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arXiv:2607.22052

Norm of resonance states in quantum scattering and electromagnetic systems

Florian Lorenz, Jan Möseritz-Schmidt, Roland Ketzmerick

quant-phnlin.CDphysics.opticsSubmitted Jul 24, 2026• Announced Jul 27, 2026• Revised Oct 29, 2026 (v2)
v1 · Jul 24, 2026v2 · Oct 29, 2026
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Abstract

Resonance states spatially diverge and are thus not square integrable. Instead, their norm is defined by the biorthogonal scalar product of left and right states. We replace the corresponding volume integral over infinite space by a convenient boundary integral in piecewise-homogeneous systems and apply this procedure to diverse physical settings. For a quantum particle in any number of dimensions we treat hard-wall and piecewise-constant potentials. For electromagnetic systems with piecewise-homogeneous material properties, we consider three-dimensional and effectively two-dimensional cavities.

Comments
30 pages, 2 figures

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Resonance states spatially diverge and are thus not square integrable. Instead, their norm is defined by the biorthogonal scalar product of left and right states. We replace the corresponding volume integral over infinite space by a convenient boundary integral in piecewise-homogeneous systems and apply this procedure to diverse physical settings. For a quantum particle in any number of dimensions we treat hard-wall and piecewise-constant potentials. For electromagnetic systems with piecewise-homogeneous material properties, we consider three-dimensional and effectively two-dimensional cavities.
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Abstract. We provide probabilistic interpretation of resonant states. This we do by showing that the integral of the modulus square of resonance wave functions (i.e., the conventional norm) over a properly expanding spatial domain is independent of time, and therefore leads to probability conservation. This is in contrast with the conventional employment of a bi-orthogonal basis that precludes probabilistic interpretation, since wave functions of resonant states diverge exponentially in space. On the other hand, resonant states decay exponentially in time, because momentum leaks out of the central scattering area. This momentum leakage is also the reason for the spatial exponential divergence of resonant state. It is by combining the opposite temporal and spatial behaviors of resonant states that we arrive at our probabilistic interpretation of these states. The physical need to normalize resonant wave functions over an expanding spatial domain arises because particles leak out of the region which contains the potential range and escape to infinity, and one has to include them in the total count of particle number.
The Hermiticity condition in quantum mechanics required for the characterization of (a) physical observables and (b) generators of unitary motions can be relaxed into a wider class of operators whose eigenvalues are real and whose eigenstates are complete. In this case, the orthogonality of eigenstates is replaced by the notion of biorthogonality that defines the relation between the Hilbert space of states and its dual space. The resulting quantum theory, which might appropriately be called 'biorthogonal quantum mechanics', is developed here in some detail in the case for which the Hilbert-space dimensionality is finite. Specifically, characterizations of probability assignment rules, observable properties, pure and mixed states, spin particles, measurements, combined systems and entanglements, perturbations, and dynamical aspects of the theory are developed. The paper concludes with a brief discussion on infinite-dimensional systems.
We seek stationary state solutions to the nonlinear Schrodinger equation (NLS) for piecewise-constant potentials V(x). In this case, we are able to find complete stationary solutions, i.e. exact solutions which are separable in time and space, in terms of Jacobi elliptic functions. These special functions can be defined as an inverse integral and are an interpolation between the circular and hyperbolic trigonometric functions. As an important mathematical application of our exact solutions, we will seek a nonlinear generalization of the transfer matrix. When this matrix, which is commonly used in electromagnetic transmission problems, is applied to the coefficient vector characterizing the solution in the region to the left of the first boundary, it yields the coefficient vector for any region to the right. To solve this general problem, we begin with the nontrivial case of two boundaries, which corresponds to a potential step or barrier. This has a direct physical application to an atom laser incident on such a potential, as occurs in Bose-Einstein condensates on a chip. Our work provides the first step towards formulating a completely general nonlinear scattering theory.
For a quantum system with observables capable of assuming an infinite number of different values we would need an infinite-dimensional complex vector space to serve as its state space. Every Cauchy sequence of vectors in a finite-dimensional complex vector space converges to a limit vector in the space the same is not true in an arbitrary infinite-dimensional complex scalar product space. Fortunately infinite-dimensional scalar product spaces of physical importance are such that all Cauchy sequences do converge. Such spaces are called Hilbert spaces. Since every Cauchy sequence in a finite-dimensional scalar product space converges they are automatically Hilbert spaces. The situation in infinite-dimensional spaces is different. With Riemann integrals for square integrability and the scalar product replaced by Lebesque integrals we would obtain a larger set of functions which are Lebesque square-integrable.