Niro, Salvatore (2026) Nonlinear Schrödinger Equation with Point Interaction on Bounded Planar Domains. [Tesi di dottorato]

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Abstract

Part I – III. This thesis investigates the ground states of the focusing nonlinear Schrödinger equation with a point interaction on a bounded planar domain, defined as constrained energy minimisers at fixed mass. The operator is the Laplacian with a δ-interaction at a point x0 ∈ Ω, realised through the theory of self-adjoint extensions, and the model domain is the disk BR(0) ⊂ R2. In the L2-subcritical regime (1 < p < 3), we prove that ground states exist for every positive mass, exhibit a logarithmic singularity at the defect, and — up to a constant phase — are positive and radially symmetric decreas- ing when x0 coincides with the centre of the disk. The spectral equation for the lowest eigenvalue of the point-interaction Hamiltonian is derived explicitly in terms of modified Bessel functions, and bilateral bounds in closed form are established. The analysis is extended to hybrid domains (disk attached to a half-line), to domain deformations, and to defocusing and combined nonlinearities. A numerical study based on Runge–Kutta shooting complements the analytical results. Part IV – V. We study the spectral and scattering theory of a hybrid quantum system consisting of a half-line R+ coupled to a deformed disk Ωε (ε-perturbation of the unit disk with h(θ) = cos(2θ)) through point interactions at the origin. Using the Kre˘ın resolvent formula combined with a Hadamard-type shape expansion of the Green’s function, we obtain a complete analytical description of the spectral structure as a function of four parameters: the half-line coupling α, the disk coupling β, the junction strength η, and the deformation amplitude ε. The main results are: (i) an explicit second-order expansion of the Green’s function on Ωε (Theorem 11.6), yielding the deformed Weyl function M ε(z) = M (z)−ς(z) ε2/(8πI0(κ)2)+ O(ε3) with the z-dependent shape coefficient ς(z) = 2κ I1(κ)/I2(κ) − 3; (ii) a closed-form eigenvalue shift formula λ(2) = ς(λ0) λ0/(2(1 − I2 0 )) under domain deformation (The- orem 12.11), with k-dependent coefficient consistent with the Faber–Krahn inequality; (iii) a birth-of-resonances theorem (Theorem 15.1) showing that the Family B eigen- values become resonances with Im(zres) < 0 upon coupling, with explicit birth rate dz/dη2|0 = 1/[(α − i√λk)(−M ε′(λk))]; (iv) a non-monotonic three-regime structure of the resonance trajectory as η varies from 0 to ∞, with convergence to the Dirichlet eigenvalues in the strong-coupling limit; (v) an exact one-channel S-matrix S(k) = (ik − Q)/(ik + Q) with Q = α + η2/(M ε(k2) − β), exhibiting Breit–Wigner peaks at resonances. All formulas are verified both analytically (elliptic regularity bootstrap, Wronskian iden- tities, implicit function theorem) and numerically (PDE residuals, spectral series conver- gence, direct solution of the spectral equation on the second Riemann sheet).

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