Can an enum have abstract methods?

Yes, you can define abstract methods in an enum declaration if and only if all enum values have custom class bodies with implementations of those methods (i.e. no concrete enum value may be lacking an implementation).

public enum Foo {
  BAR {
    public void frobnicate() {
      // do BAR stuff
    }
  },
  BAZ {
    public void frobnicate() {
      // do BAZ stuff
    }
  };

  public abstract void frobnicate();
}

Just like @lukastymo 's answer, it is possible to implement an abstract method in enum and it is preferred to implement an interface when adding a method in an enum.

From Java 8 and above, you can use lambda to implement methods in an enum for smaller code. These lambda can be executed outside the enum by exposing a public method that runs the given lambda.

public enum ScheduleRepeat {
  DAILY(date -> date.plusDays(1)),
  WEEKLY(date -> date.plusWeeks(1)),
  MONTHLY(date -> date.plusMonths(1)),
  QUARTERLY(date -> date.plusMonths(3)),
  BIANNUALLY(date -> date.plusMonths(6)),
  ANNUALLY(date -> date.plusYears(1)),
  ;

  private final Function<LocalDate, LocalDate> nextDateFunction; // or UnaryOperator<LocalDate>

  ScheduleRepeat(Function<LocalDate, LocalDate> nextDateFunction) {
    this.nextDateFunction = nextDateFunction;
  }

  public LocalDate calculateNextDate(LocalDate dateFrom) {
    return nextDateFunction.apply(dateFrom);
  }
}

Then the enum can be used like:

LocalDate today = LocalDate.of(2019, 9, 18); // 2019 Sep 18
ScheduleRepeat.DAILY.calculateNextDate(today); // 2019-09-19
ScheduleRepeat.MONTHLY.calculateNextDate(today); // 2019-10-19

This isn't exactly implementing an abstract method from the enum itself or from an interface, but I think this approach of adding method using lambda looks clean.

Yes, but you may prefer enum which implements an interface, Look here. I think It looks much better. This is example for abstract method:

public enum Animal {
    CAT {
        public String makeNoise() { return "MEOW!"; }
    },
    DOG {
        public String makeNoise() { return "WOOF!"; }
    };

    public abstract String makeNoise();
}