed enough evidence to believe that there is "probable cause" to believe the charge is true. In your example, it sounds like there is no evidence to support the confession, and the confession is either false or the result of a delusion. In these cases, if police believe the person is mentally ill, they could ask mental health authorities to assist the person, or even to get court orders to medicate or hospitalize them if the situation is dangerous. If the police believe the false confession was malicious (for example, to throw attention off of another suspect, to establish an alibi for another crime, etc), the police sometimes charge people with making false police reports, which is also a crime itself.