Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

Details Explicit and Almost Explicit Spectral Calculations for Diffusion Operators Explicit and Almost Explicit Spectral Calculations for Diffusion Operators

2008-08-22

arxiv.org/abs/0808.3044v1

Mathematics

Spectral Theory

Scientific paper

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 Schr\"odinger-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 $\inf\sigma(H_D)$ scales when $a=\nu a_0$ and $b=\gamma b_0$, where $\nu$ and $\gamma$ 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.

Affiliated with

Also associated with

No associations

Rating

Explicit and Almost Explicit Spectral Calculations for Diffusion Operators does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Explicit and Almost Explicit Spectral Calculations for Diffusion Operators, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Explicit and Almost Explicit Spectral Calculations for Diffusion Operators will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-185554