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No abstract uploaded!

No abstract uploaded!

For a fractal Schrodinger operator with a continuous potential and a coupling parameter, we obtain an analog of a semi-classical asymptotic formula for the number of bound states as the parameter tends to infinity. We also study Bohr's formula for Schrodinger operators on blowups of self-similar sets. For a Schrodinger operator defined by a fractal measure and a locally bounded potential that tends to infinity, we derive an analog of Bohr's formula under various assumptions. We demonstrate how these results can be applied to self-similar measures with overlaps, including the infinite Bernoulli convolution associated with the golden ratio, a family of convolutions of Cantor-type measures, and a family of measures that we call essentially of finite type.

We deal with a stationary problem of a reaction–diffusion system with a conservation law under the Neu-mann boundary condition. It is shown that the stationary problem turns to be the Euler–Lagrange equation of an energy functional with a mass constraint. When the domain is the finite interval (0, 1), we investigate the asymptotic profile of a strictly monotone minimizer of the energy as d, the ratio of the diffusion coef-ficient of the system, tends to zero. In view of a logarithmic function in the leading term of the potential, we get to a scaling parameter κsatisfying the relation ε:=√d=√logκ/κ2. The main result shows that a sequence of minimizers converges to a Dirac mass multiplied by the total mass and that by a scaling with κthe asymptotic profile exhibits a parabola in the nonvanishing region. We also prove the existence of an unstable monotone solution when the mass is small.