(I don't think my answer directly answers the question, but I'm hoping it would be useful.)
I assume that when you say "problem solving" you mean mathematical "problem-solving as a skill" ("being able to obtain solutions to the problems other people give you to solve," Schoenfeld, 1992).
I was unable to find any studies that answer the question "Does taking an ordinary undergraduate mathematics course improve one's ability to solve (mathematical) problems?" (where ordinary means the instruction is not explicitly targeted at improving problem solving skills).
But there have been studies that show that undergraduates taking certain "problem-solving courses" experienced "marked shifts in [their] problem solving behavior" (e.g., Schoenfeld, 1987, p. 207).
As I understand it, researchers in mathematics education usually don't consider questions of the type "does the ordinary way of teaching improve this skill/understanding?" important (where "ordinary" is usually referred to as "traditional"). They usually consider it more valuable to ask questions of the type "what way of teaching will improve this skill/understanding?"
A good reference is
Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, metacognition, and sense-making in mathematics. In D. Grouws (Ed.), Handbook for Research on Mathematics Teaching and Learning (pp. 334-370). New York: MacMillan.
which uses some material from
Schoenfeld, A. H. (Ed.). (1987). Cognitive Science and Mathematics Education. New Jersey: Erlbaum.
Chapter 2 (Foundations of cognitive theory and research for mathematics problem-solving, by E. A. Silver) and Chapter 8 (What's all the fuss about metacognition? by A. H. Schoenfeld) of the 1987 Schoenfeld book are particularly useful.