One option is to run the application as a separate process, launching the process when you want to create the application and exiting the process when the application is no longer needed (so completing a full application lifecycle). That way you will be absolutely sure that the application is not consuming any resources when it is not being used, because it won't be running.
How you would accomplish the launching and any communication between your tray service and the application would be up to you. You can research various mechanisms and, if you decide to go this route, ask some new follow up questions on accomplishing certain aspects of the task.
Some example routes you could look at are the ProcessBuilder, which is admittedly a pretty finicky and horrible API or the new Process API updates that will be available with Java 9. If wish to ensure at most a single instance of the application process is ever used, there are solutions for that. If you need to send a signal to the running application process, you could use something like RMI or run a basic HTTP REST server in your application and send messages into that.
As an aside, years ago there used to be some ongoing work on building multi-process JVMs, but there was never any wide uptake of the idea for Java. Though most modern browser clients, such as Chrome and Firefox, are multi-process architectures, the linked articles give some insight into this architecture, some of the potential implications of it and why it used for those applications.
Before going such a route, I would advise you to ensure that such an approach is truly necessary for your application (as pointed out by user npace in comments).