Explicit and Almost Explicit Spectral Calculations for Diffusion Operators
| dc.creator | Pinsky, Ross G. | |
| dc.date | 2008-08-22 | |
| dc.date.accessioned | 2026-07-07T09:57:56Z | |
| dc.date.available | 2026-07-07T09:57:56Z | |
| dc.description | The diffusion operator $$ H_D=-\frac12\frac d{dx}a\frac d{dx}-b\frac d{dx}=-\frac12\exp(-2B)\frac d{dx}a\exp(2B)\frac d{dx}, $$ where $B(x)=\int_0^x\frac ba(y)dy$, defined either on $R^+=(0,\infty)$ with the Dirichlet boundary condition at $x=0$, or on $R$, can be realized as a self-adjoint operator with respect to the density $\exp(2Q(x))dx$. The operator is unitarily equivalent to the Schrödinger-type operator $H_S=-\frac12\frac d{dx}a\frac d{dx}+V_{b,a}$, where $V_{b,a}=\frac12(\frac{b^2}a+b')$. We obtain an explicit criterion for the existence of a compact resolvent and explicit formulas up to the multiplicative constant 4 for the infimum of the spectrum and for the infimum of the essential spectrum for these operators. We give some applications which show in particular how $\infσ(H_D)$ scales when $a=νa_0$ and $b=γb_0$, where $ν$ and $γ$ are parameters, and $a_0$ and $b_0$ are chosen from certain classes of functions. We also give applications to self-adjoint, multi-dimensional diffusion operators. | |
| dc.identifier | https://arxiv.org/abs/0808.3044 | |
| dc.identifier | http://arxiv.org/abs/0808.3044 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/167528 | |
| dc.subject | Spectral Theory | |
| dc.subject | Probability | |
| dc.subject | 34L05, 47A10, 60J60 | |
| dc.title | Explicit and Almost Explicit Spectral Calculations for Diffusion Operators | |
| dc.type | text |