A thermodynamic system of N Fermions or Bosons, bound by an external potential but with almost no additional contribution of the interaction energy between the particles to the binding of the system is called a bound perfect quantum gas. Its single particle energy level density ρ (ε) depends on the properties of the external potential. This is chosen to be zero inside and infinite outside a given arbitrary simple connected closed shape. Within the leptodermous assumption A N1/3 ≫ 1 then ρ (ε) can be written explicitly as a sum of three terms which are proportional to the volume, surface, curvature tension. Its thermodynamics is developed: 1) one thermodynamic variable can be eliminated, reducing the phase space dimensions; 2) the Gibbs - Duhem relation is disfigured only by surface - and curvature terms, stating that the system is still makroscopically homogenious except in the surface area, where e.g. the particle density falls down to zero smoothly; 3) the Landsberg-definition p · V = ⅔ U still holds, confirming that our microscopically defined system is macroscopically a perfect gas in the sense of Landsberg, despite the surface phenomena. In the appendix the advantages of an operatorlike shortwriting of the partial derivative notation are demonstrated.