Trust and honesty are the cornerstones of any relationship. Jealousy has many forms and comes from many places. it could be something that happened in the individuals past, etc. Sometimes when we have a "gut" feeling about someone (partners think the other is cheating), it's been proven that it's usually 90% straight on. However, jealousy is a type of fear, and has a root cause somewhere. These type of "fear triggers" in some people can't be suppressed without first removing what is triggering the behavior.
In this case, if there is love, then there should be a desire to get that person help, and to stay with them through their therapy. Leaving them does not help. A lot of people just throw in the towel because they get enough of the "20 questions" about where they have been, etc. Not realizing that the person can not help how she or he feels, or why they cannot control it. The main thing to do is get some help, and both people attend the therapy groups so both understand what and why.
I've dealt with some of this and it is often rooted in low self esteem. They can't imagine that anyone can love them, and will probably "wander" to someone else when they find out they are not what they thought. Thing is, they are very much loved, but can't accept or see it. Revealing these kinds of weaknesses to your significant other is hard, as it makes you extremely vulnerable to the other. Since that can't be done because of the lack of trust (either real or not), then it becomes a vicious circle to the individual suffering from this, and becomes worse. It needs acute insight of the other person to realize that they are not actually being questioned of their loyalty, but that the other is dealing with something they really can't help. Jealousy is one of our emotions, but can get distorted.
The Bible says that jealousy is as cruel as the grave. In other words, it can cause great grief. Find a Christian counselor that has been aptly trained in relationships. Pray together that God helps "heal" whatever "triggers" the jealousy behavior.
However, if the spouse or significant other are actually cheating, then they have the problem.......not you.