The first regularized trace for a power of the Laplace operator on the rectangular triangle with the angle~$\pi/6$ in case of Dirichlet problem

I. V. Tomina

Fundamentalʹnaâ i prikladnaâ matematika, Tome 1 (1995) no. 2, pp. 569-572

Résumé

Consider the Hilbert space $H=L^2(D)$, where $D=\{(x,y)\mid 0\leq y\sqrt{3}\leq x\leq(2\pi-y\sqrt{3})/3\}$. Let $T$ be the self-adjoint non-negative operator from $H$ to $H$ which is generated by the spectral Dirichlet problem $\Delta u+\lambda u=0$ on $D$, $u=0$ on $\partial D$. For $p\in L^\infty(D)$ let the operator $P\colon H\to H$ take each $f\in H$ to the product $p\cdot f$. In this paper concrete formulas for the first regularized trace of the operator $T^\alpha+P$, $\alpha>3/2$, are given for different classes of essentially bounded functions $p$.